Daily Papers

Memory management is vital for LLM agents to handle long-term interaction and personalization. Most research focuses on how to organize and use memory summary, but often overlooks the initial memory extraction stage. In this paper, we argue that existing summary-based methods have two major limitations based on the recurrent processing theory. First, summarization is "ahead-of-time", acting as a blind "feed-forward" process that misses important details because it doesn't know future tasks. Second, extraction is usually "one-off", lacking a feedback loop to verify facts, which leads to the accumulation of information loss. To address these issues, we propose proactive memory extraction (namely ProMem). Unlike static summarization, ProMem treats extraction as an iterative cognitive process. We introduce a recurrent feedback loop where the agent uses self-questioning to actively probe the dialogue history. This mechanism allows the agent to recover missing information and correct errors. Our ProMem significantly improves the completeness of the extracted memory and QA accuracy. It also achieves a superior trade-off between extraction quality and token cost.

Modern sequence modeling is dominated by two families: Transformers, whose self-attention can access arbitrary elements of the visible sequence, and structured state-space models, which propagate information through an explicit recurrent state. These mechanisms face different limitations on long contexts: when attention is diffuse, the influence of individual tokens is diluted across the effective support, while recurrent state propagation can lose long-range sensitivity unless information is actively preserved. As a result, both mechanisms face challenges in preserving and selectively retrieving information over long contexts. We propose Sessa, a decoder that places attention inside a recurrent feedback path. This creates many attention-based paths through which past tokens can influence future states, rather than relying on a single attention read or a single recurrent chain. We prove that, under explicit assumptions and matched regimes, Sessa admits power-law memory tails O(ell^{-β}) for 0 < β< 1, with slower decay than in the corresponding Transformer and Mamba-style baselines. We further give an explicit construction that achieves this power-law rate. Under the same assumptions, Sessa is the only model class among those considered that realizes flexible selective retrieval, including profiles whose influence does not decay with distance. Consistent with this theoretical advantage, across matched experiments, Sessa achieves the strongest performance on long-context benchmarks while remaining competitive with Transformer and Mamba-style baselines on short-context language modeling.

Developing autonomous agents that can interact with changing environments is an open challenge in machine learning. Robustness is particularly important in these settings as agents are often fit offline on expert demonstrations but deployed online where they must generalize to the closed feedback loop within the environment. In this work, we explore the application of recurrent neural networks to tasks of this nature and understand how a parameterization of their recurrent connectivity influences robustness in closed-loop settings. Specifically, we represent the recurrent connectivity as a function of rank and sparsity and show both theoretically and empirically that modulating these two variables has desirable effects on network dynamics. The proposed low-rank, sparse connectivity induces an interpretable prior on the network that proves to be most amenable for a class of models known as closed-form continuous-time neural networks (CfCs). We find that CfCs with fewer parameters can outperform their full-rank, fully-connected counterparts in the online setting under distribution shift. This yields memory-efficient and robust agents while opening a new perspective on how we can modulate network dynamics through connectivity.

We present Recurrent Fitting (ReFit), a neural network architecture for single-image, parametric 3D human reconstruction. ReFit learns a feedback-update loop that mirrors the strategy of solving an inverse problem through optimization. At each iterative step, it reprojects keypoints from the human model to feature maps to query feedback, and uses a recurrent-based updater to adjust the model to fit the image better. Because ReFit encodes strong knowledge of the inverse problem, it is faster to train than previous regression models. At the same time, ReFit improves state-of-the-art performance on standard benchmarks. Moreover, ReFit applies to other optimization settings, such as multi-view fitting and single-view shape fitting. Project website: https://yufu-wang.github.io/refit_humans/

State tracking requires an update at every input, but the depth a Transformer applies to each token is fixed regardless of sequence length. We introduce the Recurrent Looped Transformer (RLT), which splits its layers between a parallel causal encoder and a recurrent decoder. At each token, the decoder merges the encoder output with the previous token's final decoder state, so the computation path grows with sequence length at a fixed per-token cost. On six algorithmic tasks, we compare five splits of eight layers with an eight-layer Transformer over three seeds. Trained on at most 40 bits, two RLT splits generalize parity to 256 bits with 100% accuracy in every seed, while the Transformer stays at chance. On swap-based S_5 permutation tracking at eight times the training length, RLT reaches 97% final-state accuracy versus under 1% for the Transformer, and accuracy increases with decoder depth. On modular arithmetic beyond the training lengths, RLT reaches up to 93% versus 33% for the Transformer. Ablations show that these gains depend on the feedback: removing it drops parity and swap-based S_5 to chance at every split. Updating the feedback once per four-token chunk lets known tokens in a chunk run in parallel and keeps 64-bit parity at 99%, while permutation tracking depends on per-token feedback: chunking lowers length-64 swap-based S_5 from 100% to 20%.

Reasoning has become a central capability in large language models. Recent research has shown that reasoning performance can be improved by looping an LLM's layers in the latent dimension, resulting in looped reasoning language models. Despite promising results, few works have investigated how their internal dynamics differ from those of standard feedforward models. In this paper, we conduct a mechanistic analysis of the latent states in looped language models, focusing in particular on how the stages of inference observed in feedforward models compare to those observed in looped ones. To this end, we analyze cyclic recurrence and show that for many of the studied models each layer in the cycle converges to a distinct fixed point; consequently, the recurrent block follows a consistent cyclic trajectory in the latent space. We provide evidence that as these fixed points are reached, attention-head behavior stabilizes, leading to constant behavior across recurrences. Empirically, we discover that recurrent blocks learn stages of inference that closely mirror those of feedforward models, repeating these stages in depth with each iteration. We study how recurrent block size, input injection, and normalization influence the emergence and stability of these cyclic fixed points. We believe these findings help translate mechanistic insights into practical guidance for architectural design.

The training and deployment of large language models (LLMs) create a feedback loop with human users: models learn human beliefs from data, reinforce these beliefs with generated content, reabsorb the reinforced beliefs, and feed them back to users again and again. This dynamic resembles an echo chamber. We hypothesize that this feedback loop entrenches the existing values and beliefs of users, leading to a loss of diversity and potentially the lock-in of false beliefs. We formalize this hypothesis and test it empirically with agent-based LLM simulations and real-world GPT usage data. Analysis reveals sudden but sustained drops in diversity after the release of new GPT iterations, consistent with the hypothesized human-AI feedback loop. Code and data available at https://thelockinhypothesis.com

Recurrent neural networks (RNNs) are widely used throughout neuroscience as models of local neural activity. Many properties of single RNNs are well characterized theoretically, but experimental neuroscience has moved in the direction of studying multiple interacting areas, and RNN theory needs to be likewise extended. We take a constructive approach towards this problem, leveraging tools from nonlinear control theory and machine learning to characterize when combinations of stable RNNs will themselves be stable. Importantly, we derive conditions which allow for massive feedback connections between interacting RNNs. We parameterize these conditions for easy optimization using gradient-based techniques, and show that stability-constrained "networks of networks" can perform well on challenging sequential-processing benchmark tasks. Altogether, our results provide a principled approach towards understanding distributed, modular function in the brain.

Recurrent neural networks (RNNs) have recently demonstrated strong performance and faster inference than Transformers at comparable parameter budgets. However, the recursive gradient computation with the backpropagation through time (or BPTT) algorithm remains the major computational bottleneck. In this work, we propose a novel method that replaces BPTT with a fixed gradient feedback mechanism, yielding an efficient approximation of the exact gradient propagation based on the assumption of time stationarity. Our approach leverages state-space model (SSM) principles to define a structured feedback matrix that directly propagates gradients from future time steps. This formulation bypasses the need for recursive gradient backpropagation, significantly reducing training overhead while preserving the network's ability to capture long-term dependencies. The experiments on language modeling benchmarks exhibit competitive perplexity scores, while significantly reducing the training costs. These promising results suggest that designing a feedback method like an SSM can fully exploit the efficiency advantages of RNNs for many practical applications.

Recurrent neural networks (RNNs) provide a powerful approach in neuroscience to infer latent dynamics in neural populations and to generate hypotheses about the neural computations underlying behavior. However, past work has focused on relatively simple, input-driven, and largely deterministic behaviors - little is known about the mechanisms that would allow RNNs to generate the richer, spontaneous, and potentially stochastic behaviors observed in natural settings. Modeling with Hidden Markov Models (HMMs) has revealed a segmentation of natural behaviors into discrete latent states with stochastic transitions between them, a type of dynamics that may appear at odds with the continuous state spaces implemented by RNNs. Here we first show that RNNs can replicate HMM emission statistics and then reverse-engineer the trained networks to uncover the mechanisms they implement. In the absence of inputs, the activity of trained RNNs collapses towards a single fixed point. When driven by stochastic input, trajectories instead exhibit noise-sustained dynamics along closed orbits. Rotation along these orbits modulates the emission probabilities and is governed by transitions between regions of slow, noise-driven dynamics connected by fast, deterministic transitions. The trained RNNs develop highly structured connectivity, with a small set of "kick neurons" initiating transitions between these regions. This mechanism emerges during training as the network shifts into a regime of stochastic resonance, enabling it to perform probabilistic computations. Analyses across multiple HMM architectures - fully connected, cyclic, and linear-chain - reveal that this solution generalizes through the modular reuse of the same dynamical motif, suggesting a compositional principle by which RNNs can emulate complex discrete latent dynamics.

Countless learning tasks require dealing with sequential data. Image captioning, speech synthesis, and music generation all require that a model produce outputs that are sequences. In other domains, such as time series prediction, video analysis, and musical information retrieval, a model must learn from inputs that are sequences. Interactive tasks, such as translating natural language, engaging in dialogue, and controlling a robot, often demand both capabilities. Recurrent neural networks (RNNs) are connectionist models that capture the dynamics of sequences via cycles in the network of nodes. Unlike standard feedforward neural networks, recurrent networks retain a state that can represent information from an arbitrarily long context window. Although recurrent neural networks have traditionally been difficult to train, and often contain millions of parameters, recent advances in network architectures, optimization techniques, and parallel computation have enabled successful large-scale learning with them. In recent years, systems based on long short-term memory (LSTM) and bidirectional (BRNN) architectures have demonstrated ground-breaking performance on tasks as varied as image captioning, language translation, and handwriting recognition. In this survey, we review and synthesize the research that over the past three decades first yielded and then made practical these powerful learning models. When appropriate, we reconcile conflicting notation and nomenclature. Our goal is to provide a self-contained explication of the state of the art together with a historical perspective and references to primary research.

Recurrent Neural Networks (RNNs) are popular models of brain function. The typical training strategy is to adjust their input-output behavior so that it matches that of the biological circuit of interest. Even though this strategy ensures that the biological and artificial networks perform the same computational task, it does not guarantee that their internal activity dynamics match. This suggests that the trained RNNs might end up performing the task employing a different internal computational mechanism, which would make them a suboptimal model of the biological circuit. In this work, we introduce a novel training strategy that allows learning not only the input-output behavior of an RNN but also its internal network dynamics, based on sparse neural recordings. We test the proposed method by training an RNN to simultaneously reproduce internal dynamics and output signals of a physiologically-inspired neural model. Specifically, this model generates the multiphasic muscle-like activity patterns typically observed during the execution of reaching movements, based on the oscillatory activation patterns concurrently observed in the motor cortex. Remarkably, we show that the reproduction of the internal dynamics is successful even when the training algorithm relies on the activities of a small subset of neurons sampled from the biological network. Furthermore, we show that training the RNNs with this method significantly improves their generalization performance. Overall, our results suggest that the proposed method is suitable for building powerful functional RNN models, which automatically capture important computational properties of the biological circuit of interest from sparse neural recordings.

Looped Transformers repeatedly apply the same set of Transformer layers, giving them a recurrent architecture for latent computation. Their strong performance on iterative reasoning and length-generalization tasks suggests an appealing explanation: recurrence may provide an inductive bias that lets the model reuse a learned algorithm across loops. However, weight sharing alone does not imply that every loop performs the same operation. This raises a basic question: is each loop actually repeating the same computation, and if not, what routes the shared parameters to different operations? We study this question using graph walks as a test case. In the model's native trajectories, decoded predictions can advance by different numbers of graph steps or remain at a reached target, showing that recurrent progress need not follow a fixed one-loop-one-step pattern. We then show that a frozen loop can be steered toward different transitions by modifying its entering hidden state: a learned linear layer J selects the desired transition without changing the shared Transformer layers. To test how this steering works, we use activation patching and find that attention patterns can recover its effects and switch the selected transition. Across five matched pairs of graph models, changing intermediate supervision during backbone training changes which transitions J can induce. This suggests that J selects computations learned by the backbone rather than creating new algorithms. Together, these results show that the hidden state can control shared computation, with attention routing as a causal pathway.

Looped language models (LoopLMs) increase computational depth through parameter sharing, offering a path to scale inference computation without adding parameters. However, it remains unclear when additional recurrence is beneficial and how architectural choices affect its effectiveness. Through controlled experiments, we systematically examine (1) when recurrence helps, (2) where it should be applied, and (3) how its conditioning affects performance. Our evaluation covers inference budgets below, within, and beyond the training horizon under knowledge and reasoning tasks. (1) We find that recurrence can improve reasoning beyond the training horizon while degrading knowledge performance, but harder reasoning instances do not consistently benefit more. (2) Performance also depends on how distinct layers and recurrent iterations are allocated, showing that effective depth alone is insufficient to predict behavior. Non-recurrent output layers improve robustness to under-unrolling, while the preferred placement of input and output layers varies with inference budget. (3) Finally, we find that conventional initial-state injection offers limited robustness to varying recurrence depth. We therefore propose history-state injection as an alternative, and show that channel-wise history-state injection combined with timestep conditioning offers a low-cost and more effective design, better preserving knowledge under extended unrolling while improving robustness across inference budgets. Overall, our results clarify when recurrent computation helps, where it fails, and offer practical guidelines for designing LoopLMs across variable inference budgets.

Quantization is widely used to reduce the computational and memory demands of neural-network inference. In recurrent networks, however, the quantized state is stored and returned at the next time step, so the rule used to store that state can alter subsequent computations. Here, we introduce recurrent-state write-back to denote this rule and isolate its effect in a compact GRU encoder--decoder for fluorescence lifetime imaging, a molecular imaging modality used in quantitative biological imaging. A central task is estimating two lifetime parameters, the short-lived component τ1 and the long-lived component τ2, from high-noise time-resolved fluorescence signals. Holding the trained model fixed, replacing continuous state propagation with deterministic 4-bit state storage increases estimation errors for τ1 and τ2 by approximately 70x and 300x, respectively. Failure occurs when repeated small updates remain below the write threshold, leaving the stored state nearly fixed while the network continues to propose change. Error feedback, residual memory, and direction memory carry information from these suppressed updates across time and recover accuracy without retraining. Precision sweeps show that increasing state precision can worsen a fixed recurrent solution, while matched training shows that compatibility with the state interface can be learned. To test whether this behavior extends beyond the GRU, we repeat the post-training intervention in an independently trained LSTM, where coarse write-back reproduces the failure, error feedback restores accuracy, and state-specific interventions reveal greater sensitivity of the cell state than the hidden state. Our results establish recurrent-state write-back as a key determinant of low-precision recurrent dynamics and identify the state-storage interface as a central design consideration for quantized recurrent inference.

This paper addresses the general problem of reinforcement learning (RL) in partially observable environments. In 2013, our large RL recurrent neural networks (RNNs) learned from scratch to drive simulated cars from high-dimensional video input. However, real brains are more powerful in many ways. In particular, they learn a predictive model of their initially unknown environment, and somehow use it for abstract (e.g., hierarchical) planning and reasoning. Guided by algorithmic information theory, we describe RNN-based AIs (RNNAIs) designed to do the same. Such an RNNAI can be trained on never-ending sequences of tasks, some of them provided by the user, others invented by the RNNAI itself in a curious, playful fashion, to improve its RNN-based world model. Unlike our previous model-building RNN-based RL machines dating back to 1990, the RNNAI learns to actively query its model for abstract reasoning and planning and decision making, essentially "learning to think." The basic ideas of this report can be applied to many other cases where one RNN-like system exploits the algorithmic information content of another. They are taken from a grant proposal submitted in Fall 2014, and also explain concepts such as "mirror neurons." Experimental results will be described in separate papers.

Language models influence the external world: they query APIs that read and write to web pages, generate content that shapes human behavior, and run system commands as autonomous agents. These interactions form feedback loops: LLM outputs affect the world, which in turn affect subsequent LLM outputs. In this work, we show that feedback loops can cause in-context reward hacking (ICRH), where the LLM at test-time optimizes a (potentially implicit) objective but creates negative side effects in the process. For example, consider an LLM agent deployed to increase Twitter engagement; the LLM may retrieve its previous tweets into the context window and make them more controversial, increasing engagement but also toxicity. We identify and study two processes that lead to ICRH: output-refinement and policy-refinement. For these processes, evaluations on static datasets are insufficient -- they miss the feedback effects and thus cannot capture the most harmful behavior. In response, we provide three recommendations for evaluation to capture more instances of ICRH. As AI development accelerates, the effects of feedback loops will proliferate, increasing the need to understand their role in shaping LLM behavior.

Recent work identifies a mid-depth band of verbalisable, causally potent representations in a standard feedforward transformer --- a functional analogue of a global workspace. Whether the same workspace functionality emerges when depth is implemented through recurrence rather than a stack of distinct layers remains unknown. Looped and depth-recurrent transformers provide a direct test of this question because they reuse the same weights across depth. We extend the Jacobian lens to iterated architectures using a virtual-unrolling adapter. We apply the full workspace suite --- lens fitting, readout, and eleven causal experiment families --- to Ouro-2.6B (48 layers looped 4 times, deeply supervised) and Huginn-0125 (a 4-layer core recurred 16 times, trained for latent reasoning), using Qwen3.6-27B (64 untied layers) as the standard baseline. We find that a workspace forms in the iterated part of each architecture, but that recurrence changes how it can be accessed. Ouro reconstructs workspace content in every loop, and linear transport cannot carry that content across loop boundaries; writes and ablations must therefore span every remaining loop. Huginn carries content forward across all sixteen recurrences, while reads, writes, and ablations act only within a sliding window of roughly two recurrences. Whether newly injected content can be verbalised tracks explicit per-iteration supervision; whether existing content can be steered does not.

Conventional deep networks rely on one-way backpropagation that overlooks reconciling high-level predictions with lower-level representations. We propose Contextual Feedback Loops (CFLs), a lightweight mechanism that re-injects top-down context into earlier layers for iterative refinement. Concretely, CFLs map the network's prediction to a compact context vector, which is fused back into each layer via gating adapters. Unrolled over multiple feedback steps, CFLs unify feed-forward and feedback-driven inference, letting top-level outputs continually refine lower-level features. Despite minimal overhead, CFLs yield consistent gains on tasks including CIFAR-10, ImageNet-1k, SpeechCommands, and GLUE SST-2. Moreover, by a Banach Fixed Point argument under mild Lipschitz conditions, these updates converge stably. Overall, CFLs show that even modest top-down feedback can substantially improve deep models, aligning with cognitive theories of iterative perception.

Generative AI search is reshaping information retrieval by offering end-to-end answers to complex queries, reducing users' reliance on manually browsing and summarizing multiple web pages. However, while this paradigm enhances convenience, it disrupts the feedback-driven improvement loop that has historically powered the evolution of traditional Web search. Web search can continuously improve their ranking models by collecting large-scale, fine-grained user feedback (e.g., clicks, dwell time) at the document level. In contrast, generative AI search operates through a much longer search pipeline, spanning query decomposition, document retrieval, and answer generation, yet typically receives only coarse-grained feedback on the final answer. This introduces a feedback loop disconnect, where user feedback for the final output cannot be effectively mapped back to specific system components, making it difficult to improve each intermediate stage and sustain the feedback loop. In this paper, we envision NExT-Search, a next-generation paradigm designed to reintroduce fine-grained, process-level feedback into generative AI search. NExT-Search integrates two complementary modes: User Debug Mode, which allows engaged users to intervene at key stages; and Shadow User Mode, where a personalized user agent simulates user preferences and provides AI-assisted feedback for less interactive users. Furthermore, we envision how these feedback signals can be leveraged through online adaptation, which refines current search outputs in real-time, and offline update, which aggregates interaction logs to periodically fine-tune query decomposition, retrieval, and generation models. By restoring human control over key stages of the generative AI search pipeline, we believe NExT-Search offers a promising direction for building feedback-rich AI search systems that can evolve continuously alongside human feedback.

AI agents encounter learning opportunities in every episode they run, and discard nearly all of them: the underlying models are frozen at deployment, so an agent that resolves a difficult request today starts from zero when it recurs tomorrow. Yet ordinary operation already produces feedback, in the form of outcome verdicts and after-the-fact corrections. We show that this feedback is a sufficient signal for continual learning when the frozen model is paired with an external memory that distils each episode into retrievable natural-language rules. On the banking domain of τ-bench, against a static-RAG control retrieving over the complete policy corpus, learning from the one-bit outcome verdict lifts single-trial success to 1.6times the baseline, and learning from corrections to 2.6times, converting 22 of the 84 tasks the baseline never solves. The result spans the deployment spectrum, measured on Mistral Large, an open-weights model that organisations with data sovereignty requirements can self-host, and replicated on a frontier model, Claude Sonnet 5. The accumulated memory also transfers: each model, reading the store built by the other, rises above its own no-memory baseline. The harness, protocol, and data are released.

Humans and machines often solve harder problems by spending more time on computation. In deep learning, looped models implement this idea during inference by recurrently updating a hidden state. In practice, however, their training backpropagates through only one or a few updates, making it hard to train early updates to support future ones. We propose looped flows, an approach that sidesteps this issue by training the recurrence with local denoising objectives. By imposing temporal association across denoising objectives through progressively decreasing noise levels and shared noise, the model is incentivized to learn recurrent states that transfer useful computation over time, even when gradients cover only a few updates. We then formulate inference as integrating the velocity of a probability flow parameterized by the learned denoiser, coupled with recurrent states. This allows solving harder problems by spending more computation through a finer temporal grid and enables multiple valid predictions from different initial noise samples. Across six reasoning benchmarks including two multi-solution benchmarks, looped flows outperform prior state-of-the-art looped models overall, achieving 58.8% test accuracy on ARC-AGI-1 and 12.2% on ARC-AGI-2.

Real-time recurrent learning (RTRL) for sequence-processing recurrent neural networks (RNNs) offers certain conceptual advantages over backpropagation through time (BPTT). RTRL requires neither caching past activations nor truncating context, and enables online learning. However, RTRL's time and space complexity make it impractical. To overcome this problem, most recent work on RTRL focuses on approximation theories, while experiments are often limited to diagnostic settings. Here we explore the practical promise of RTRL in more realistic settings. We study actor-critic methods that combine RTRL and policy gradients, and test them in several subsets of DMLab-30, ProcGen, and Atari-2600 environments. On DMLab memory tasks, our system trained on fewer than 1.2 B environmental frames is competitive with or outperforms well-known IMPALA and R2D2 baselines trained on 10 B frames. To scale to such challenging tasks, we focus on certain well-known neural architectures with element-wise recurrence, allowing for tractable RTRL without approximation. Importantly, we also discuss rarely addressed limitations of RTRL in real-world applications, such as its complexity in the multi-layer case.

A latent recurrent language model applies a shared block of layers repeatedly, so one checkpoint can trade inference compute for accuracy by changing the loop count. At one loop, however, such models typically fall well short of a non-recurrent model of the same size, a one-loop tax that makes a separate non-recurrent model the better choice whenever latency matters. We present Elastic Latent Recurrence (ELR), a recipe for training looped language models without this tax. At one loop, an ELR model costs the same to serve as a non-recurrent model of the same size and, trained on the same data, reaches similar accuracy. Each additional loop then buys more accuracy for more inference compute. Three choices make this work: a gated state update with a learned step size, loop counts mixed within each optimizer update, and conditioning each pass on its loop index and budget. The first two preserve one-loop accuracy, and the third lowers loss at every additional loop. Mixed depths can be introduced late in pretraining to limit the added training cost, and with zero-initialized conditioning added at the switch, this late schedule reaches lower loss than mixed-depth training from scratch. In MoE models, reusing first-loop expert routes on later loops cuts mixed-depth update time by 11.7% with no measurable change in accuracy. At inference, keeping only the last loop's attention KV cache holds attention KV memory at its one-loop level while retaining most of the gain from additional loops. On 4B dense and 30B-A3B MoE hybrid models, ELR at one loop is 0.52 points below and 1.04 points above its matched baseline on a ten-task average, and four loops add 7.25 and 4.75 points. Carried through post-training, the recipe yields Koi-4B, a single checkpoint that scores an estimated 18 on the Artificial Analysis Intelligence Index at one loop, matching Qwen3.5-4B, and 29 at four loops.

Looped Transformers introduce recurrent depth as a new scaling axis for LLMs: by repeatedly applying shared Transformer blocks, they increase effective depth without increasing parameter count. However, the benefits of looping remain unclear for large MoE LLMs under FLOPs-matched comparisons. The main reason is that the gains from additional iterations diminish quickly and can even turn into degradation, so the extra FLOPs spent on looping yield little substantial improvement. Consequently, prior work typically settles on two loops. We identify two main obstacles to scaling looped MoE. First, looping inherits and amplifies the curse of depth: hidden-state variance grows with each iteration as residual updates accumulate, which destabilizes deep recurrence and causes representations to drift. Second, looped MoE suffers from expert selection collapse: routers repeatedly select the same experts across loops, so extra iterations add computation without adding computational diversity. Guided by this diagnosis, we propose LOOM, built on a single principle: each loop should contribute new computation while keeping the recurrent state stable. LOOM stabilizes recurrence by scaling residual updates to bound variance growth and re-injecting the input embedding at every loop, and diversifies it through per-loop routers that engage different experts and a Looping Residual that carries earlier outputs forward. Experiments across 100M-1.7B models show stable scaling to 9-12 loops. Under near-iso-FLOP, the 700M model performs best at 5 loops, reducing perplexity from 18.36 to 16.54 and improving average zero-shot accuracy from 38.84% to 39.53% over the non-looped baseline. Without FLOP matching, the 1.7B model trained on 60B tokens peaks at 9 loops, reducing perplexity from 9.62 to 7.77 and improving average zero-shot accuracy from 42.4% to 47.7%. Code is available https://github.com/hed-ucas/LOOM.

Recurrent neural networks (RNNs) are a cornerstone of sequence modeling across various scientific and industrial applications. Owing to their versatility, numerous RNN variants have been proposed over the past decade, aiming to improve the modeling of long-term dependencies and to address challenges such as vanishing and exploding gradients. However, no central library is available to test these variations, and reimplementing diverse architectures can be time-consuming and error-prone, limiting reproducibility and exploration. Here, we introduce three open-source libraries in Julia and Python that centralize numerous recurrent cell implementations and higher-level recurrent architectures. torchrecurrent, RecurrentLayers.jl, and LuxRecurrentLayers.jl offer a consistent framework for constructing and extending RNN models, providing built-in mechanisms for customization and experimentation. All packages are available under the MIT license and actively maintained on GitHub.

We introduce a new class of time-continuous recurrent neural network models. Instead of declaring a learning system's dynamics by implicit nonlinearities, we construct networks of linear first-order dynamical systems modulated via nonlinear interlinked gates. The resulting models represent dynamical systems with varying (i.e., liquid) time-constants coupled to their hidden state, with outputs being computed by numerical differential equation solvers. These neural networks exhibit stable and bounded behavior, yield superior expressivity within the family of neural ordinary differential equations, and give rise to improved performance on time-series prediction tasks. To demonstrate these properties, we first take a theoretical approach to find bounds over their dynamics and compute their expressive power by the trajectory length measure in latent trajectory space. We then conduct a series of time-series prediction experiments to manifest the approximation capability of Liquid Time-Constant Networks (LTCs) compared to classical and modern RNNs. Code and data are available at https://github.com/raminmh/liquid_time_constant_networks

In the realm of artificial intelligence, where a vast majority of data is unstructured, obtaining substantial amounts of labeled data to train supervised machine learning models poses a significant challenge. To address this, we delve into few-shot and active learning, where are goal is to improve AI models with human feedback on a few labeled examples. This paper focuses on understanding how a continuous feedback loop can refine models, thereby enhancing their accuracy, recall, and precision through incremental human input. By employing Large Language Models (LLMs) such as GPT-3.5, BERT, and SetFit, we aim to analyze the efficacy of using a limited number of labeled examples to substantially improve model accuracy. We benchmark this approach on the Financial Phrasebank, Banking, Craigslist, Trec, Amazon Reviews datasets to prove that with just a few labeled examples, we are able to surpass the accuracy of zero shot large language models to provide enhanced text classification performance. We demonstrate that rather than needing to manually label millions of rows of data, we just need to label a few and the model can effectively predict the rest.

Looped language models can run more iterations at test time, but additional iterations need not help them solve longer problems. For sequential state tracking, final-answer supervision does not specify how far each iteration should advance, and repeated updates can change the scale of the recurrent state. We introduce Progress-Aligned Recurrence (PAR), which trains the prediction at iteration k to match the state after the first k operations and rescales recurrent states to their first-iteration norms. On swap-S_5, a Qwen2.5-0.5B model trained on at most 32 moves achieves 95.5\% mean full-state accuracy at 512 moves, or 16times the training length, compared with 55.7\% reported for RLT and 0.8\% for explicit chain of thought on the same backbone. Controlled S_5 ablations show that removing norm anchoring reduces the extrapolation factor from 16times to below 2.2times, and replacing progress-aligned targets with final-answer supervision reduces it to at most 8times. A learned completion readout selects when to answer without a supplied stopping depth. When intermediate labels are unavailable, prefix self-distillation induces the correspondence from shorter problem prefixes using only final answers. The gains apply across sequential state-tracking tasks, establishing the task structure required for depth extrapolation. These findings support training intermediate predictions and state scale together to make additional recurrent depth useful for longer problems.

Repeating a small block of middle layers increases a language model's effective inference depth without adding parameters or generating extra tokens, and recent work shows that this latent recurrence improves reasoning. However, two design choices limit these gains. Each iteration sees only the previous output and cannot directly access earlier computations. Moreover, a fixed loop count wastes depth on easy inputs while leaving hard ones with too little computation. We introduce RecurTrace, which addresses both limitations using the loop's own trajectory. Specifically, Loop Memory Attention lets each looped layer attend to its own states from previous iterations along the loop-time axis, so the model can revisit earlier computations instead of relying on the latest state alone. A halting head then reads the loop state and predicts whether to continue, with supervision from an oracle that identifies when additional depth still reduces loss. In a controlled MathQA comparison on the same looped backbone, RecurTrace achieves 56.9% accuracy with an average of 2.0 loops, exceeding the best fixed loop depth by 2.2 points at matched compute. By comparison, ACT and PonderNet collapse to one loop, and CALM reaches only 54.1% with 5.6 loops, while the stronger LoopUS-Conf and TaH-Mismatch baselines reach 55.3% at 3.2 loops and 55.7% at 2.1 loops. Finally, RecurTrace improves generation accuracy over same-budget fine-tuned baselines at 0.6B, 1.7B, 4B, and 8B, with the gain growing with model size from 0.6 to 3.4 points.

Recurrent connectivity in the visual cortex is believed to aid object recognition for challenging conditions such as occlusion. Here we investigate if and how artificial neural networks also benefit from recurrence. We compare architectures composed of bottom-up, lateral and top-down connections and evaluate their performance using two novel stereoscopic occluded object datasets. We find that classification accuracy is significantly higher for recurrent models when compared to feedforward models of matched parametric complexity. Additionally we show that for challenging stimuli, the recurrent feedback is able to correctly revise the initial feedforward guess.

Recursive self-improvement requires models not only to acquire useful training experience, but also to decide when their current experience has ceased to be useful. Existing methods often focus on what experience to generate next, while paying less attention to when fresh experience should be introduced. Many also construct training targets through self-consistency or majority voting, making supervision depend on the model's own uncertain predictions. We address these two questions separately: training dynamics determine when to acquire new experience, while the answers to new tasks remain independently checkable. A-RSI monitors reward variation and accuracy in training logs to refresh a task pool when its learning signal declines, without consulting validation or test scores. A static library of model-authored programs produces multimodal questions and derives their reference answers; we specify the task-generation contract, automatically verify chosen properties of the reward and refresh rules, and separately audit the generated answers. The training loop uses no external training examples or human-written answer labels. On the primary eleven-benchmark suite, A-RSI raises the performance average from 57.77 to 61.05 with 800 fresh tasks.

Recently, recurrent models such as state space models and linear attention have become popular due to their linear complexity in the sequence length. Thanks to their recurrent nature, in principle they can process arbitrarily long sequences, but their performance sometimes drops considerably beyond their training context lengths-i.e. they fail to length generalize. In this work, we provide comprehensive empirical and theoretical analysis to support the unexplored states hypothesis, which posits that models fail to length generalize when during training they are only exposed to a limited subset of the distribution of all attainable states (i.e. states that would be attained if the recurrence was applied to long sequences). Furthermore, we investigate simple training interventions that aim to increase the coverage of the states that the model is trained on, e.g. by initializing the state with Gaussian noise or with the final state of a different input sequence. With only 500 post-training steps (sim 0.1% of the pre-training budget), these interventions enable length generalization for sequences that are orders of magnitude longer than the training context (e.g. 2klongrightarrow 128k) and show improved performance in long context tasks, thus presenting a simple and efficient way to enable robust length generalization in general recurrent models.

Sequential dynamics are a key feature of many modern recommender systems, which seek to capture the `context' of users' activities on the basis of actions they have performed recently. To capture such patterns, two approaches have proliferated: Markov Chains (MCs) and Recurrent Neural Networks (RNNs). Markov Chains assume that a user's next action can be predicted on the basis of just their last (or last few) actions, while RNNs in principle allow for longer-term semantics to be uncovered. Generally speaking, MC-based methods perform best in extremely sparse datasets, where model parsimony is critical, while RNNs perform better in denser datasets where higher model complexity is affordable. The goal of our work is to balance these two goals, by proposing a self-attention based sequential model (SASRec) that allows us to capture long-term semantics (like an RNN), but, using an attention mechanism, makes its predictions based on relatively few actions (like an MC). At each time step, SASRec seeks to identify which items are `relevant' from a user's action history, and use them to predict the next item. Extensive empirical studies show that our method outperforms various state-of-the-art sequential models (including MC/CNN/RNN-based approaches) on both sparse and dense datasets. Moreover, the model is an order of magnitude more efficient than comparable CNN/RNN-based models. Visualizations on attention weights also show how our model adaptively handles datasets with various density, and uncovers meaningful patterns in activity sequences.

Recurrent neural networks are a widely used class of neural architectures. They have, however, two shortcomings. First, they are often treated as black-box models and as such it is difficult to understand what exactly they learn as well as how they arrive at a particular prediction. Second, they tend to work poorly on sequences requiring long-term memorization, despite having this capacity in principle. We aim to address both shortcomings with a class of recurrent networks that use a stochastic state transition mechanism between cell applications. This mechanism, which we term state-regularization, makes RNNs transition between a finite set of learnable states. We evaluate state-regularized RNNs on (1) regular languages for the purpose of automata extraction; (2) non-regular languages such as balanced parentheses and palindromes where external memory is required; and (3) real-word sequence learning tasks for sentiment analysis, visual object recognition and text categorisation. We show that state-regularization (a) simplifies the extraction of finite state automata that display an RNN's state transition dynamic; (b) forces RNNs to operate more like automata with external memory and less like finite state machines, which potentiality leads to a more structural memory; (c) leads to better interpretability and explainability of RNNs by leveraging the probabilistic finite state transition mechanism over time steps.

Continual learning, the setting where a learning agent is faced with a never ending stream of data, continues to be a great challenge for modern machine learning systems. In particular the online or "single-pass through the data" setting has gained attention recently as a natural setting that is difficult to tackle. Methods based on replay, either generative or from a stored memory, have been shown to be effective approaches for continual learning, matching or exceeding the state of the art in a number of standard benchmarks. These approaches typically rely on randomly selecting samples from the replay memory or from a generative model, which is suboptimal. In this work, we consider a controlled sampling of memories for replay. We retrieve the samples which are most interfered, i.e. whose prediction will be most negatively impacted by the foreseen parameters update. We show a formulation for this sampling criterion in both the generative replay and the experience replay setting, producing consistent gains in performance and greatly reduced forgetting. We release an implementation of our method at https://github.com/optimass/Maximally_Interfered_Retrieval.

The fixed-size context of Transformer makes GPT models incapable of generating arbitrarily long text. In this paper, we introduce RecurrentGPT, a language-based simulacrum of the recurrence mechanism in RNNs. RecurrentGPT is built upon a large language model (LLM) such as ChatGPT and uses natural language to simulate the Long Short-Term Memory mechanism in an LSTM. At each timestep, RecurrentGPT generates a paragraph of text and updates its language-based long-short term memory stored on the hard drive and the prompt, respectively. This recurrence mechanism enables RecurrentGPT to generate texts of arbitrary length without forgetting. Since human users can easily observe and edit the natural language memories, RecurrentGPT is interpretable and enables interactive generation of long text. RecurrentGPT is an initial step towards next-generation computer-assisted writing systems beyond local editing suggestions. In addition to producing AI-generated content (AIGC), we also demonstrate the possibility of using RecurrentGPT as an interactive fiction that directly interacts with consumers. We call this usage of generative models by ``AI As Contents'' (AIAC), which we believe is the next form of conventional AIGC. We further demonstrate the possibility of using RecurrentGPT to create personalized interactive fiction that directly interacts with readers instead of interacting with writers. More broadly, RecurrentGPT demonstrates the utility of borrowing ideas from popular model designs in cognitive science and deep learning for prompting LLMs. Our code is available at https://github.com/aiwaves-cn/RecurrentGPT and an online demo is available at https://www.aiwaves.org/recurrentgpt.

We measure how much one extra recurrence is worth to a looped (depth-recurrent) language model, in equivalent unique parameters. From an iso-depth sweep of 116 pretraining runs across recurrence counts r in {1, 2, 4, 8} spanning {sim}50times in training compute, we fit a joint scaling law L = E + A,(N_once + r^φ N_rec)^{-α} + B,D^{-β} and recover a new recurrence-equivalence exponent φ= 0.46. Intuitively, φ tells us whether looping a block r times is equivalent in validation loss to r unique blocks of a non-looped model (full equivalence, φ{=}1) or to a single block run repeatedly with no capacity gain (φ{=}0). Our φ= 0.46 sits in between, so each additional recurrence predictably increases validation loss at matched training compute. For example, at r{=}4 a 410M looped model performs on par with a 580M non-looped model, but incurs the training cost of a 1B non-looped one. We demonstrate the utility of φ as a measurement tool on two probes. Truncated backpropagation lowers φ to 0.38, indicating that the loop mechanism is poorly trained under truncation, even though validation loss decreases. Conversely, hyperconnections raise φ to 0.65, a genuine capacity gain. Our method applies to any looped LM and separates true loop improvements from token-budget gains.

Looped Language Models (LoopLMs) enable efficient latent reasoning through depth recurrence, yet exhibit unreliable test-time scaling behavior: performance often peaks at a certain iteration depth and then collapses with further recurrence. Through latent dynamics analysis, we find an inherent trade-off between stability and effectiveness in existing architectures and strategies. By conceptualizing reasoning as uncertainty reduction, we propose that convergence toward stable fixed points while preserving effectiveness represents a promising way. To this end, we propose STARS (STAbility-driven Recurrent Scaling), a training framework that constrains latent states to approach asymptotically stable fixed points. This is realized via efficient Jacobian Spectral Radius Regularization with random loop sampling, enabling STARS to maximize effectiveness while ensuring rigorous stability. Experiments on arithmetic tasks show that STARS achieves reliable test-time scaling, and on complex mathematical reasoning it substantially mitigates performance degradation as recurrence depth increases while also improving peak performance.

A Recurrent Neural Network (RNN) was used to perform video frame prediction of microbial growth for a population of two mutants of Pseudomonas aeruginosa. The RNN was trained on videos of 20 frames that were acquired using fluorescence microscopy and microfluidics. The network predicted the last 10 frames of each video, and the accuracy's of the predictions was assessed by comparing raw images, population curves, and the number and size of individual colonies. Overall, we found the predictions to be accurate using this approach. The implications this result has on designing autonomous experiments in microbiology, and the steps that can be taken to make the predictions even more accurate, are discussed.

Animals ranging from rats to humans can demonstrate cognitive map capabilities. We evolved weights in a biologically plausible recurrent neural network (RNN) using an evolutionary algorithm to replicate the behavior and neural activity observed in rats during a spatial and working memory task in a triple T-maze. The rat was simulated in the Webots robot simulator and used vision, distance and accelerometer sensors to navigate a virtual maze. After evolving weights from sensory inputs to the RNN, within the RNN, and from the RNN to the robot's motors, the Webots agent successfully navigated the space to reach all four reward arms with minimal repeats before time-out. Our current findings suggest that it is the RNN dynamics that are key to performance, and that performance is not dependent on any one sensory type, which suggests that neurons in the RNN are performing mixed selectivity and conjunctive coding. Moreover, the RNN activity resembles spatial information and trajectory-dependent coding observed in the hippocampus. Collectively, the evolved RNN exhibits navigation skills, spatial memory, and working memory. Our method demonstrates how the dynamic activity in evolved RNNs can capture interesting and complex cognitive behavior and may be used to create RNN controllers for robotic applications.

Recurrent reasoning models have attracted growing attention for scaling test-time computation, typically by iteratively refining latent states with shared parameters. However, these models apply each learned update with a fixed unit scale, which can be conservative when updates make persistent progress and overly aggressive when they fluctuate, limiting the benefit of additional loops. To understand how the scale should vary along the trajectory, we first analyze the sensitivity of terminal loss to recurrent update scale. We show that its temporal average admits an exact decomposition into persistent-progress and centered-fluctuation contributions. Based on this, we introduce the Trajectory Adaptive Progress-Fluctuation Scheduler (TAPS), which tracks their balance across recurrent updates and adapts the step size online. Theoretically, we establish sufficient conditions under which TAPS reduces expected terminal loss and reaches a target quality in fewer recurrent loops. Empirically, we show that TAPS improves terminal accuracy across structured reasoning tasks without retraining. By further incorporating the progress-fluctuation principle into training, TAPS yields additional accuracy gains with up to 1.56 times wall-clock speedup at matched baseline accuracy. The broad applicability of TAPS is supported by its effectiveness across diverse recurrent architectures and inference strategies. Together, these results establish update scale as complementary control axis of recurrent inference alongside architecture and depth.

Linear Recurrence has proven to be a powerful tool for modeling long sequences efficiently. In this work, we show that existing models fail to take full advantage of its potential. Motivated by this finding, we develop GateLoop, a foundational sequence model that generalizes linear recurrent models such as S4, S5, LRU and RetNet, by employing data-controlled state transitions. Utilizing this theoretical advance, GateLoop empirically outperforms existing models for auto-regressive language modeling. Our method comes with a low-cost O(l) recurrent mode and an efficient O(l log_{2} l) parallel mode making use of highly optimized associative scan implementations. Furthermore, we derive an O(l^2) surrogate attention mode, revealing remarkable implications for Transformer and recently proposed architectures. Specifically, we prove that our approach can be interpreted as providing data-controlled relative-positional information to Attention. While many existing models solely rely on data-controlled cumulative sums for context aggregation, our findings suggest that incorporating data-controlled complex cumulative products may be a crucial step towards more powerful sequence models.

Recurrent neural network is a powerful model that learns temporal patterns in sequential data. For a long time, it was believed that recurrent networks are difficult to train using simple optimizers, such as stochastic gradient descent, due to the so-called vanishing gradient problem. In this paper, we show that learning longer term patterns in real data, such as in natural language, is perfectly possible using gradient descent. This is achieved by using a slight structural modification of the simple recurrent neural network architecture. We encourage some of the hidden units to change their state slowly by making part of the recurrent weight matrix close to identity, thus forming kind of a longer term memory. We evaluate our model in language modeling experiments, where we obtain similar performance to the much more complex Long Short Term Memory (LSTM) networks (Hochreiter & Schmidhuber, 1997).

Music generation has generally been focused on either creating scores or interpreting them. We discuss differences between these two problems and propose that, in fact, it may be valuable to work in the space of direct it performance generation: jointly predicting the notes it and it also their expressive timing and dynamics. We consider the significance and qualities of the data set needed for this. Having identified both a problem domain and characteristics of an appropriate data set, we show an LSTM-based recurrent network model that subjectively performs quite well on this task. Critically, we provide generated examples. We also include feedback from professional composers and musicians about some of these examples.

Looped Language Models (LoopLMs) offer a parameter efficient approach to scaling reasoning by reusing shared parameters across recurrent computation steps. Despite their promise, effective post-training of LoopLMs remains challenging. Existing approaches either provide reward based supervision that is sparse or costly to extend across loops, or rely on external teachers or privileged information, leading to limited teacher availability or teacher-student context mismatch. To address these limitations, we introduce LoopOPD, a cross-loop on-policy distillation framework that uses additional recurrent computation within a LoopLM as its own source of supervision. LoopOPD uses a frozen terminal loop policy as a compute privileged teacher for an intermediate loop student on student generated rollouts, providing dense supervision without an external teacher or privileged information. We further propose Dynamic LoopOPD (D-LoopOPD), which continually refreshes the terminal loop teacher as the shared model parameters are updated, enabling recurrent self-improvement. We characterize how distillation updates propagate across loop depths and derive sufficient conditions under which a single update yields simultaneous local improvement at both loop depths. Experiments on Ouro-Thinking models show that LoopOPD improves mathematical reasoning, while D-LoopOPD yields further gains through dynamic teacher updates. Despite being trained only on mathematical data, the resulting models also improve on general reasoning and code generation benchmarks, demonstrating that recurrent computation can serve as an effective source of supervision for LoopLMs. Our code and model checkpoints will be released upon acceptance.

Learning to remember over long timescales is fundamentally challenging for recurrent neural networks (RNNs). While much prior work has explored why RNNs struggle to learn long timescales and how to mitigate this, we still lack a clear understanding of the dynamics involved when RNNs learn long timescales via gradient descent. Here we build a mathematical theory of the learning dynamics of linear RNNs trained to integrate white noise. We show that when the initial recurrent weights are small, the dynamics of learning are described by a low-dimensional system that tracks a single outlier eigenvalue of the recurrent weights. This reveals the precise manner in which the long timescale associated with white noise integration is learned. We extend our analyses to RNNs learning a damped oscillatory filter, and find rich dynamical equations for the evolution of a conjugate pair of outlier eigenvalues. Taken together, our analyses build a rich mathematical framework for studying dynamical learning problems salient for both machine learning and neuroscience.

Learning continuously during all model lifetime is fundamental to deploy machine learning solutions robust to drifts in the data distribution. Advances in Continual Learning (CL) with recurrent neural networks could pave the way to a large number of applications where incoming data is non stationary, like natural language processing and robotics. However, the existing body of work on the topic is still fragmented, with approaches which are application-specific and whose assessment is based on heterogeneous learning protocols and datasets. In this paper, we organize the literature on CL for sequential data processing by providing a categorization of the contributions and a review of the benchmarks. We propose two new benchmarks for CL with sequential data based on existing datasets, whose characteristics resemble real-world applications. We also provide a broad empirical evaluation of CL and Recurrent Neural Networks in class-incremental scenario, by testing their ability to mitigate forgetting with a number of different strategies which are not specific to sequential data processing. Our results highlight the key role played by the sequence length and the importance of a clear specification of the CL scenario.

The human brain is autonomously active, being characterized by a self-sustained neural activity which would be present even in the absence of external sensory stimuli. Here we study the interrelation between the self-sustained activity in autonomously active recurrent neural nets and external sensory stimuli. There is no a priori semantical relation between the influx of external stimuli and the patterns generated internally by the autonomous and ongoing brain dynamics. The question then arises when and how are semantic correlations between internal and external dynamical processes learned and built up? We study this problem within the paradigm of transient state dynamics for the neural activity in recurrent neural nets, i.e. for an autonomous neural activity characterized by an infinite time-series of transiently stable attractor states. We propose that external stimuli will be relevant during the sensitive periods, {\it viz} the transition period between one transient state and the subsequent semi-stable attractor. A diffusive learning signal is generated unsupervised whenever the stimulus influences the internal dynamics qualitatively. For testing we have presented to the model system stimuli corresponding to the bars and stripes problem. We found that the system performs a non-linear independent component analysis on its own, being continuously and autonomously active. This emergent cognitive capability results here from a general principle for the neural dynamics, the competition between neural ensembles.

Drifting Models generate a sample with a single function evaluation, making them attractive for fast parallel language generation, but their one-step formulation does not support iterative refinement. We propose Recurrent Language Generation via Drifting (ReLaDrift), which uses a shared Drifting Model to generate an initial draft from noise and iteratively refine it in a continuous latent space. We train this recurrence by sampling the number of rounds and applying the drifting loss only at the final round, stopping gradients through preceding rounds so that only one round requires backpropagation. Unlike the one-step Drifting Model, whose accuracy does not improve when iterated, ReLaDrift enables a trade-off between test-time computation and generation quality, as diffusion and flow language models do. On Sudoku Hard and GSM8K, it achieves competitive or superior performance compared with these iterative-generation approaches across a range of inference budgets. Our code is available at https://github.com/Yuki-Ots/ReLaDrift

Recurrent Neural Networks (RNNs), and specifically a variant with Long Short-Term Memory (LSTM), are enjoying renewed interest as a result of successful applications in a wide range of machine learning problems that involve sequential data. However, while LSTMs provide exceptional results in practice, the source of their performance and their limitations remain rather poorly understood. Using character-level language models as an interpretable testbed, we aim to bridge this gap by providing an analysis of their representations, predictions and error types. In particular, our experiments reveal the existence of interpretable cells that keep track of long-range dependencies such as line lengths, quotes and brackets. Moreover, our comparative analysis with finite horizon n-gram models traces the source of the LSTM improvements to long-range structural dependencies. Finally, we provide analysis of the remaining errors and suggests areas for further study.

State Space Models (SSMs) are inherently recurrent along the sequence dimension, yet depth-recurrence - reusing the same block repeatedly across layers, as recently applied in looped transformers - has not been explored in this model family. We show that a looped SSM with k parameters iterated L times consistently closely matches or outperforms a standard SSM with k cdot L independent parameters across four architectures (LRU, S5, LinOSS, LrcSSM) and six time series classification benchmarks, despite operating within a strictly smaller hypothesis space, as we formally establish. Since the larger model contains the looped model as a special case, this dominance cannot be explained by expressivity and instead points to parameter sharing across depth as a beneficial inductive bias that simplifies optimization. These results demonstrate that depth-recurrence is orthogonal to sequence-recurrence and independently beneficial. We further show that input reshaping is an equally neglected design axis: concatenating timesteps for low-dimensional inputs, or flattening and rechunking the joint feature-time dimension for high-dimensional ones, yields accuracy gains of 1-6% across all models, confirmed over 5 random seeds. Both techniques provide standalone improvements that compound when combined, suggesting that depth and input reshaping are two independent and underexplored design axes for SSMs on time series.

In this paper, we argue that looped Transformers need their own residual connections to prevent performance degradation as the number of iterations grows. We observe that increasing loop iterations can reduce reasoning accuracy: noisy state updates overwrite correct intermediate deductions and even undo completed solutions. This leaves subsequent iterations to recover lost information from an already degraded representation: once an error arises in an earlier loop, often as a result of long-range propagation through the recurrence, later loops find it difficult to correct. In this paper, we introduce InfiLoop, a loop-native residual connection that learns which past computations to retain and how much to accept from each new update. InfiLoop combines content-based weighting with learned temporal decay to maintain a running summary of recurrent states. An exact streaming recurrence keeps its persistent aggregation memory constant as the loop count grows. The resulting adaptive update suppresses unreliable proposals and preserves useful intermediate states. Across extensive reasoning tasks, a 7M-parameter InfiLoop model outperforms existing recursive architectures, reaching 97.9% exact accuracy on Sudoku-Extreme, and 13.6% pass@2 on ARC-AGI-2. Notably, on Sudoku-Extreme, InfiLoop continues to improve with test-time looping beyond 20,000 effective steps, showing that added depth translates directly into stronger reasoning. Our code is available at https://github.com/pixeli99/InfiLoop.

Seismic advances in generative AI algorithms for imagery, text, and other data types has led to the temptation to use synthetic data to train next-generation models. Repeating this process creates an autophagous (self-consuming) loop whose properties are poorly understood. We conduct a thorough analytical and empirical analysis using state-of-the-art generative image models of three families of autophagous loops that differ in how fixed or fresh real training data is available through the generations of training and in whether the samples from previous generation models have been biased to trade off data quality versus diversity. Our primary conclusion across all scenarios is that without enough fresh real data in each generation of an autophagous loop, future generative models are doomed to have their quality (precision) or diversity (recall) progressively decrease. We term this condition Model Autophagy Disorder (MAD), making analogy to mad cow disease.

Knowledge tracing---where a machine models the knowledge of a student as they interact with coursework---is a well established problem in computer supported education. Though effectively modeling student knowledge would have high educational impact, the task has many inherent challenges. In this paper we explore the utility of using Recurrent Neural Networks (RNNs) to model student learning. The RNN family of models have important advantages over previous methods in that they do not require the explicit encoding of human domain knowledge, and can capture more complex representations of student knowledge. Using neural networks results in substantial improvements in prediction performance on a range of knowledge tracing datasets. Moreover the learned model can be used for intelligent curriculum design and allows straightforward interpretation and discovery of structure in student tasks. These results suggest a promising new line of research for knowledge tracing and an exemplary application task for RNNs.

Large language models (LLMs) are increasingly used to automate AI research, including language-model (LM) development, offering a potential path toward recursive self-improvement. However, turning experimental feedback into a learned ability to propose effective improvements across models faces two challenges. First, training such an ability with real feedback is expensive: reinforcement learning (RL) requires evaluating tens of thousands of proposals, each potentially requiring a full model-training run. Even at ten GPU-hours per run, reward collection alone would consume hundreds of thousands of GPU-hours, motivating a cheaper surrogate. Second, surrogate feedback can become unreliable as policy optimization favors overestimated proposals when the proposal distribution shifts. To address these challenges, we propose LM2LM, which trains LLMs with RL to generate effective LM improvement proposals. To provide scalable feedback, we use a research world model (RWM), a frozen LLM that predicts proposal gains from real experimental records provided in context, as the reward model for RL. To maintain feedback reliability, we periodically execute current-policy proposals ranked highly by the RWM and add their measured outcomes to its context. Across eight autoregressive and diffusion LM development environments, real-execution evaluations on held-out tasks show that LM2LM enables Qwen3.8-27B to outperform GPT-6-sol, even when GPT-6-sol is given 300 real experimental records in context, while requiring 33% lower data-generation cost. LM2LM also generalizes to a model family unseen during training.

Brain-like intelligent systems need brain-like learning methods. Equilibrium Propagation (EP) is a biologically plausible learning framework with strong potential for brain-inspired computing hardware. However, existing im-plementations of EP suffer from instability and prohibi-tively high computational costs. Inspired by the structure and dynamics of the brain, we propose a biologically plau-sible Feedback-regulated REsidual recurrent neural network (FRE-RNN) and study its learning performance in EP framework. Feedback regulation enables rapid convergence by reducing the spectral radius. The improvement in con-vergence property reduces the computational cost and train-ing time of EP by orders of magnitude, delivering perfor-mance on par with backpropagation (BP) in benchmark tasks. Meanwhile, residual connections with brain-inspired topologies help alleviate the vanishing gradient problem that arises when feedback pathways are weak in deep RNNs. Our approach substantially enhances the applicabil-ity and practicality of EP in large-scale networks that un-derpin artificial intelligence. The techniques developed here also offer guidance to implementing in-situ learning in physical neural networks.

LLM agents can improve across tasks without updating model parameters by storing experience from their own executions in external memory and reusing it to guide later decisions. A central design question is whether to retain only observed trajectories and outcomes or also the rules, summaries, and skills inferred from them. Retaining these abstractions allows agents to reuse strategies without repeatedly deriving them from raw trajectories. However, errors in stored abstractions can propagate to subsequent decisions through retrieval and reuse. They can also shape execution trajectories, which then become the basis for new abstractions that reproduce the original errors. We term this feedback mechanism error reproduction: errors are regenerated through a cycle of abstraction, execution, and re-abstraction, extending their persistence beyond direct reuse of the original faulty knowledge. We propose Anti-Persistence, a memory design that accumulates trajectories and outcomes while reconstructing task-specific guidance on demand. An adaptive selector uses prior outcomes to decide whether and how to synthesize guidance, which is discarded after execution. This prevents its direct reuse as a stored object, although indirect feedback through retained trajectories remains possible. Across different benchmarks and backbones, Anti-Persistence achieves higher overall held-out success and the lowest average LLM-call cost among the evaluated memory designs. These results support separating experience accumulation from abstraction persistence in memorybased self-improvement.

Training recurrent neural networks (RNNs) is a high-dimensional process that requires updating numerous parameters. Therefore, it is often difficult to pinpoint the underlying learning mechanisms. To address this challenge, we propose to gain mechanistic insights into the phenomenon of abrupt learning by studying RNNs trained to perform diverse short-term memory tasks. In these tasks, RNN training begins with an initial search phase. Following a long period of plateau in accuracy, the values of the loss function suddenly drop, indicating abrupt learning. Analyzing the neural computation performed by these RNNs reveals geometric restructuring (GR) in their phase spaces prior to the drop. To promote these GR events, we introduce a temporal consistency regularization that accelerates (bioplausible) training, facilitates attractor formation, and enables efficient learning in strongly connected networks. Our findings offer testable predictions for neuroscientists and emphasize the need for goal-agnostic secondary mechanisms to facilitate learning in biological and artificial networks.

We define an atomic generation fact f=(u,tau,omega,z;rho), recording the origin, realized transformation, concrete occurrence, generated result and relation role. Compiled into a Generation-Fact Graph (GFG), these facts provide an AI-native, compilable scientific fact substrate preserving generation histories. We establish a GFG-based recursive scientific process in which analysis, intervention, replay and validation form facts for later cycles. Using nanoGPT, we establish unified training-learning dynamics. Training is the evolution of a parameter-optimizer system with state and memory: each actual training action enters the receiving state and produces a finite-amplitude nonlinear functional response conditioned by that state and target-specific update geometry. Learning is the persistent reorganization of distributed functional support by these responses; capability formation, maintenance, decline or recovery becomes observable when target-specific states are evaluated against their readout boundaries. Three primary coordinates - target-boundary state, target-specific update geometry and parameter-Adam receiving state - yield a second-order predictor operating before post-update outputs are read. On held-out runs, it achieved 91.43% accuracy and 91.49% macro-averaged recall across four transitions. We further establish inference as a frozen projection of training-learning dynamics. Component gating and rollback show causal recruitment and non-additive combination of query-conditioned support formed during training, deriving organizational conditions realized by Attention. Controlled feedback indicates possible double-edged reinforcement effects. ResNet/CIFAR-100 and diffusion/CIFAR-10 experiments confirm receiving-state-conditioned responses, persistent support reorganization and frozen inference projection beyond nanoGPT.

Solving problems through repeated attempts is a sequential modeling task: at each step, the solver receives feedback and decides how to revise their solutions. Predicting whether performance improves, plateaus, or regresses across attempts is central to understanding any iterative problem-solving process in both human learners and autonomous agents. Beyond outcomes, modeling what errors persist and how strategies shift across attempts provides deeper insight into the mechanics of sequential learning. Studying these dynamics requires observing many solvers as they attempt, receive feedback, and revise. Programming courses with automated grading provide this setting, as students iteratively submit code to test suites and receive feedback on every attempt. We therefore curate CodeInsight, a large-scale dataset of over 3 million submissions from 3,286 undergraduates across 2 introductory C++ courses in 2 academic years, with test-case-level outcomes, timestamps, and source code. On this dataset, we build a benchmark that evaluates models spanning parametric, sequential, and generative traditions under a shared calibration-and-scoring protocol, including a Recurrent State Space Model (RSSM) adapted to track solver characteristics through discrete latent variables and an LLM-based predictor that generates explicit solutions. The adapted RSSM achieves the strongest predictive accuracy on three of the four courses. The LLM predictor is less accurate but produces full submissions at each attempt, enabling direct analysis of failure modes. We find that the model's coding proficiency is inversely related to predictive performance in this setting, with the LLM better understood as a generative solver conditioned on context rather than a faithful predictor of solver behavior. We publicly release our code and the dataset on request to facilitate future research.

Providing high-quality feedback is crucial for student success but is constrained by time, cost, and limited data availability. We introduce Synthetic Educational Feedback Loops (SEFL), a novel framework designed to deliver immediate, on-demand feedback at scale without relying on extensive, real-world student data. In SEFL, two large language models (LLMs) operate in teacher--student roles to simulate assignment completion and formative feedback, generating abundant synthetic pairs of student work and corresponding critiques. We then fine-tune smaller, more computationally efficient LLMs on these synthetic pairs, enabling them to replicate key features of high-quality, goal-oriented feedback. Unlike personalized tutoring approaches that offer multi-turn, individualized instruction, SEFL specifically focuses on replicating the teacher-->student feedback loop for diverse assignments. Through both LLM-as-a-judge and human evaluations, we demonstrate that SEFL-tuned models outperform their non-tuned counterparts in feedback quality, clarity, and timeliness. These findings reveal SEFL's potential to transform feedback processes for higher education and beyond, offering an ethical and scalable alternative to conventional manual feedback cycles.

An agent can improve its behavior within an unresolved problem by changing how it acts and which experience it retains. We propose MMLA-RSI, a recursive self-improvement scheme with two implementations in Memory-Mediated Learning Architecture: Reasoning-Time Training (RTT) changes the policy state Φ, while Reasoning-Time Memory Adaptation (RTMA) changes authoritative memory M. Each route also makes its adaptation method an object of improvement. A current numerical updater trains the policy proposing its successor; a current memory manager chooses the records available to the process proposing its successor. An independently admitted successor takes over the same role in the next generation. We give bounded executable rule languages, shared task/proposal interfaces, and one algorithm for either route or their composition. The RTT instance changes scalar schedules in a clipped-momentum update. The RTMA instance chooses NULL or a complete-row write with possible eviction. A fixed interpreter, trusted committer, and independent evaluator bound what can change. State versions, read closures, and conditional properties describe causal execution, accepted transitions, and finite work. Existing numerical-component studies motivate the construction. In four-candidate program selection, retaining LARC feedback updates reduces expected error by 24.65 and 36.65 percentage points for two trained initializations. In chronological continuous-integration replay, feedback binding is positive while retention increases loss. Completed SQL studies improve substantially over an unadapted substrate but do not establish the required additional online benefit. These results characterize their declared components. The two recursive routes and their joint realization are specified here; their beneficial recursive chains and discovery efficiency remain to be evaluated.

Biological neurons transmit information with stereotyped electrical impulses called ``spikes'', sensitive to coincident timings. Spiking Neural Networks (SNNs), introduced in the nineties, have gained popularity in AI for their energy efficiency and competitive performance with deep learning. Among them, Recurrent SNNs (RSNNs) are particularly appealing for their ability to learn long-term dependencies and exhibit rich dynamics. In SNNs, each connection can have a weight and a transmission delay, both plastic in the brain. While theory has long suggested that trainable delays enhance a network's expressivity, practical learning methods emerged only recently and remain mostly limited to feedforward delays. Here, we introduce DelRec, the first method to jointly optimize recurrent delays with synaptic weights in RSNNs via surrogate gradient learning, compatible with any spiking neuron model. DelRec works in discrete time, leveraging differentiable interpolation to handle non-integer delays with well-defined gradients at training time. Using simple neurons, DelRec outperforms all baselines on a chaotic time-series prediction task, and achieves competitive performance on four challenging temporal datasets. Analysis of trained networks show that recurrent delay optimization builds structured, depth-dependent spatio-temporal receptive fields that the network actively exploits, with delay-weight joint optimization reshaping temporal selectivity. Beyond accuracy, we show that recurrent delay learning can reduce the memory footprint and energy consumption of a network, and shapes its robustness to temporal perturbations of its inputs. This work establishes recurrent delay optimization as a promising framework for both biological circuit modeling and neuromorphic computing.

Language model agents increasingly operate over long interaction streams, requiring online adaptation to distribution shifts through continual learning. However, existing agent harnesses typically either evolve their states only through forward information flow or rely on additional replay or validation data for harness optimization. To address this, we propose ReHarness, a recurrent harness framework for agentic continual learning that complements forward read and write with backward revision under a strictly online streaming protocol. We ask what information could have been captured differently in the harness state to better inform the current instance in the stream. Specifically, after each instance, we leverage the resulting trajectory as feedback to revise the reflections retrieved for that instance, thereby propagating feedback backward through the textual harness state. Across three continual learning benchmarks and three language model backbones, ReHarness consistently outperforms strong raw-context baselines, memory- and reflection-based methods, and prompt optimization approaches. Further analysis shows that ReHarness produces source-grounded and meaningful revisions, corrects overgeneralized or outdated experience under distribution shifts, and improves both stability and plasticity during continual learning.

Cortical circuits exhibit intricate recurrent architectures that are remarkably similar across different brain areas. Such stereotyped structure suggests the existence of common computational principles. However, such principles have remained largely elusive. Inspired by gated-memory networks, namely long short-term memory networks (LSTMs), we introduce a recurrent neural network in which information is gated through inhibitory cells that are subtractive (subLSTM). We propose a natural mapping of subLSTMs onto known canonical excitatory-inhibitory cortical microcircuits. Our empirical evaluation across sequential image classification and language modelling tasks shows that subLSTM units can achieve similar performance to LSTM units. These results suggest that cortical circuits can be optimised to solve complex contextual problems and proposes a novel view on their computational function. Overall our work provides a step towards unifying recurrent networks as used in machine learning with their biological counterparts.

Yes. Can it cure doom loops? Probably not. The Gemma 4 instruction-tuned models share a reproducible failure: on long factual enumeration prompts, such as listing every episode of a TV series, the 88 IAU constellations, or the 151 original Pokemon, they collapse into repetition, either a tight verbatim loop or a list whose entries decay onto a single answer. These loops occur at rates as high as 95% and survive prompt rewording, inference-engine changes, and most sampling adjustments. In this paper we explore whether this behavior is localized enough to remove by weight edits. To localize the cause, we use per-layer ablation and per-neuron attribution, then confirm the strongest candidates with full-generation sweeps. The loops trace to a small set of MLP neurons (or, in the 26B-A4B Mixture-of-Experts model, a few routed experts) which we suppress with static weight edits. These "surgeries" can be as small as a single sign-inverted neuron (in the E2B model). The size of the effective edits grows with model scale, but in all cases, the loop patterns can be addressed at normal generation budgets while preserving general-purpose benchmark scores. However, the edits do not solve everything: we also study longer thinking budgets, where the two larger models most visibly enter doom looping, i.e. a non-convergent regime in which the model self-corrects in circles over a fact it cannot recall, exhausting the budget without committing to a final answer. We show this residual failure is reduced but not eliminated by the same edits, and argue it is fundamentally a knowledge-precision problem rather than a removable circuit; weight surgery can delete a loop, but it cannot supply a missing fact. Our results are both a feasibility demonstration, that is, evidence that a concrete generation pathology can be localized to a few parameters and edited out, and a delineation of where that approach stops.

Attention mechanisms have revolutionized sequence learning but suffer from quadratic computational complexity. This paper introduces Lattice, a novel recurrent neural network (RNN) mechanism that leverages the inherent low-rank structure of K-V matrices to efficiently compress the cache into a fixed number of memory slots, achieving sub-quadratic complexity. We formulate this compression as an online optimization problem and derive a dynamic memory update rule based on a single gradient descent step. The resulting recurrence features a state- and input-dependent gating mechanism, offering an interpretable memory update process. The core innovation is the orthogonal update: each memory slot is updated exclusively with information orthogonal to its current state hence incorporation of only novel, non-redundant data, which minimizes the interference with previously stored information. The experimental results show that Lattice achieves the best perplexity compared to all baselines across diverse context lengths, with performance improvement becoming more pronounced as the context length increases.

We study auto research as a closed empirical loop driven by external measurement. Each submitted trial carries a hypothesis, an executable code edit, an evaluator-owned outcome, and feedback that shapes the next proposal. The output is not a generated paper or a single model checkpoint, but an auditable trajectory of proposals, code diffs, experiments, scores, and failure labels. We instantiate this loop with specialist agents that partition recipe surfaces and share measured lineage across trials. The central empirical finding is that lineage feedback lets agents turn evaluator outcomes, including crashes, budget overruns, size failures, and accuracy-gate misses, into later program-level recipe edits rather than one-shot suggestions. Across 1,197 headline-run trials plus 600 Parameter Golf control trials after one-time setup and launch, humans did not choose proposals, edit recipes, override scores, or repair failed trials during the search. In the three headline runs, the same submitted-trial loop reduces Parameter Golf validation bpb by 0.81%, raises NanoChat-D12 CORE by 38.7%, and reduces CIFAR-10 Airbench96 wallclock by 4.59%, with each task measured by its own external evaluator and legality checks. The trace includes a strict architecture-domain audit of 157 headline-run submissions and program rewrites such as a NanoChat attention-kernel path change. Within this scope the loop autonomously writes code, submits experiments, absorbs feedback, applies and combines known techniques inside each environment, and improves public starting recipes.

This study introduces PV-RNN, a novel variational RNN inspired by the predictive-coding ideas. The model learns to extract the probabilistic structures hidden in fluctuating temporal patterns by dynamically changing the stochasticity of its latent states. Its architecture attempts to address two major concerns of variational Bayes RNNs: how can latent variables learn meaningful representations and how can the inference model transfer future observations to the latent variables. PV-RNN does both by introducing adaptive vectors mirroring the training data, whose values can then be adapted differently during evaluation. Moreover, prediction errors during backpropagation, rather than external inputs during the forward computation, are used to convey information to the network about the external data. For testing, we introduce error regression for predicting unseen sequences as inspired by predictive coding that leverages those mechanisms. The model introduces a weighting parameter, the meta-prior, to balance the optimization pressure placed on two terms of a lower bound on the marginal likelihood of the sequential data. We test the model on two datasets with probabilistic structures and show that with high values of the meta-prior the network develops deterministic chaos through which the data's randomness is imitated. For low values, the model behaves as a random process. The network performs best on intermediate values, and is able to capture the latent probabilistic structure with good generalization. Analyzing the meta-prior's impact on the network allows to precisely study the theoretical value and practical benefits of incorporating stochastic dynamics in our model. We demonstrate better prediction performance on a robot imitation task with our model using error regression compared to a standard variational Bayes model lacking such a procedure.

Recent work on looped language models suggests that many reasoning problems benefit from greater computational depth rather than from additional independent parameters. Existing studies, however, focus almost exclusively on Transformer backbones, leaving open whether this principle also applies to state-space language models. We investigate Looped Mamba and Looped Hybrid Mamba-Transformer architectures, which repeatedly apply a shared Mamba (or hybrid) block to introduce explicit finite-depth recurrent computation. On two controlled reasoning tasks-Mano (modular-arithmetic manipulation) and p-hop induction-Looped Mamba consistently outperforms parameter-matched non-looped baselines and, in several settings, matches or exceeds non-looped models of equal effective depth. We then extend the study to language model pre-training under matched iso-parameter and iso-FLOPs protocols, which jointly disentangle the effects of parameter sharing and effective depth: looped models remain competitive on downstream benchmarks with substantially fewer distinct parameters, although deeper non-looped models retain an advantage in validation perplexity under strict iso-FLOPs comparisons. Finally, we adapt Ouro's two-stage exit gate to Looped Mamba for threshold-controlled selection among recurrent-step outputs. Since all recurrent steps are still executed, the selected exit step represents prediction depth rather than reduced wall-clock computation. At the scales studied, adaptive exit-state selection improves downstream performance at intermediate depths, while actual inference-time savings require additional state-handling mechanisms.

Random Neural Networks (RNNs) are a class of Neural Networks (NNs) that can also be seen as a specific type of queuing network. They have been successfully used in several domains during the last 25 years, as queuing networks to analyze the performance of resource sharing in many engineering areas, as learning tools and in combinatorial optimization, where they are seen as neural systems, and also as models of neurological aspects of living beings. In this article we focus on their learning capabilities, and more specifically, we present a practical guide for using the RNN to solve supervised learning problems. We give a general description of these models using almost indistinctly the terminology of Queuing Theory and the neural one. We present the standard learning procedures used by RNNs, adapted from similar well-established improvements in the standard NN field. We describe in particular a set of learning algorithms covering techniques based on the use of first order and, then, of second order derivatives. We also discuss some issues related to these objects and present new perspectives about their use in supervised learning problems. The tutorial describes their most relevant applications, and also provides a large bibliography.

Multiple extensions of Recurrent Neural Networks (RNNs) have been proposed recently to address the difficulty of storing information over long time periods. In this paper, we experiment with the capacity of Neural Turing Machines (NTMs) to deal with these long-term dependencies on well-balanced strings of parentheses. We show that not only does the NTM emulate a stack with its heads and learn an algorithm to recognize such words, but it is also capable of strongly generalizing to much longer sequences.

In this paper, we investigate the long-term memory learning capabilities of state-space models (SSMs) from the perspective of parameterization. We prove that state-space models without any reparameterization exhibit a memory limitation similar to that of traditional RNNs: the target relationships that can be stably approximated by state-space models must have an exponential decaying memory. Our analysis identifies this "curse of memory" as a result of the recurrent weights converging to a stability boundary, suggesting that a reparameterization technique can be effective. To this end, we introduce a class of reparameterization techniques for SSMs that effectively lift its memory limitations. Besides improving approximation capabilities, we further illustrate that a principled choice of reparameterization scheme can also enhance optimization stability. We validate our findings using synthetic datasets and language models.

Standard training pipelines for large language models (LLMs) are typically unidirectional, progressing from pre-training to post-training. However, the potential for a bidirectional process--where insights from post-training retroactively improve the pre-trained foundation--remains unexplored. We aim to establish a self-reinforcing flywheel: a cycle in which reinforcement learning (RL)-tuned model strengthens the base model, which in turn enhances subsequent post-training performance, requiring no specially trained teacher or reference model. To realize this, we analyze training dynamics and identify the mid-training (annealing) phase as a critical turning point for model capabilities. This phase typically occurs at the end of pre-training, utilizing high-quality corpora under a rapidly decaying learning rate. Building upon this insight, we introduce ReMiT (Reinforcement Learning-Guided Mid-Training). Specifically, ReMiT leverages the reasoning priors of RL-tuned models to dynamically reweight tokens during the mid-training phase, prioritizing those pivotal for reasoning. Empirically, ReMiT achieves an average improvement of 3\% on 10 pre-training benchmarks, spanning math, code, and general reasoning, and sustains these gains by over 2\% throughout the post-training pipeline. These results validate an iterative feedback loop, enabling continuous and self-reinforcing evolution of LLMs.

Constructing states from sequences of observations is an important component of reinforcement learning agents. One solution for state construction is to use recurrent neural networks. Back-propagation through time (BPTT), and real-time recurrent learning (RTRL) are two popular gradient-based methods for recurrent learning. BPTT requires complete trajectories of observations before it can compute the gradients and is unsuitable for online updates. RTRL can do online updates but scales poorly to large networks. In this paper, we propose two constraints that make RTRL scalable. We show that by either decomposing the network into independent modules or learning the network in stages, we can make RTRL scale linearly with the number of parameters. Unlike prior scalable gradient estimation algorithms, such as UORO and Truncated-BPTT, our algorithms do not add noise or bias to the gradient estimate. Instead, they trade off the functional capacity of the network for computationally efficient learning. We demonstrate the effectiveness of our approach over Truncated-BPTT on a prediction benchmark inspired by animal learning and by doing policy evaluation of pre-trained policies for Atari 2600 games.

Transformers have become the dominant architecture for sequence modeling by using self-attention to enable expressive and highly parallel processing. However, the resulting quadratic time and memory costs limit efficiency in long-context settings. Recurrent models such as LSTMs provide explicit nonlinear state updates and strong state-tracking capabilities, yet their strictly sequential computation limits parallelism. We introduce the Parallel Recursive LSTM (PR-LSTM), a hierarchical recurrent architecture that replaces left-to-right recurrence with recursive nonlinear state composition over a balanced computation tree. Tokens are first mapped independently to latent states, which are then recursively merged by a learned gated composition block. This structure uses the reduction pattern underlying parallel scans as a fixed execution schedule, rather than assuming an associative recurrence. As a result, PR-LSTM retains nonlinear gated state representations while reducing recurrent parallel depth from linear to logarithmic. Empirically, PR-LSTM achieves strong sequence-length generalization on formal-language benchmarks, solving more tasks than standard RNN, LSTM, and Transformer baselines, while avoiding the quadratic scaling of attention. These results suggest that recurrent computation can be reorganized hierarchically to expose parallelism without restricting the transition dynamics to linear or associative forms.

Sequential prediction problems such as imitation learning, where future observations depend on previous predictions (actions), violate the common i.i.d. assumptions made in statistical learning. This leads to poor performance in theory and often in practice. Some recent approaches provide stronger guarantees in this setting, but remain somewhat unsatisfactory as they train either non-stationary or stochastic policies and require a large number of iterations. In this paper, we propose a new iterative algorithm, which trains a stationary deterministic policy, that can be seen as a no regret algorithm in an online learning setting. We show that any such no regret algorithm, combined with additional reduction assumptions, must find a policy with good performance under the distribution of observations it induces in such sequential settings. We demonstrate that this new approach outperforms previous approaches on two challenging imitation learning problems and a benchmark sequence labeling problem.

Vision-language-action (VLA) models predict chunks of future actions from the current observation, an assumption that fails under partial observability, where decisions depend on information no longer visible. Existing memory-augmented VLAs simultaneously introduce recurrence, retrieval, compression modules, auxiliary objectives, hierarchical memory, or task-specific architectural changes, so the contribution of recurrence itself remains entangled with surrounding machinery. We present a controlled isolation study of recurrence in a strong pretrained VLA backbone. Our formulation augments the transformer with a small set of learnable memory tokens carried across timesteps and updated through self-attention, trained end to end with truncated backpropagation through time, with no auxiliary losses and no architectural changes. We instantiate this as μVLA, a family of OpenVLA-OFT variants parameterized by memory width m, TBPTT length K, and the memory update rule (cross-step gradients or a detached EMA), so that recurrence is the only varying factor. On MIKASA-Robo, μVLA improves average success rate on five training tasks from 0.42 to 0.84 at the strongest setting and reaches 0.23 on held-out tasks with the same memory structure versus 0.07 for the memoryless baseline. On tasks requiring different memory structure, performance remains near baseline. On LIBERO, the strongest recurrent variant achieves 96.2% average success, indicating no regression under full observability. We interpret these results as a calibration of the capability envelope of minimal in-backbone recurrence, identifying the regime in which it is sufficient and the regime where additional memory structure is required. Demos and videos can be found in https://avanturist322.github.io/mu-vla/.

We prove an inverse approximation theorem for the approximation of nonlinear sequence-to-sequence relationships using recurrent neural networks (RNNs). This is a so-called Bernstein-type result in approximation theory, which deduces properties of a target function under the assumption that it can be effectively approximated by a hypothesis space. In particular, we show that nonlinear sequence relationships that can be stably approximated by nonlinear RNNs must have an exponential decaying memory structure - a notion that can be made precise. This extends the previously identified curse of memory in linear RNNs into the general nonlinear setting, and quantifies the essential limitations of the RNN architecture for learning sequential relationships with long-term memory. Based on the analysis, we propose a principled reparameterization method to overcome the limitations. Our theoretical results are confirmed by numerical experiments. The code has been released in https://github.com/radarFudan/Curse-of-memory

Language models with recurrent depth, also referred to as universal or looped when considering transformers, are defined by the capacity to increase their computation through the repetition of layers. Recent efforts in pretraining have demonstrated that these architectures can scale to modern language modeling tasks while exhibiting advantages in reasoning tasks. In this work, we examine the relationship between recurrent-depth models and diffusion language models. Building on their similarities, we develop a new diffusion forcing sampler for these models to accelerate generation. The sampler advances by decoding new tokens at every forward pass of the model, while the latent states of these tokens can be further refined in parallel through recurrence. Theoretically, generation with our sampler is strictly more expressive than the baseline autoregressive generation using the same time budget on modern hardware. Moreover, this sampler, based on principles from diffusion literature, can be directly applied to existing 3.5B recurrent-depth transformers without any tuning, leading to up to a 5x speedup. Consequently, our findings not only provide an efficient mechanism for parallelizing the extra computation in recurrent-depth models at inference, but also suggest that such models can be naturally viewed as strong continuous, though causal, diffusion language models.

In mid-2026 a slogan reorganized how practitioners talk about coding agents: stop prompting your agent, start designing the loop that prompts it. We take this claim seriously and give it a careful treatment. We call the object of the new practice the loop specification: a bounded, reusable artifact, made of a trigger, a goal, a verification step, a stopping rule and a memory, that a human hands to an agent harness (such as Claude Code or Codex) so the agent pursues a goal on its own, in place of step-by-step prompting. We distinguish this external loop specification from two things it is often confused with: an ordinary programming loop, and the internal perceive-act-observe cycle that the harness already provides as plumbing. We position loop engineering as a new layer in the progression from prompt to context to harness to loop, and we argue, against the stronger headlines, that it does not retire prompt engineering; loop and prompt are distinct tools with distinct uses. We offer four contributions: a definition and scope for the discipline; an anatomy and taxonomy of loop specifications organized around trigger, goal type, a five-level verification ladder, architecture, and named terminal states; a descriptive analysis of the Loop Library, a public corpus of fifty real loops that we code by hand; and a set of design principles and anti-patterns grounded in the scientific literature on self-correction, reward hacking and model-as-judge fragility. The corpus shows that practice has matured most where the discipline says it matters: seventy percent of loops verify in the autonomous zone of the ladder and seventy-four percent name their terminal states, while automated triggering and durable memory remain comparatively underdeveloped. We close with the limits the practice must respect, including the verification burden, comprehension debt and the risk of cognitive surrender.

We seek a hierarchical resolution of residual error under a composite learning objective as a principled mechanism for identifying and exploiting additional information, particularly under distribution shift and perturbation. Inspired by the System 1/System 2 paradigm, we formulate this principle as a two-loop mechanism serving a common inference goal. A fast, System 1-like loop resolves the residual addressable by the available evidence; the remaining epistemic residual triggers a medium speed, System 2-like loop that actively acquires additional evidence and measures its informational value through residual reduction. Over nested solution spaces, the medium-loop optimum is provably no worse than the fast-loop optimum. Experiments show that the resulting two-loop model consistently improves upon the fast loop at approximately half the computation of the medium loop, including under unseen corruptions, and, when provided a cleaner observation, outperforms staged classifiers and learning-to-defer methods at equal computational cost.

We propose a stronger formulation of RL on top of RWKV-style recurrent sequence models, in which the fixed-size recurrent state is explicitly interpreted as a belief state rather than an opaque hidden vector. Instead of conditioning policy and value on a single summary h_t, we maintain a compact uncertainty-aware state b_t = (μ_t, Σ_t) derived from RWKV-style recurrent statistics and let control depend on both memory and uncertainty. This design targets a key weakness of plain fixed-state policies in partially observed settings: they may store evidence, but not necessarily confidence. We present the method, a theoretical program, and a pilot RL experiment with hidden episode-level observation noise together with a test-time noise sweep. The pilot shows that belief-state policies nearly match the best recurrent baseline overall while slightly improving return on the hardest in-distribution regime and under a held-out noise shift. Additional ablations show that this simple belief readout is currently stronger than two more structured extensions, namely gated memory control and privileged belief targets, underscoring the need for richer benchmarks.

AI agents are beginning to automate research and development across the AI stack, from improving training efficiency to optimizing inference. A natural next step is to improve the research efficiency of the agents themselves. When an AI research agent's own code is the object of optimization, each accepted rewrite becomes the agent that the next round edits. We refer to this loop as recursive self-improvement. Its significance lies in a long-standing trend, in which increased cumulative spending on R&D yields diminishing returns. Sustained self-improvement offers a way to counter this trend. We present AIDE^2, a system that implements this loop for a frontier AI research agent. It proposes changes to its own code, benchmarks modified versions of itself on a suite of AI R&D tasks, and keeps the changes that perform best on hidden evaluations. In an autonomous 8-day run, AIDE^2 discovered seven successive improvements, ranging from a new search policy to memory mechanisms that compress and manage the agent's growing context. These gains generalize to four held-out benchmarks spanning machine learning engineering, heuristic algorithm engineering, and physics-based weather forecasting, the last of which is out of distribution from the selection tasks. On all four, the strongest discovered agent matches or exceeds a human-engineered production research agent that ranks among the strongest on FML-Bench. On a separate held-out task family, the discovered agents also exhibit reduced reward hacking, a property the loop never explicitly optimized for: the rate falls from 55% to 32% during the run, 7 percentage points below the human-engineered agent. Together, these results show that an AI research agent can improve its own research efficiency through recursive self-improvement, and that these gains transfer to tasks and domains the loop never encountered.

Humans learn continually throughout their lifespan by accumulating diverse knowledge and fine-tuning it for future tasks. When presented with a similar goal, neural networks suffer from catastrophic forgetting if data distributions across sequential tasks are not stationary over the course of learning. An effective approach to address such continual learning (CL) problems is to use hypernetworks which generate task dependent weights for a target network. However, the continual learning performance of existing hypernetwork based approaches are affected by the assumption of independence of the weights across the layers in order to maintain parameter efficiency. To address this limitation, we propose a novel approach that uses a dependency preserving hypernetwork to generate weights for the target network while also maintaining the parameter efficiency. We propose to use recurrent neural network (RNN) based hypernetwork that can generate layer weights efficiently while allowing for dependencies across them. In addition, we propose novel regularisation and network growth techniques for the RNN based hypernetwork to further improve the continual learning performance. To demonstrate the effectiveness of the proposed methods, we conducted experiments on several image classification continual learning tasks and settings. We found that the proposed methods based on the RNN hypernetworks outperformed the baselines in all these CL settings and tasks.

Interactively learning from observation and language feedback is an increasingly studied area driven by the emergence of large language model (LLM) agents. While impressive empirical demonstrations have been shown, so far a principled framing of these decision problems remains lacking. In this paper, we formalize the Learning from Language Feedback (LLF) problem, assert sufficient assumptions to enable learning despite latent rewards, and introduce transfer eluder dimension as a complexity measure to characterize the hardness of LLF problems. We show that transfer eluder dimension captures the intuition that information in the feedback changes the learning complexity of the LLF problem. We demonstrate cases where learning from rich language feedback can be exponentially faster than learning from reward. We develop a no-regret algorithm, called HELiX, that provably solves LLF problems through sequential interactions, with performance guarantees that scale with the transfer eluder dimension of the problem. Across several empirical domains, we show that HELiX performs well even when repeatedly prompting LLMs does not work reliably. Our contributions mark a first step towards designing principled interactive learning algorithms from generic language feedback.

In recent years, attention-like mechanisms have been used to great success in the space of large language models, unlocking scaling potential to a previously unthinkable extent. "Attention Is All You Need" famously claims RNN cells are not needed in conjunction with attention. We challenge this view. In this paper, we point to existing proofs that architectures with fully parallelizable forward or backward passes cannot represent classes of problems specifically interesting for long-running agentic tasks. We further conjecture a critical time t beyond which non-recurrence-complete models fail to aggregate inputs correctly, with concrete implications for agentic systems (e.g., software engineering agents). To address this, we introduce a recurrence-complete architecture and train it on GitHub-derived action sequences. Loss follows a power law in the trained sequence length while the parameter count remains fixed. Moreover, longer-sequence training always amortizes its linearly increasing wall-time cost, yielding lower loss as a function of wall time.

Looped language models have shown promising results on reasoning benchmarks, yet their potential for agentic tool use remains largely unexplored. We study this question in compositional tool-calling settings, where models must coordinate multiple API calls, maintain intermediate state, and preserve dependencies across tool interactions. We evaluate native and retrofitted looped language models on API-Bank, BFCL, and NESTful, comparing looped and non-looped models trained under matched supervised fine-tuning recipes and varying recurrent depth at inference time. In controlled experiments, recurrent computation generally benefits compositional and dependency-aware tool use, while providing smaller and more model-dependent gains on isolated API invocation. Accuracy on multi-step tool use generally increases with recurrent depth; adaptive inference, however, achieves a more favorable compute-performance trade-off by allocating additional computation only when needed. Our results suggest that looped language models are a promising architecture for agentic systems that require reliable planning, coordination, and execution of compositional tool use workflows.

Humans continuously learn from experience, whereas conventional large language model (LLM) evaluations ignore the models' ability to improve through inference-time interaction. In this paper, we study how LLMs learn from iterative experience at test time, a setting we refer to as Chain-of-Experience (CoE), where models accumulate experiential traces through iterative interactions with self or environmental feedback to form a continual improvement loop beyond zero-shot inference. We instantiate CoE with diverse feedback mechanisms, including model self-feedback and environmental signals such as correctness or public coding test pass rates, and evaluate across math, coding, and knowledge domains using 8 LLMs, including GPT-5, Gemini-2.5 Pro, Claude-4.5 Sonnet. Our study shows that leveraging iterative experience consistently outperforms feedback-free baselines, achieving substantial gains with self feedback alone, alongside a 5.6% overall improvement and 19% lower API cost across tasks and models. We further show that combining complementary feedback channels (e.g., model and correctness signals) yields additional gains, and that CoE delivers higher accuracy per token than existing test-time strategies. We observe a positive correlation between LLM base ability and improvement capacity, and show that models remain robust under weak or spurious feedback, with different feedback contributing to distinct improvement aspects and most gains emerging early in the iterations.

Koopman representations aim to learn features of nonlinear dynamical systems (NLDS) which lead to linear dynamics in the latent space. Theoretically, such features can be used to simplify many problems in modeling and control of NLDS. In this work we study autoencoder formulations of this problem, and different ways they can be used to model dynamics, specifically for future state prediction over long horizons. We discover several limitations of predicting future states in the latent space and propose an inference-time mechanism, which we refer to as Periodic Reencoding, for faithfully capturing long term dynamics. We justify this method both analytically and empirically via experiments in low and high dimensional NLDS.

Reconstructing nonlinear dynamical systems (DS) from data (DSR) is a fundamental challenge in science and engineering, but it inherently relies on sequential models. Recent breakthroughs for sequential models have produced algorithms that parallelize computation along sequence length T, achieving logarithmic time complexity, O(log T). Since sequence lengths have been practically limited due to the linear runtime complexity O(T) of classical backpropagation through time, this opens new avenues for DSR. This paper studies two prominent classes of parallel-in-time algorithms for this task, both of which leverage parallel associative scans as their core computational primitive. The first class comprises models with linear yet non-autonomous dynamics and a nonlinear readout, such as modern State Space Models (SSMs), while the second consists of general nonlinear models which can be parallelized using the DEER framework. We find that the linear training-time recurrence of the first class of models imposes limitations that often hinder learning of accurate nonlinear dynamics. To address this, we augment DEER with Generalized Teacher Forcing (GTF), a novel variant within the more general nonlinear framework that ensures stable and effective learning of nonlinear dynamics across arbitrary sequence lengths. Using GTF-DEER, we investigate the benefits of training on extremely long sequences (T>10^4) for DSR. Our results show that access to such long trajectories significantly improves DSR if the data features long time scales. This work establishes GTF-DEER as a robust tool for data-driven discovery and underscores the largely untapped potential of long-sequence learning in modeling complex DS.

Models that can simulate how environments change in response to actions can be used by agents to plan and act efficiently. We improve on previous environment simulators from high-dimensional pixel observations by introducing recurrent neural networks that are able to make temporally and spatially coherent predictions for hundreds of time-steps into the future. We present an in-depth analysis of the factors affecting performance, providing the most extensive attempt to advance the understanding of the properties of these models. We address the issue of computationally inefficiency with a model that does not need to generate a high-dimensional image at each time-step. We show that our approach can be used to improve exploration and is adaptable to many diverse environments, namely 10 Atari games, a 3D car racing environment, and complex 3D mazes.

The brain solves the credit assignment problem remarkably well. For credit to be assigned across neural networks they must, in principle, wait for specific neural computations to finish. How the brain deals with this inherent locking problem has remained unclear. Deep learning methods suffer from similar locking constraints both on the forward and feedback phase. Recently, decoupled neural interfaces (DNIs) were introduced as a solution to the forward and feedback locking problems in deep networks. Here we propose that a specialised brain region, the cerebellum, helps the cerebral cortex solve similar locking problems akin to DNIs. To demonstrate the potential of this framework we introduce a systems-level model in which a recurrent cortical network receives online temporal feedback predictions from a cerebellar module. We test this cortico-cerebellar recurrent neural network (ccRNN) model on a number of sensorimotor (line and digit drawing) and cognitive tasks (pattern recognition and caption generation) that have been shown to be cerebellar-dependent. In all tasks, we observe that ccRNNs facilitates learning while reducing ataxia-like behaviours, consistent with classical experimental observations. Moreover, our model also explains recent behavioural and neuronal observations while making several testable predictions across multiple levels. Overall, our work offers a novel perspective on the cerebellum as a brain-wide decoupling machine for efficient credit assignment and opens a new avenue between deep learning and neuroscience.

Multivariate time-series data often exhibit complex temporal dependencies, irregular sampling, and heterogeneous dynamics across multiple time scales, making accurate sequence modeling particularly challenging. Traditional recurrent neural networks (RNNs), such as Long Short-Term Memory (LSTM) networks, operate in discrete time and may struggle to effectively capture continuous and irregular temporal behaviors. Liquid Neural Networks (LNNs) address some of these limitations through continuous-time dynamics, but standard LNN architectures typically rely on a single dynamical system, limiting their ability to model heterogeneous temporal patterns. To address these challenges, we propose a Multi-Rate Mixture-of-Experts (MR-MoE) framework built on top of Liquid Neural Networks. In the proposed architecture, multiple LNN-based experts operate at distinct time scales, enabling the model to explicitly separate fast-changing dynamics from slow-evolving temporal trends. A gating network further enables adaptive expert specialization based on input conditions. In addition, we incorporate both feature-level and temporal attention mechanisms to improve robustness, interpretability, and long-range dependency modeling. Feature-level attention suppresses noisy or irrelevant variables, while temporal attention selectively focuses on informative historical states. We evaluate the proposed framework on a complex multivariate time-series prediction task and compare it against strong baselines, including LSTM, monolithic LNN, and standard MoE models. Experimental results demonstrate that the proposed MR-MoE framework consistently achieves improved AUROC and AUPRC performance while maintaining favorable computational efficiency. These results highlight the effectiveness of combining continuous-time dynamics, multi-scale expert decomposition, and adaptive attention mechanisms for time-series modeling.

We introduce WARP (Weight-space Adaptive Recurrent Prediction), a simple yet powerful model that unifies weight-space learning with linear recurrence to redefine sequence modeling. Unlike conventional recurrent neural networks (RNNs) which collapse temporal dynamics into fixed-dimensional hidden states, WARP explicitly parametrizes its hidden state as the weights and biases of a distinct auxiliary neural network, and uses input differences to drive its recurrence. This brain-inspired formulation enables efficient gradient-free adaptation of the auxiliary network at test-time, in-context learning abilities, and seamless integration of domain-specific physical priors. Empirical validation shows that WARP matches or surpasses state-of-the-art baselines on diverse classification tasks, featuring in the top three in 5 out of 6 real-world challenging datasets. Furthermore, extensive experiments across sequential image completion, multivariate time series forecasting, and dynamical system reconstruction demonstrate its expressiveness and generalisation capabilities. Remarkably, a physics-informed variant of our model outperforms the next best model by more than 10x. Ablation studies confirm the architectural necessity of key components, solidifying weight-space linear RNNs as a transformative paradigm for adaptive machine intelligence.

Industrial recommender systems face the challenge of operating in non-stationary environments, where data distribution shifts arise from evolving user behaviors over time. To tackle this challenge, a common approach is to periodically re-train or incrementally update deployed deep models with newly observed data, resulting in a continual training process. However, the conventional learning paradigm of neural networks relies on iterative gradient-based updates with a small learning rate, making it slow for large recommendation models to adapt. In this paper, we introduce ReLoop2, a self-correcting learning loop that facilitates fast model adaptation in online recommender systems through responsive error compensation. Inspired by the slow-fast complementary learning system observed in human brains, we propose an error memory module that directly stores error samples from incoming data streams. These stored samples are subsequently leveraged to compensate for model prediction errors during testing, particularly under distribution shifts. The error memory module is designed with fast access capabilities and undergoes continual refreshing with newly observed data samples during the model serving phase to support fast model adaptation. We evaluate the effectiveness of ReLoop2 on three open benchmark datasets as well as a real-world production dataset. The results demonstrate the potential of ReLoop2 in enhancing the responsiveness and adaptiveness of recommender systems operating in non-stationary environments.

Task-trained recurrent neural networks (RNNs) are widely used in neuroscience and machine learning to model dynamical computations. To gain mechanistic insight into how neural systems solve tasks, prior work often reverse-engineers individual trained networks. However, different RNNs trained on the same task and achieving similar performance can exhibit strikingly different internal solutions, a phenomenon known as solution degeneracy. Here, we develop a unified framework to systematically quantify and control solution degeneracy across three levels: behavior, neural dynamics, and weight space. We apply this framework to 3,400 RNNs trained on four neuroscience-relevant tasks: flip-flop memory, sine wave generation, delayed discrimination, and path integration, while systematically varying task complexity, learning regime, network size, and regularization. We find that higher task complexity and stronger feature learning reduce degeneracy in neural dynamics but increase it in weight space, with mixed effects on behavior. In contrast, larger networks and structural regularization reduce degeneracy at all three levels. These findings empirically validate the Contravariance Principle and provide practical guidance for researchers seeking to tune the variability of RNN solutions, either to uncover shared neural mechanisms or to model the individual variability observed in biological systems. This work provides a principled framework for quantifying and controlling solution degeneracy in task-trained RNNs, offering new tools for building more interpretable and biologically grounded models of neural computation.

Despite the success of test-time scaling, Large Reasoning Models (LRMs) frequently encounter repetitive loops that lead to computational waste and inference failure. In this paper, we identify a distinct failure mode termed Circular Reasoning. Unlike traditional model degeneration, this phenomenon manifests as a self-reinforcing trap where generated content acts as a logical premise for its own recurrence, compelling the reiteration of preceding text. To systematically analyze this phenomenon, we introduce LoopBench, a dataset designed to capture two distinct loop typologies: numerical loops and statement loops. Mechanistically, we characterize circular reasoning as a state collapse exhibiting distinct boundaries, where semantic repetition precedes textual repetition. We reveal that reasoning impasses trigger the loop onset, which subsequently persists as an inescapable cycle driven by a self-reinforcing V-shaped attention mechanism. Guided by these findings, we employ the Cumulative Sum (CUSUM) algorithm to capture these precursors for early loop prediction. Experiments across diverse LRMs validate its accuracy and elucidate the stability of long-chain reasoning.

Going beyond predicting robot actions, World Action Models (WAMs) can also generate future visual observations. We build on this generative capability to propose Recurrent Generative Replay (REGEN), a continual imitation learning framework that synthesizes pseudo-replay trajectories, enabling a robot policy to rehearse previously learned tasks without storing their original human demonstrations. During continual adaptation, REGEN recursively queries the WAM to synthesize pseudo-replay trajectories conditioned only on prior task instructions and current-task observations. Experiments in both simulation and real-world manipulation settings show that REGEN reduces catastrophic forgetting by up to 50% relative to sequential fine-tuning, while approaching the performance of privileged experience replay methods that require access to real replay data. Finally, we analyze the factors limiting generated replay, identifying long-horizon visual degradation and action-observation inconsistency as the primary bottlenecks. Our results establish WAMs as a promising foundation for continual robot learning without stored demonstrations.

Despite recent advances in CUAs, their development remains constrained by the limited diversity, controllability, and reliability of interaction data. Moreover, conventional optimization pipelines lack iterative feedback, while existing recursive improvement approaches often rely on self-generated experiences, limiting ability expansion and yielding diminishing returns. To address these challenges, we introduce CUA-Master, a nested double-loop recursive improvement framework. At the system level, an outer world-data-model flywheel continuously incorporates open-world corpus and leverages evolving model weaknesses to guide subsequent data construction and optimization iterations. At the data level, an inner flywheel couples open-world seeding and autocatalytic propagation. The former mines authentic workflows as seed tasks from instructional videos, and employs visual-semantic dual-path trajectory reconstruction; The latter transforms these seeds into reusable primitives for controllable synthesis of diverse, complex, and verifiable tasks. To enable efficient and reliable trajectory collection, we further introduce a feedback-guided snapshot-rollback retry mechanism in various real environments. Extensive experiments demonstrate that CUA-Master-27B achieves competitive performance across GUI-only and GUI-CLI hybrid benchmarks, scoring 80.2 on OSWorld Verified, 36.9 on OSWorld 2.0, 43.9 on WeaveBench, and 25.0 on Agents' Last Exam. Our framework also substantially improves the quality and efficiency of data synthesis, and exhibits continuous gains across recursive iterations.

As Vision Transformers (ViTs) become standard vision backbones, a mechanistic account of their computational phenomenology is essential. Despite architectural cues that hint at dynamical structure, there is no settled framework that interprets Transformer depth as a well-characterized flow. In this work, we introduce the Block-Recurrent Hypothesis (BRH), arguing that trained ViTs admit a block-recurrent depth structure such that the computation of the original L blocks can be accurately rewritten using only k ll L distinct blocks applied recurrently. Across diverse ViTs, between-layer representational similarity matrices suggest few contiguous phases. To determine whether these phases reflect genuinely reusable computation, we train block-recurrent surrogates of pretrained ViTs: Recurrent Approximations to Phase-structured TransfORmers (Raptor). In small-scale, we demonstrate that stochastic depth and training promote recurrent structure and subsequently correlate with our ability to accurately fit Raptor. We then provide an empirical existence proof for BRH by training a Raptor model to recover 96% of DINOv2 ImageNet-1k linear probe accuracy in only 2 blocks at equivalent computational cost. Finally, we leverage our hypothesis to develop a program of Dynamical Interpretability. We find i) directional convergence into class-dependent angular basins with self-correcting trajectories under small perturbations, ii) token-specific dynamics, where cls executes sharp late reorientations while patch tokens exhibit strong late-stage coherence toward their mean direction, and iii) a collapse to low rank updates in late depth, consistent with convergence to low-dimensional attractors. Altogether, we find a compact recurrent program emerges along ViT depth, pointing to a low-complexity normative solution that enables these models to be studied through principled dynamical systems analysis.

Connectivity structure shapes neural computation, but inferring this structure from population recordings is degenerate: multiple connectivity structures can generate identical dynamics. Recent work uses low-rank recurrent neural networks (lrRNNs) to infer low-dimensional latent dynamics and connectivity from observed activity, enabling a mechanistic interpretation of the dynamics. However, standard approaches for training lrRNNs can recover spurious structures irrelevant to the underlying dynamics. We first characterize the identifiability of connectivity structures in lrRNNs and determine conditions under which a unique solution exists. To find such solutions, we develop an inference framework based on maximum entropy and continuous normalizing flows (CNFs), trained via flow matching. Instead of estimating a single connectivity matrix, our method learns a distribution over connection weights that is maximally unbiased over unidentifiable components while matching the observed dynamics. This approach captures complex yet necessary distributions such as heavy-tailed connectivity found in empirical data. We validate our method on synthetic datasets with connectivity structures that generate multistable attractors, limit cycles, and ring attractors, and demonstrate its applicability in recordings from rat frontal cortex during decision-making. Our framework shifts circuit inference from recovering connectivity to identifying which connectivity structures are computationally required, and which are artifacts of underconstrained inference.

As large language models (LLMs) are increasingly deployed in diverse user facing applications, aligning them with real user preferences becomes essential. Existing methods like Reinforcement Learning from Human Feedback (RLHF) rely on expert annotators trained on manually defined guidelines, whose judgments may not reflect the priorities of everyday users. We introduce Reinforcement Learning from User Feedback (RLUF), a framework for aligning LLMs directly to implicit signals from users in production. RLUF addresses key challenges of user feedback: user feedback is often binary (e.g., emoji reactions), sparse, and occasionally adversarial. We train a reward model, P[Love], to predict the likelihood that an LLM response will receive a Love Reaction, a lightweight form of positive user feedback, and integrate P[Love] into a multi-objective policy optimization framework alongside helpfulness and safety objectives. In large-scale experiments, we show that P[Love] is predictive of increased positive feedback and serves as a reliable offline evaluator of future user behavior. Policy optimization using P[Love] significantly raises observed positive-feedback rates, including a 28% increase in Love Reactions during live A/B tests. However, optimizing for positive reactions introduces reward hacking challenges, requiring careful balancing of objectives. By directly leveraging implicit signals from users, RLUF offers a path to aligning LLMs with real-world user preferences at scale.

In this work, we propose Retentive Network (RetNet) as a foundation architecture for large language models, simultaneously achieving training parallelism, low-cost inference, and good performance. We theoretically derive the connection between recurrence and attention. Then we propose the retention mechanism for sequence modeling, which supports three computation paradigms, i.e., parallel, recurrent, and chunkwise recurrent. Specifically, the parallel representation allows for training parallelism. The recurrent representation enables low-cost O(1) inference, which improves decoding throughput, latency, and GPU memory without sacrificing performance. The chunkwise recurrent representation facilitates efficient long-sequence modeling with linear complexity, where each chunk is encoded parallelly while recurrently summarizing the chunks. Experimental results on language modeling show that RetNet achieves favorable scaling results, parallel training, low-cost deployment, and efficient inference. The intriguing properties make RetNet a strong successor to Transformer for large language models. Code will be available at https://aka.ms/retnet.

Recurrent-depth language models, such as looped Transformers, repeatedly apply shared network blocks to refine latent representations without generating explicit intermediate reasoning tokens. However, each step recomputes full attention over the entire context, repeating costly global routing. We study how attention routing evolves across recurrent depth and find a consistent separation in convergence timescales: attention support and distributions stabilize substantially earlier than hidden states and attention outputs. This suggests two stages of recurrent inference: early discovery of a sparse working set, followed by representation refinement over largely stable routing support. Motivated by this finding, we introduce WISE (Working-set Inference with Support Exploitation), a training-free method that uses unrestricted attention during early recurrent steps to discover a block-structured working set, then reuses its support in later steps while keeping attention weights and recurrent refinement dynamic. Controlled interventions show that multi-step discovery yields more effective working sets than first-step selection, and that support reuse better preserves model behavior than more restrictive forms of attention reuse. Across multi-hop QA benchmarks, WISE largely preserves full-attention performance. Matched context-scaling experiments reveal an increasingly favorable quality-efficiency tradeoff as routing support becomes sparser with longer contexts. A sparse-attention implementation achieves up to a 1.76x late-step attention speedup over native FlashAttention at 4K context. Code: https://github.com/tbn5pj/WISE_code.

Neural networks are very effective when trained on large datasets for a large number of iterations. However, when they are trained on non-stationary streams of data and in an online fashion, their performance is reduced (1) by the online setup, which limits the availability of data, (2) due to catastrophic forgetting because of the non-stationary nature of the data. Furthermore, several recent works (Caccia et al., 2022; Lange et al., 2023) arXiv:2205.13452 showed that replay methods used in continual learning suffer from the stability gap, encountered when evaluating the model continually (rather than only on task boundaries). In this article, we study the effect of model ensembling as a way to improve performance and stability in online continual learning. We notice that naively ensembling models coming from a variety of training tasks increases the performance in online continual learning considerably. Starting from this observation, and drawing inspirations from semi-supervised learning ensembling methods, we use a lightweight temporal ensemble that computes the exponential moving average of the weights (EMA) at test time, and show that it can drastically increase the performance and stability when used in combination with several methods from the literature.

In recommender system or crowdsourcing applications of online learning, a human's preferences or abilities are often a function of the algorithm's recent actions. Motivated by this, a significant line of work has formalized settings where an action's loss is a function of the number of times that action was recently played in the prior m timesteps, where m corresponds to a bound on human memory capacity. To more faithfully capture decay of human memory with time, we introduce the Weighted Tallying Bandit (WTB), which generalizes this setting by requiring that an action's loss is a function of a weighted summation of the number of times that arm was played in the last m timesteps. This WTB setting is intractable without further assumption. So we study it under Repeated Exposure Optimality (REO), a condition motivated by the literature on human physiology, which requires the existence of an action that when repetitively played will eventually yield smaller loss than any other sequence of actions. We study the minimization of the complete policy regret (CPR), which is the strongest notion of regret, in WTB under REO. Since m is typically unknown, we assume we only have access to an upper bound M on m. We show that for problems with K actions and horizon T, a simple modification of the successive elimination algorithm has O left( KT + (m+M)K right) CPR. Interestingly, upto an additive (in lieu of mutliplicative) factor in (m+M)K, this recovers the classical guarantee for the simpler stochastic multi-armed bandit with traditional regret. We additionally show that in our setting, any algorithm will suffer additive CPR of Omega left( mK + M right), demonstrating our result is nearly optimal. Our algorithm is computationally efficient, and we experimentally demonstrate its practicality and superiority over natural baselines.

Large language models are strong sequence predictors, yet standard inference relies on immutable context histories. After making an error at generation step t, the model lacks an updatable memory mechanism that improves predictions for step t+1. We propose LLM-as-RNN, an inference-only framework that turns a frozen LLM into a recurrent predictor by representing its hidden state as natural-language memory. This state, implemented as a structured system-prompt summary, is updated at each timestep via feedback-driven text rewrites, enabling learning without parameter updates. Under a fixed token budget, LLM-as-RNN corrects errors and retains task-relevant patterns, effectively performing online learning through language. We evaluate the method on three sequential benchmarks in healthcare, meteorology, and finance across Llama, Gemma, and GPT model families. LLM-as-RNN significantly outperforms zero-shot, full-history, and MemPrompt baselines, improving predictive accuracy by 6.5% on average, while producing interpretable, human-readable learning traces absent in standard context accumulation.

While reasoning over long context is crucial for various real-world applications, it remains challenging for large language models (LLMs) as they suffer from performance degradation as the context length grows. Recent work MemAgent has tried to tackle this by processing context chunk-by-chunk in an RNN-like loop and updating a textual memory for final answering. However, this naive recurrent memory update faces two crucial drawbacks: (i) memory can quickly explode because it can update indiscriminately, even on evidence-free chunks; and (ii) the loop lacks an exit mechanism, leading to unnecessary computation after even sufficient evidence is collected. To address these issues, we propose GRU-Mem, which incorporates two text-controlled gates for more stable and efficient long-context reasoning. Specifically, in GRU-Mem, the memory only updates when the update gate is open and the recurrent loop will exit immediately once the exit gate is open. To endow the model with such capabilities, we introduce two reward signals r^{update} and r^{exit} within end-to-end RL, rewarding the correct updating and exiting behaviors respectively. Experiments on various long-context reasoning tasks demonstrate the effectiveness and efficiency of GRU-Mem, which generally outperforms the vanilla MemAgent with up to 400\% times inference speed acceleration.

Multi-turn problem solving is critical yet challenging for Large Reasoning Models (LRMs) to reflect on their reasoning and revise from feedback. Existing Reinforcement Learning (RL) methods train large reasoning models on a single-turn paradigm with verifiable rewards. However, we observe that models trained with existing RL paradigms often lose their ability to solve problems across multiple turns and struggle to revise answers based on contextual feedback, leading to repetitive responses. We ask: can LRMs learn to reflect their answers in a multi-turn context? In this work, we find that training models with multi-turn RL using only unary feedback (e.g., "Let's try again") after wrong answers can improve both single-turn performance and multi-turn reasoning. We introduce Unary Feedback as Observation (UFO) for reinforcement learning, which uses minimal yet common unary user feedback during iterative problem solving. It can be easily applied to existing single-turn RL training setups. Experimental results show that RL training with UFO keeps single-turn performance and improves multi-turn reasoning accuracy by up to 14%, enabling language models to better react to feedback in multi-turn problem solving. To further minimize the number of turns needed for a correct answer while encouraging diverse reasoning when mistakes occur, we design reward structures that guide models to produce careful and deliberate answers in each turn. Code: https://github.com/lichengliu03/unary-feedback

Dynamical systems (DS) theory is fundamental for many areas of science and engineering. It can provide deep insights into the behavior of systems evolving in time, as typically described by differential or recursive equations. A common approach to facilitate mathematical tractability and interpretability of DS models involves decomposing nonlinear DS into multiple linear DS separated by switching manifolds, i.e. piecewise linear (PWL) systems. PWL models are popular in engineering and a frequent choice in mathematics for analyzing the topological properties of DS. However, hand-crafting such models is tedious and only possible for very low-dimensional scenarios, while inferring them from data usually gives rise to unnecessarily complex representations with very many linear subregions. Here we introduce Almost-Linear Recurrent Neural Networks (AL-RNNs) which automatically and robustly produce most parsimonious PWL representations of DS from time series data, using as few PWL nonlinearities as possible. AL-RNNs can be efficiently trained with any SOTA algorithm for dynamical systems reconstruction (DSR), and naturally give rise to a symbolic encoding of the underlying DS that provably preserves important topological properties. We show that for the Lorenz and R\"ossler systems, AL-RNNs discover, in a purely data-driven way, the known topologically minimal PWL representations of the corresponding chaotic attractors. We further illustrate on two challenging empirical datasets that interpretable symbolic encodings of the dynamics can be achieved, tremendously facilitating mathematical and computational analysis of the underlying systems.

Chaotic dynamical systems (DS) are ubiquitous in nature and society. Often we are interested in reconstructing such systems from observed time series for prediction or mechanistic insight, where by reconstruction we mean learning geometrical and invariant temporal properties of the system in question (like attractors). However, training reconstruction algorithms like recurrent neural networks (RNNs) on such systems by gradient-descent based techniques faces severe challenges. This is mainly due to exploding gradients caused by the exponential divergence of trajectories in chaotic systems. Moreover, for (scientific) interpretability we wish to have as low dimensional reconstructions as possible, preferably in a model which is mathematically tractable. Here we report that a surprisingly simple modification of teacher forcing leads to provably strictly all-time bounded gradients in training on chaotic systems, and, when paired with a simple architectural rearrangement of a tractable RNN design, piecewise-linear RNNs (PLRNNs), allows for faithful reconstruction in spaces of at most the dimensionality of the observed system. We show on several DS that with these amendments we can reconstruct DS better than current SOTA algorithms, in much lower dimensions. Performance differences were particularly compelling on real world data with which most other methods severely struggled. This work thus led to a simple yet powerful DS reconstruction algorithm which is highly interpretable at the same time.

Current Vision-Language-Action (VLA) models rely on fixed computational depth, expending the same amount of compute on simple adjustments and complex multi-step manipulation. While Chain-of-Thought (CoT) prompting enables variable computation, it scales memory linearly and is ill-suited for continuous action spaces. We introduce Recurrent-Depth VLA (RD-VLA), an architecture that achieves computational adaptivity via latent iterative refinement rather than explicit token generation. RD-VLA employs a recurrent, weight-tied action head that supports arbitrary inference depth with a constant memory footprint. The model is trained using truncated backpropagation through time (TBPTT) to efficiently supervise the refinement process. At inference, RD-VLA dynamically allocates compute using an adaptive stopping criterion based on latent convergence. Experiments on challenging manipulation tasks show that recurrent depth is critical: tasks that fail entirely (0 percent success) with single-iteration inference exceed 90 percent success with four iterations, while simpler tasks saturate rapidly. RD-VLA provides a scalable path to test-time compute in robotics, replacing token-based reasoning with latent reasoning to achieve constant memory usage and up to 80x inference speedup over prior reasoning-based VLA models. Project page: https://rd-vla.github.io/

Large language models (LLMs) show the promise in supporting scientific research implementation, yet their ability to generate correct and executable code remains limited. Existing works largely adopt one-shot settings, ignoring the iterative and feedback-driven nature of realistic workflows of scientific research development. To address this gap, we present RECODE-H, a benchmark of 102 tasks from research papers and repositories that evaluates LLM agents through multi-turn interactions with LLM-simulated human feedback. It includes structured instructions,unit tests, and a five-level feedback hierarchy to reflect realistic researcher-agent collaboration. We further present ReCodeAgent, a framework that integrates feedback into iterative code generation. Experiments with leading LLMs, including GPT-5, Claude-Sonnet-4, DeepSeek-V3.1, and Gemini 2.5, show substantial performance gains with richer feedback, while also highlighting ongoing challenges in the generation of complex research code. RECODE-H establishes a foundation for developing adaptive, feedback-driven LLM agents in scientific research implementation

How to better utilize sequential information has been extensively studied in the setting of recommender systems. To this end, architectural inductive biases such as Markov-Chains, Recurrent models, Convolutional networks and many others have demonstrated reasonable success on this task. This paper proposes a new neural architecture, multi-scale Quasi-RNN for next item Recommendation (QR-Rec) task. Our model provides the best of both worlds by exploiting multi-scale convolutional features as the compositional gating functions of a recurrent cell. The model is implemented in a multi-scale fashion, i.e., convolutional filters of various widths are implemented to capture different union-level features of input sequences which influence the compositional encoder. The key idea aims to capture the recurrent relations between different kinds of local features, which has never been studied previously in the context of recommendation. Through extensive experiments, we demonstrate that our model achieves state-of-the-art performance on 15 well-established datasets, outperforming strong competitors such as FPMC, Fossil and Caser absolutely by 0.57%-7.16% and relatively by 1.44%-17.65% in terms of MAP, Recall@10 and NDCG@10.

In open-ended environments, exploration is fundamental for autonomous agents, yet current language model agents struggle with this. Effective exploration requires memory, but retaining raw interaction histories is computationally expensive over long trajectories. While latent memory offers a solution to compress interaction histories, its training lacks reliable supervisory signals. We introduce Joint Agent Memory and Exploration Learning (JAMEL), a framework that trains agentic memory and exploration policy together through novelty-driven interaction. We observe that memory and exploration form a mutually dependent loop: sustained exploration requires memory to distinguish exhausted behaviors from unseen ones, while novelty-seeking interaction provides the supervision needed to make memory useful for future exploration. By utilizing deterministic and persistent novelty signals such as code coverage in the GUI domain, we provide natural, annotation-free supervision for the memory module. Empirical evaluations demonstrate that \ours successfully generalizes to unseen environments. Its exploration capability outperforms open-weight baselines and rivals the exploration depth of a closed-source model while reducing token consumption. Our code and model are open-sourced at https://github.com/MobileLLM/JAMEL.

Generative recommendation (GenRec) models typically model user behavior via full attention, but scaling to lifelong sequences is hindered by prohibitive computational costs and noise accumulation from stochastic interactions. To address these challenges, we introduce Rec2PM, a framework that compresses long user interaction histories into compact Preference Memory tokens. Unlike traditional recurrent methods that suffer from serial training, Rec2PM employs a novel self-referential teacher-forcing strategy: it leverages a global view of the history to generate reference memories, which serve as supervision targets for parallelized recurrent updates. This allows for fully parallel training while maintaining the capability for iterative updates during inference. Additionally, by representing memory as token embeddings rather than extensive KV caches, Rec2PM achieves extreme storage efficiency. Experiments on large-scale benchmarks show that Rec2PM significantly reduces inference latency and memory footprint while achieving superior accuracy compared to full-sequence models. Analysis reveals that the Preference Memory functions as a denoising Information Bottleneck, effectively filtering interaction noise to capture robust long-term interests.

A recurrent neural network (RNN) is a widely used deep-learning network for dealing with sequential data. Imitating a dynamical system, an infinite-width RNN can approximate any open dynamical system in a compact domain. In general, deep networks with bounded widths are more effective than wide networks in practice; however, the universal approximation theorem for deep narrow structures has yet to be extensively studied. In this study, we prove the universality of deep narrow RNNs and show that the upper bound of the minimum width for universality can be independent of the length of the data. Specifically, we show that a deep RNN with ReLU activation can approximate any continuous function or L^p function with the widths d_x+d_y+2 and max{d_x+1,d_y}, respectively, where the target function maps a finite sequence of vectors in R^{d_x} to a finite sequence of vectors in R^{d_y}. We also compute the additional width required if the activation function is tanh or more. In addition, we prove the universality of other recurrent networks, such as bidirectional RNNs. Bridging a multi-layer perceptron and an RNN, our theory and proof technique can be an initial step toward further research on deep RNNs.

In Recurrent Neural Networks (RNNs), encoding information in a suboptimal or erroneous way can impact the quality of representations based on later elements in the sequence and subsequently lead to wrong predictions and a worse model performance. In humans, challenging cases like garden path sentences (an instance of this being the infamous "The horse raced past the barn fell") can lead their language understanding astray. However, they are still able to correct their representation accordingly and recover when new information is encountered. Inspired by this, I propose an augmentation to standard RNNs in form of a gradient-based correction mechanism: This way I hope to enable such models to dynamically adapt their inner representation of a sentence, adding a way to correct deviations as soon as they occur. This could therefore lead to more robust models using more flexible representations, even during inference time. I conduct different experiments in the context of language modeling, where the impact of using such a mechanism is examined in detail. To this end, I look at modifications based on different kinds of time-dependent error signals and how they influence the model performance. Furthermore, this work contains a study of the model's confidence in its predictions during training and for challenging test samples and the effect of the manipulation thereof. Lastly, I also study the difference in behavior of these novel models compared to a standard LSTM baseline and investigate error cases in detail to identify points of future research. I show that while the proposed approach comes with promising theoretical guarantees and an appealing intuition, it is only able to produce minor improvements over the baseline due to challenges in its practical application and the efficacy of the tested model variants.