Daily Papers

Every recurrence of a looped language model adds cost in training, decoding, prefill, and reinforcement learning (RL). The closer recurrent states get to fixed points, the less the path to them matters. This enables truncated backpropagation in training; terminal key-value (KV) sharing for decoding with almost no loss in accuracy; a distilled student that prefills up to 1.79x faster; and RL updates that compute gradients from saved rollout states, 2x faster than backpropagating through the replayed trajectory. We therefore improve the two components of training that shape these fixed points: the depth prior and input injection. Fixed-depth training breaks KV sharing, and Huginn's broad depth prior supports sharing but dilutes supervision at the target depth more than sharing requires; we learn the prior from prediction feedback, with an entropy term that keeps it broad. Existing injection schemes let the state's component along the input amplify or cancel the injection; we remove this component with orthogonal injection. From 100M to 1.6B parameters, the learned prior and orthogonal injection lower perplexity at every scale relative to Huginn's prior and existing injection schemes, respectively. At 1.6B, the learned prior with a 3x smaller KV cache matches the downstream average of fixed-depth training with the full cache.

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.

Looped language models reuse physical Transformer blocks across logical depth, but established architectures entangle recurrence placement, input injection, and recurrent state organization. We isolate these axes in models trained from scratch at matched parameter scale, logical depth, and token budgets, and then test whether the resulting design principles transfer to resident- and active-parameter-matched Mixture-of-Experts (MoE) models. The prelude-recurrent block-coda sandwich architecture consistently improves performance on context extraction and reasoning-intensive tasks. Input injection primarily enhances knowledge retrieval and context utilization, whereas random initialization and separate H/L recurrent states provide mixed or negative returns. Taken together, these results favor the Huginn-style design: at the 730M-parameter and 336B-token scale, the complete Huginn architecture outperforms Ouro across all ten benchmarks. The advantages of Huginn over Ouro transfer to MoE models. At the 8BA0.8B (32BA3B when unrolled) scale trained on 500B tokens, Huginn outperforms Ouro on nearly all benchmarks spanning knowledge, reasoning, and coding, while also exhibiting more balanced expert routing. Under matched training and inference FLOPs, the Huginn-style model uses 75% fewer resident parameters than a 112-layer feedforward MoE baseline. Despite its smaller physical parameters, it outperforms the feedforward baseline on DROP and GSM8K, matches it on MATH500, and remains competitive on the other benchmarks.

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.

Looped language models apply the same stack of layers T times to each token, which deepens the model without adding parameters but multiplies its key-value (KV) cache by T. The larger cache limits how many sequences a GPU can decode at once and slows each decoding step, which reads the whole cache. We propose Hybrid Latent Attention (HLA), which keeps exact keys and values within a sliding window of W recent tokens and stores each older token as a compact latent that the query of each loop reads directly, without reconstructing keys and values. We uptrain HLA on Ouro looped models (T=4) with 1.4B and 2.6B parameters, keeping the pretrained weights frozen and training only the added parameters to reproduce the original attention. The cache shrinks by 10.7x per token, fitting 4.0-8.8x as many concurrent sequences per GPU, and decoding throughput improves by 2.5x at 1K-token contexts and by up to 7.4x at 16K. HLA retains over 97% of the original accuracy on math, knowledge and reasoning benchmarks, and 96-100% on long-context retrieval up to 16K tokens. After supervised fine-tuning, it performs on par with the fine-tuned original model on competition-level math.

Traditional fixed-depth architectures scale quality by increasing training FLOPs, typically through increased parameterization, at the expense of a higher memory footprint, or data. A potential alternative is looped architectures, which instead increase FLOPs by sending activations through a block of layers in a loop. While promising, existing recipes for training looped architectures can be unstable, suffering from residual explosion and loss spikes. We address these challenges by recasting looping as a nonlinear time-variant dynamical system over the residual stream. Via a linear approximation to this system, we find that instability occurs in existing looped architectures as a result of large spectral norms in their injection parameters. To address these instability issues, we propose Parcae, a novel stable, looped architecture that constrains the spectral norm of the injection parameters via discretization of a negative diagonal parameterization. As a result, Parcae achieves up to 6.3% lower validation perplexity over prior large-scale looped models. Using our stable looped architecture, we investigate the scaling properties of looping as a medium to improve quality by increasing FLOPs in training and test-time. For training, we derive predictable power laws to scale FLOPs while keeping parameter count fixed. Our initial scaling laws suggest that looping and data should be increased in tandem, given a fixed FLOP budget. At test-time, we find that Parcae can use looping to scale compute, following a predictable, saturating exponential decay. When scaled up to 1.3B parameters, we find that Parcae improves CORE and Core-Extended quality by 2.99 and 1.18 points when compared to strong Transformer baselines under a fixed parameter and data budget, achieving a relative quality of up to 87.5% a Transformer twice the size.

Looped language models repeat a set of transformer layers through depth, reducing memory costs and providing natural early-exit points at loop boundaries. However, looped models do not scale as favorably as standard transformers with unique layers. We compare standard and Mixture-of-Experts (MoE) transformers, with and without looping, and find two main results. First, we find Looped-MoE models scale better than the standard baseline while dense looped models do not. We trace this to routing divergence between loops: in Looped-MoE models, different experts are activated on each pass through the same shared layers, recovering expressivity without additional parameters. Our second finding is that looped models have better compute-quality trade-offs with early exits than standard models. Because each loop ends with the same layers that produce the final output, loop boundaries are superior exit points, as confirmed by earlier output convergence at these points. In sum, we provide a clear direction for scaling looped models: a Looped-MoE model with early exits can not only beat standard transformers at scale, but also enable significant memory and inference savings with minimal degradation in quality.

Decision models answer typed questions about an input with one probability per option, computed in a single forward pass without generating text. Like the fast System 1 of dual-process theory, they answer at once, but their depth is fixed at training time, so they spend the same computation on easy and hard questions. Models that reason in generated text can think longer, but they pay for every extra step in decoded tokens. We present LEV, a decision model that thinks deeper by rerunning its layers in a loop instead of generating tokens. LEV builds on a language model pretrained as a loop, reads the options after every loop step and trains all steps jointly, so one checkpoint can answer at any depth. On three looped backbones, accuracy rises with depth on the same weights, mostly on hard questions, and a 2.6B LEV goes from 63.2% at depth one to 76.2% at depth four on JevBench. On-demand depth, set per question by a small stopper with a threshold fitted on unlabelled requests, saves 15% to 24% of the computation at a cost of under one accuracy point, level with or slightly above a fixed mix of depths. On a locked test set, the 1.4B and 3B models are 5.4 to 8.8 points more accurate than dense (non-looped) models and cascades of the same latency below 50 ms.

Large language models have shown remarkable reasoning abilities and scaling laws suggest that large parameter count, especially along the depth axis, is the primary driver. In this work, we make a stronger claim -- many reasoning problems require a large depth but not necessarily many parameters. This unlocks a novel application of looped models for reasoning. Firstly, we show that for many synthetic reasoning problems like addition, p-hop induction, and math problems, a k-layer transformer looped L times nearly matches the performance of a kL-layer non-looped model, and is significantly better than a k-layer model. This is further corroborated by theoretical results showing that many such reasoning problems can be solved via iterative algorithms, and thus, can be solved effectively using looped models with nearly optimal depth. Perhaps surprisingly, these benefits also translate to practical settings of language modeling -- on many downstream reasoning tasks, a language model with k-layers looped L times can be competitive to, if not better than, a kL-layer language model. In fact, our empirical analysis reveals an intriguing phenomenon: looped and non-looped models exhibit scaling behavior that depends on their effective depth, akin to the inference-time scaling of chain-of-thought (CoT) reasoning. We further elucidate the connection to CoT reasoning by proving that looped models implicitly generate latent thoughts and can simulate T steps of CoT with T loops. Inspired by these findings, we also present an interesting dichotomy between reasoning and memorization, and design a looping-based regularization that is effective on both fronts.

Looped transformers have demonstrated promising parameter efficiency by reusing layers for latent computation. Prior studies compare looped and non-looped models at matched parameters or per-token FLOPs. However, to the best of our knowledge, whether looping improves test-time scaling as outputs grow longer remains underexplored. Through post-training looped transformers, we study the accuracy-compute slope, measured as the accuracy gain per doubling of test-time decoding FLOPs. We find that existing looped transformers often yield steeper slopes than their non-looped baseline, yet underperform it at matched compute. While fixed-depth looping spends extra iterations on every token, our analysis shows that many tokens do not benefit from extra iterations. We therefore propose TaH2, which enables the model to focus extra iterations on the tokens that benefit from looping. It jointly post-trains the backbone and an iteration decider through lookahead depth supervision, which uses online labels indicating whether further iteration improves the prediction. TaH2 improves both the efficiency and attainable accuracy of test-time scaling. On challenging AIME benchmarks, TaH2 improves the accuracy-compute slope by 53% (2.74 vs. 1.79) over the non-looped baseline, exceeding the baseline's peak accuracy by about 3.4 points at matched test-time compute. As the maximum iteration depth increases, existing looped models largely plateau, while TaH2's gain over the non-looped baseline continues to grow from +2.8 points at depth 2 to +3.9 points at depth 8. Our code is available at https://github.com/thu-nics/TaH.

Looped transformers and Mixture-of-Experts (MoE) offer complementary routes to efficient scaling: recurrence increases computational depth at fixed parameters, while MoE sparsity expands total capacity at fixed active compute. Yet existing scaling laws model recurrence or sparsity in isolation. In this work, we introduce Loop Scaling Laws, the first scaling law to jointly model recurrence and sparsity alongside model size and data. At its core is a bounded, sparsity-conditional recurrence mapping that characterizes the effective-parameter gain from looping and how sparsity raises this gain. The laws predict the held-out loss of looped models more accurately than prior alternatives, and recover the standard dense and MoE scaling laws as special cases. Beyond prediction, the fitted laws provide a principled foundation for designing looped MoE models under compute and memory constraints. Downstream evaluations further demonstrate the complementary benefits of the two axes: sparsity delivers ~3x active-parameter efficiency, recurrence yields ~2x total-parameter efficiency on reasoning, and joint scaling further advances the performance frontier. As a practical extension, we show these gains hold at trillion-token scale: at matched training compute, a looped MoE with law-derived recurrence matches a ~2x larger non-looped MoE on the reasoning benchmarks, while enabling test-time scaling through recurrence.

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.

Looped transformers are a parameter-efficient alternative to standard transformers for language modeling. By reapplying the same layers across multiple loop steps, recent looped language models match the performance of non-looped models several times their size. While this architecture effectively reduces parameter memory, each loop step still stores its own keys and values, so the key-value (KV) cache grows linearly with the number of loop steps and becomes the main memory bottleneck. We observe that the KV cache of looped models is highly redundant along the loop axis and that the loop steps differ substantially in their sensitivity to quantization. Based on these observations, we propose LoopVQ, a KV cache vector quantization method for looped transformers that codes channels of multiple loop steps jointly, and fine-tunes the codebooks to match the output of the full-precision model. At 0.5 bits per element, LoopVQ outperforms existing KV cache vector quantization methods for standard LLMs on general, long chain-of-thought reasoning, and long-context retrieval tasks with Ouro and Huginn models. Its design is orthogonal to these methods, and combined with TaSQ, it retains much of the BF16 accuracy even at 0.25 bits per element. We also implement LoopVQ in SGLang, where it nearly halves the decode latency at a 32K-token context and raises the peak serving throughput by up to 4.3times and 6.1times on the 1.4B and 2.6B Ouro models compared to the BF16 baseline.

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.

Looped Transformers offer a promising alternative to purely feed-forward computation by iteratively refining latent representations, improving language modeling and reasoning. Yet recurrent architectures remain unstable to train, costly to optimize and deploy, and constrained to small, fixed recurrence depths. We introduce Attractor Models, in which a backbone module first proposes output embeddings, then an attractor module refines them by solving for the fixed point, with gradients obtained through implicit differentiation. Thus, training memory remains constant in effective depth, and iterations are chosen adaptively by convergence. Empirically, Attractor Models outperform existing models across two regimes, large-scale language-model pretraining and reasoning with tiny models. In language modeling, Attractor Models deliver a Pareto improvement over standard Transformers and stable looped models across sizes, improving perplexity by up to 46.6% and downstream accuracy by up to 19.7% while reducing training cost. Notably, a 770M Attractor Model outperforms a 1.3B Transformer trained on twice as many tokens. On challenging reasoning tasks, we show that our model with only 27M parameters and approximately 1000 examples achieves 91.4% accuracy on Sudoku-Extreme and 93.1% on Maze-Hard, scaling favorably where frontier models like Claude and GPT o3, fail completely, and specialized recursive reasoners collapse at larger sizes. Lastly, we show that Attractor Models exhibit a novel phenomenon, which we call equilibrium internalization: fixed-point training places the model's initial output embedding near equilibrium, allowing the solver to be removed at inference time with little degradation. Together, these results suggest that Attractor Models make iterative refinement scalable by turning recurrence into a computation the model can learn to internalize.

Scaling model performance typically requires increasing model size. Looped Transformer offers a compelling alternative by iteratively reusing the same Transformer blocks, trading additional computation for improved performance without increasing parameter count or context length. Because the number of loop iterations can be adjusted at inference, it also provides a natural mechanism for balancing performance and test-time compute. However, Looped Transformer still suffers from training instability when the number of loop iterations increases. Our analysis reveals that this instability stems from two sources: gradient oscillation and residual explosion. To address these two problems, we propose the Fully Looped Transformer, which introduces two parameter-free modifications: (1) Fully Looped Architecture, which distributes inter-loop signals across all layers to mitigate residual explosion; (2) Attention Injection, which reuses the existing attention block to suppress gradient oscillation. These modifications stabilize training dynamics, enabling the Fully Looped Transformer to be trained stably up to 12 loop iterations, whereas other baseline looped models collapse in this regime. In milder settings where Looped Transformer does not collapse, Fully Looped Transformer still improves average downstream-task performance by up to 13.2\%. Overall, our experiments demonstrate that Fully Looped Transformer improves training stability, enhances downstream performance, and provides preliminary adaptability under different test-time compute budgets by varying loop iterations at inference.

Looped Transformers apply the same layers several times per token, adding compute to improve quality without more parameters. Each recursion, however, writes its own key-value cache, so memory still grows with compute. Inference-time techniques can shrink this cache at a cost in quality. We pretrain looped language models to share memory: only the first recursion writes a cache, and later recursions read it while keeping a short window of their own. Surprisingly, we find that sharing memory does not cost quality and instead improves it. At 150M-1B parameters, our Looped Prediction Transformer (LPT) and its hybrid variant set a new quality-memory frontier for looped models: with five recursions, the hybrid lowers validation perplexity on FineWeb-Edu by 1.12-1.82 relative to a same-size standard Transformer while using 76-79% less context memory. Through an extensive analysis, we investigate why memory sharing helps. Shared and local memory develop different representations, and later recursions attend mostly to the shared memory, which also acts as a gradient highway to the first recursion.

Looped transformers increase effective depth by repeatedly applying a shared set of transformer layers, offering a parameter-efficient way to scale computation. However, standard residual connections aggregate information across recurrent depth into a single stream, providing limited control over which intermediate representations from preceding layers should be reused. In this paper, we introduce ARLoop, a looped transformer architecture based on attention residuals, where the input of each layer is selectively aggregated through attention over model depth. Distinct from standard transformers, we find that looped transformers do not require access to the full residual history: sliding-window depth attention matches or outperforms full-history attention at larger recurrent depths. Across controlled experiments with multiple parameter scales (140M and 370M) and recurrent depths (R = 1 to 5), ARLoop consistently outperforms prior residual designs for looped transformers. More importantly, we further scale ARLoop to 0.6B and 1.2B parameters under a realistic LLM pretraining recipe, demonstrating that the gains transfer to realistic pretraining, with particularly strong improvements on math and coding tasks. Our 1.2B model, pretrained on only 0.5T tokens with three recurrences, outperforms Huginn-3.5B (the only fully-open looped LLM as of our release date) across all evaluated tasks and is competitive with similar-sized non-looped models such as Qwen3.5-2B despite being trained on substantially less data.

Looping is a powerful inductive bias for reasoning. By repeatedly iterating a recurrent block, looped models can scale their effective depth and enable multi-step reasoning in latent space. However, existing looped models are not inherently guided by an explicit measure of progress, making it difficult to determine when to stop looping and how to improve upon the current state. To enable this, we introduce Energy-Based Reasoners (EBRs), which learn an energy function over latent reasoning states and formulate looped updates as gradient descent on this energy. The energy scores latent states, while its gradient guides refinement and provides a signal for stopping. To train EBRs effectively, we further introduce exploration-based training to discover additional modes and to better shape the energy landscape over a broader set of states. Across four algorithmic reasoning benchmarks, EBRs match or outperform prior recurrent reasoning models, achieving 99.9% accuracy on Sudoku-Extreme while using 345times less inference compute than the next-best model and maintaining comparable training compute. On all multi-solution benchmarks, EBRs achieve state-of-the-art coverage of distinct valid solutions while maintaining near-100% single-sample accuracy. These results suggest that energy-based models are a promising approach for strong reasoning performance.

Chain-of-thought (CoT) prompting enables reasoning in language models but requires explicit verbalization of intermediate steps. Looped transformers offer an alternative by iteratively refining representations within hidden states. This parameter efficiency comes at a cost, as looped models lack the storage capacity of deeper models which use unique weights per layer. In this work, we investigate transformer models that feature both adaptive per-layer looping, where each transformer block learns to iterate its hidden state via a learned halting mechanism, and gated memory banks, that provide additional learned storage. We find that looping primarily benefits mathematical reasoning, while memory banks help recover performance on commonsense tasks compared to parameter and FLOP matched models. Combining both mechanisms yields a model that outperforms an iso-FLOP baseline, with three times the number of layers, across math benchmarks. Analysis of model internals reveals layer specialization: early layers learn to loop minimally and access memory sparingly, while later layers do both more heavily.

Looped Language Models (LoopLMs) perform multi-step latent reasoning prior to token generation and outperform conventional LLMs on reasoning benchmarks at smaller parameter budgets. However, attempts to further improve LoopLM reasoning with reinforcement learning have failed - standard objectives such as Group Relative Policy Optimization (GRPO) only assign credit to the final latent state, creating a fundamental mismatch with the model's internal computation. To resolve this, we introduce RLTT (Reward Latent Thought Trajectories), a reinforcement learning framework which distributes reward across the full latent reasoning trajectory. RLTT provides dense, trajectory-level credit assignment without relying on external verifiers and can directly replace GRPO with negligible overhead. Across extensive experiments with Ouro-2.6B-Thinking under identical training and inference conditions, RLTT yields substantial improvements over GRPO on challenging mathematical reasoning benchmarks, improving accuracy by +14.4% on MATH-500, +16.6% on AIME24, and +10.0% on BeyondAIME. Despite being trained exclusively on mathematics, RLTT also transfers effectively to non-mathematical reasoning benchmarks, demonstrating the effectiveness of trajectory-level credit assignment for reinforcement learning in LoopLMs.

Looped language models (LoopLMs) perform iterative latent computation to refine internal representations, offering a promising alternative to explicit chain-of-thought (CoT) reasoning. However, existing reinforcement learning (RL) paradigms primarily target output tokens, creating a structural mismatch with looped architectures whose reasoning unfolds implicitly. In this work, we propose LoopRPT, a reinforcement pre-training framework tailored for LoopLMs. By reframing next-token prediction as a next-token reasoning task, LoopRPT assigns reinforcement signals directly to latent steps using an EMA teacher reference and noisy latent rollouts. This formulation enables RL to directly shape intermediate representations, compressing effective reasoning into fewer iterations. We instantiate LoopRPT on the Ouro architecture across multiple model scales. Results demonstrate that LoopRPT consistently improves per-step representation quality, achieving Pareto dominance in accuracy-computation trade-offs. Notably, significant gains on hard tokens indicate that LoopRPT enhances early-stage reasoning rather than merely encouraging premature exits. Our findings highlight reinforcement pre-training as a principled paradigm for learning efficient latent reasoning in LoopLMs.

Current world models face a fundamental tension: faithful long-horizon simulation demands deep computation, but deeper models are expensive to deploy and prone to compounding errors. We resolve this by introducing Looped World Models (LoopWM), which are the first looped architectures for world modelling. Our method iteratively refines latent environment states through a parameter-shared transformer block. This yield up to 100x parameter efficiency over conventional approaches with adaptive computation that automatically scales depth to match the complexity of each prediction step. Orthogonal to scaling model size and training data, LoopWM establishes iterative latent depth as a new scaling axis for world simulation, which might significantly push the community forward.

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.

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.

Looped language models repeatedly apply a weight-shared block to increase effective depth without increasing parameter count, but the resulting T sequential recurrent-block calls per generated token substantially increase decoding latency. To address the issue, we introduce Wavefront Decoding (WFD), a training-free self-speculative decoding framework designed for looped language models. WFD exploits two properties of these architectures: intermediate recurrence outputs provide effective draft predictions, and weight sharing allows token states at different positions and recurrence depths to be processed in one batched recurrent-block call. WFD organizes these mixed-depth states into a diagonal wavefront, continuously drafting new positions at shallow depth while advancing earlier positions toward full-depth verification. Unlike the phase-separated draft-then-verify schedule, WFD therefore concurrently batches drafting and verification within the same recurrent calls, while rejected drafts are corrected using full-depth predictions. Across six Spec-Bench task categories, WFD achieves 2.42x speedup on Ouro-2.6B and 3.54x on Huginn-3.5B over autoregressive decoding, consistently outperforming draft-then-verify. Cross-recurrence KV sharing further reduces wavefront KV traffic and increases WFD's speedup to 4.81x on Huginn-3.5B. The code is available at https://github.com/summerbro-hhj/wavefront-decoding.

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.

Looped language models separate computational depth from parameter count by repeatedly applying the same transformer block. Adapting these models requires a shared update that remains effective as hidden states evolve throughout the recurrent computation. Our empirical analysis reveals a pronounced late-loop bias in standard low-rank adaptation (LoRA): the shared update is more effective at later loop positions. This imbalance motivates training shared updates under varying combinations of their applications. Randomly omitting adapter applications alone, however, does not improve task performance; it reduces expected update strength during training while leaving inference unchanged. We introduce Loop Dropout, which couples stochastic masking of adapter applications with inverse-survival rescaling to preserve expected update strength and promote effective adaptation across loops. Extensive experiments demonstrate improved mathematical reasoning across model sizes, adapter ranks and training recipes, with benefits extending to general instruction tuning and code generation. Loop Dropout outperforms existing LoRA variants and adapter regularizers, while further analysis shows stronger early-loop adaptation. Every backbone loop remains active, and inference applies the adapter at all loops using standard LoRA without additional trainable parameters or inference computation.

Looped Language Models (LoopLMs) perform "latent reasoning" by recursively refining internal latent representations with shared weights, offering a more effective alternative to explicit verbal reasoning. Despite their effectiveness, we find that LoopLMs remain prone to loop instability: unstable refinement across iterations can produce localized uncertain "hard" tokens associated with reasoning errors. To address this, we propose LoopCD, loop-wise contrastive decoding that enhances the reasoning performance of LoopLMs by intervening on these tokens at inference time. Specifically, we exploit the internal dynamics of LoopLMs and contrast the logits from earlier iterations with logits from the last refined iteration to form the final sampling distribution. We find that this strategy is highly efficient, introducing only negligible inference overhead and requiring no additional training, while effectively improving reasoning performance by naturally refining reasoning-critical hard tokens. Extensive experiments show that our method improves the performance of recent representative LoopLMs across various reasoning tasks.

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.

Looped Language Models (LoopLMs) provide a parameter efficient approach to scaling model capabilities through repeated use of shared parameters across recurrent steps. Since each recurrent depth can be read out independently, a single LoopLM exposes a broader output space across inference depths, raising an important question: whether safety is preserved throughout recurrent computation. Prior evaluations suggest that deeper recurrence can improve safety on harmful queries, but robustness under jailbreak attacks remains unclear. We therefore conduct a comprehensive safety evaluation of LoopLMs under jailbreak attacks targeting different recurrent depths. We find that attack success can increase at deeper inference depths, the same query can elicit different safety behaviors across depths, and attacks constructed against one depth can transfer to others. Moreover, SFT and preference alignment do not eliminate these safety gaps, motivating an alignment method designed for LoopLMs. We introduce SafeBridge, which combines lightweight depth specific control of shared recurrent layers, selective state bridging, and joint safety supervision across recurrent depths. Across model scales, multiple attack methods, and safety benchmarks, SafeBridge substantially reduces attack success for both matched-depth and cross-depth attacks. It also improves general utility over the vanilla models while maintaining comparable over-refusal behavior. Our results show that the safety of a LoopLM cannot be inferred from a single recurrent depth, motivating safety alignment across recurrent computation. Our code and model checkpoints will be released upon acceptance.

Looped language models increase effective depth by repeatedly applying a shared block of layers, but existing large-scale recipes require multi-stage training over trillions of tokens, while the benefits of recurrence remain difficult to separate from differences in data and training. In this work, we establish practical training recipes for looped language models, with three main results. (1) We develop a compute-efficient from-scratch pipeline that reduces the training budget from 7.7T tokens in Ouro to 310B tokens while retaining strong reasoning performance. Pretraining followed by high-quality mid-training, together with learning-rate warmup and stronger exit-gate regularization, enables stable recurrent training without prior multi-stage schedules. (2) Under controlled comparisons, our 1.4B LoopLM outperforms a parameter-matched dense model trained on the same data and token budget on all 12 evaluated benchmarks, including +14 points on GSM8K, +10 on MATH, and +22 on DROP. At matched inference compute, it approaches a 3.9B dense model on mathematical reasoning and reading comprehension while using only 36\% as many parameters. (3) We introduce a minimal recipe for converting pretrained dense models into looped ones: a single learned input-mixing scalar and a smoothed exit loss, with no step-specific parameters. Applied to Qwen3-1.7B-Base, Looped Qwen improves over an identically continued dense baseline on every evaluated benchmark across two data regimes, with statistically clear gains on GSM8K, MATH, and MMLU-Pro on the curated mixture. Together, these results make looped language models substantially cheaper to train from scratch and practical to introduce into existing pretrained checkpoints, while isolating the gains due to recurrence itself.

Looped Mixture-of-Experts (MoE) models combine parameter reuse with sparse computation to improve performance under matched budgets. Prior work demonstrates that different loop visits can play different roles even when they share parameters. To probe the contributions of different loop visits, we learn separate residual coefficients in a model that applies a shared block twice, observing a smaller scale in the first visit. Fixed coefficient experiments show that reducing the first visit's scale preserves or improves performance, whereas the same reductions to the second hurt it. This asymmetry motivates us to explore asymmetric compute allocation. We propose LatentLoop, which compresses the first visit's sequence, retains full resolution in the second, and reallocates saved computation to increase the hidden size. We further learn how strongly the first visit updates the full sequence, and share MoE weights across visits while using separate attention parameters to accommodate their different sequence resolutions. Across 15 configurations spanning 1.3B--7B parameters, with the largest budget exceeding 10^{21} FLOPs, LatentLoop consistently lowers validation NLL. At a fitted NLL of 1.988, predicted compute savings are 19.0\% and 7.9\% relative to vanilla MoE and the looped baseline. It also improves average scores on nine benchmarks at all sizes in 0-shot and 5-shot settings. These results suggest asymmetric compute allocation is a promising direction for looped MoE models.

Recurrent LLM architectures have emerged as a promising approach for improving reasoning, as they enable multi-step computation in the embedding space without generating intermediate tokens. Models such as Ouro perform reasoning by iteratively updating internal representations while retaining a standard Key-Value (KV) cache across iterations, causing memory consumption to grow linearly with reasoning depth. Consequently, increasing the number of reasoning iterations can lead to prohibitive memory usage, limiting the practical scalability of such architectures. In this work, we propose Memory-Efficient Looped Transformer (MELT), a novel architecture that decouples reasoning depth from memory consumption. Instead of using a standard KV cache per layer and loop, MELT maintains a single KV cache per layer that is shared across reasoning loops. This cache is updated over time via a learnable gating mechanism. To enable stable and efficient training under this architecture, we propose to train MELT using chunk-wise training in a two phase procedure: interpolated transition, followed by attention-aligned distillation, both from the LoopLM starting model to MELT. Empirically, we show that MELT models fine-tuned from pretrained Ouro parameters outperform standard LLMs of comparable size, while maintaining a memory footprint comparable to those models and dramatically smaller than Ouro's. Overall, MELT achieves constant-memory iterative reasoning without sacrificing LoopLM performance, using only a lightweight post-training procedure.

Modern LLMs are trained to "think" primarily via explicit text generation, such as chain-of-thought (CoT), which defers reasoning to post-training and under-leverages pre-training data. We present and open-source Ouro, named after the recursive Ouroboros, a family of pre-trained Looped Language Models (LoopLM) that instead build reasoning into the pre-training phase through (i) iterative computation in latent space, (ii) an entropy-regularized objective for learned depth allocation, and (iii) scaling to 7.7T tokens. Ouro 1.4B and 2.6B models enjoy superior performance that match the results of up to 12B SOTA LLMs across a wide range of benchmarks. Through controlled experiments, we show this advantage stems not from increased knowledge capacity, but from superior knowledge manipulation capabilities. We also show that LoopLM yields reasoning traces more aligned with final outputs than explicit CoT. We hope our results show the potential of LoopLM as a novel scaling direction in the reasoning era. Our model could be found in: http://ouro-llm.github.io.

A main promise of looped language models is depth-adaptive inference. By looping a block of shared layers a variable number of times, the model can use less compute for "easy" tokens and more for "hard" ones. However, tokens with different numbers of loops cannot share a uniform forward pass and therefore cannot be handled by standard batching systems such as vLLM. The practical value of depth-adaptive inference thus hinges on whether batching can be made efficient. We introduce the first efficient method for depth-adaptive looped LMs via continuous depth batching (CDB), which forms new batches between loop steps. Our method dynamically schedules looped and non-looped parts of the architecture, manages looped KV-caching, and predicts which tokens will exit the loop in advance so it can prepare batches asynchronously. Experiments on Ouro 1.4B and Huginn 3.5B show that fully looped architectures are best suited to depth-adaptive inference, as large non-looped layers outside the recurrent core (e.g., token embedding, LM head, and unshared transformer blocks) slow down and complicate scheduling. Overall, CDB realizes up to 99% of the estimated maximum speedup available, leaving further gains primarily dependent on model architecture and exit behavior.

Can a language model read the quality of its ongoing computation, and can an external intervention turn that readout into better outcomes? We test both questions in a frozen 2.6B looped transformer, Ouro-RLTT. On GSM8K, a strict pre-answer probe excludes the answer region and gold value yet predicts success: hidden states plus length/log-probability features reach AUROC 0.797 versus 0.731 for those surface features alone (increment +0.066; task-clustered 95% CI [+0.021,+0.112]; 170 tasks). On Horizon Logic, a prospectively extended task-disjoint study gives an increment of +0.111 (CI [+0.056,+0.169]), independently replicated on the new cohort (+0.095) and robust to an adversarial malformed-sibling shortcut. Recurrence also moves candidate-quality readability to progressively earlier physical depth; the trend replicates across the Ouro family and qualitatively in out-of-family Huginn, although their transfer geometry differs. The readout converts into validated decision-level gains. Hidden-state-based scores improve risk-coverage over shortcut-only scores in four sealed selective-prediction arms, and terminal selection beats matched random even when every candidate is well formed (27/32 correct selections versus 64.8% expected; p = 0.0086). Generative control does not convert: directional steering is negative, a branch screen is bounded, and exact-compute loop allocation and minimal LoRA direction-binding detect no gain. These tests run through bit-exact branch/carry/prune machinery over Ouro's 192-slot recurrent cache, including a suffix-recompute splice saving up to 88% of per-branch layer passes. We call this decision-usable but not generatively controllable property operational proto-introspection. All load-bearing values use source-item-disjoint splits and antisymmetrized pairwise evaluation.

How do later loops in a pretrained looped language model repair two-hop retrieval? We study Ouro-1.4B and Ouro-2.6B by transplanting attention queries between loops at the same layer, head and final prompt position. On synthetic in-context questions, injecting the layer-indexed collection of loop-3 queries throughout loop 1 rescues 90% and 97% of questions answered incorrectly at loop 1 and correctly at loop 3, versus 24% and 21% for shuffled donors. The rescue also holds under greedy generation in Ouro-2.6B; in Ouro-1.4B it is limited to four-candidate ranking. Crossed donor-recipient swaps favor retrieval through the donor's bridge entity over copying its answer, although a substantial positional component remains in Ouro-2.6B. Late keys and values rescue far fewer targets. Freezing queries at their loop-1 values also damages two-hop accuracy more than a matched question that supplies the bridge. On natural questions, rescue is smaller and replicates only in Ouro-2.6B, and crossed swaps do not establish bridge redirection. These results identify a transferable collection of late-loop queries that supports bridge-conditioned retrieval on synthetic scenes; they do not locate where the bridge is computed or establish a single-vector handoff.

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 Transformers scale sequential computation by applying a compact stack of physical blocks for multiple rounds, increasing unrolled depth without increasing stored parameters. This reuse changes the residual-scaling problem: in an untied Transformer, each residual branch receives and applies its own parameter update, whereas in a looped Transformer one shared update aggregates gradients from repeated visits and is read back by those same visits in the next linearized forward pass. We formalize this tied-depth effect through a first-order perturbation bound controlled by a visit-alignment coefficient κ_R. The bound recovers the DeepNorm exponent when visits decorrelate, but in the conservative aligned regime it requires the exponent to increase from 1/4 to 1/2 as loop count grows at fixed physical depth. The resulting method, DeepLoop, keeps the Post-LN DeepNorm architecture and sets α=(2N)^{1/2} and β=(8N)^{-1/2} for unrolled depth N. On GPT-style looped language models at GPT-2 small and GPT-2 medium scale, DeepLoop is neutral when no physical block is revisited and improves validation loss and downstream accuracy once recurrent depth is activated. These results show that stable recurrent depth requires residual scaling rules that account for parameter visits, not only nominal layer count.

Looped Transformers have attracted substantial attention as a parameter-efficient approach to increasing computational depth through repeated application of shared Transformer blocks. However, their practical advantages over conventional Transformers remain under debate: each additional loop incurs another Transformer pass and requires caching another set of KV states, causing inference FLOPs and KV-cache memory to grow continuously with loop depth. This overhead becomes particularly severe at large loop counts and long context, preventing the parameter efficiency of Looped Transformers from translating into practical inference efficiency. In this paper, we find that much of the additional computation and storage introduced by looping is redundant. As recurrence proceeds, state changes become increasingly concentrated on a small subset of tokens; attention-output differences are dominated by a sparse and stable subset of key columns; and KV residuals between adjacent loops become progressively more amenable to low-bit quantization. Building on these observations, we introduce FlashLoop, a training-free inference framework that reduces cross-loop redundancy through token-sparse updates, sparse attention, and KV-residual quantization. Across several Looped Transformers models, FlashLoop delivers lossless accuracy while achieving up to 1.64times end-to-end speedup and up to 6times KV-cache memory reduction, substantially improving the practicality of scaling Looped Transformers to greater computational depths and longer context.

Looped Transformers improve parameter efficiency by repeatedly applying shared Transformer blocks over multiple recurrent loops, increasing computational depth without increasing the parameter count. However, KV cache memory still scales with the number of loops, becoming a key memory bottleneck that limits batch size and inference throughput. KV cache quantization can alleviate this bottleneck, but existing methods often suffer substantial accuracy degradation at aggressive low-precision regimes. We observe that looped Transformers offer a unique opportunity: KV states across loops are highly similar. Based on this observation, we propose ResidualQuant, which uses the final-loop KV states as a reference and represents the remaining loops with low-precision residuals. Our method further combines least-square scaling and rotations applied to the residuals, as well as loop-wise mixed precision, to enable accurate quantization down to INT2 while retaining efficient reconstruction. Across multiple looped Transformer models and mathematical reasoning and code generation benchmarks, ResidualQuant consistently improves the accuracy-memory tradeoff over state-of-the-art rotation-based KV quantization. In particular, our method retains accuracy close to BF16 under mixed-precision settings while reducing theoretical KV storage by 80.7%, achieving up to 13.0% higher accuracy than the rotation-based baseline at the same memory budget. On an RTX 5090, the reduced KV memory traffic improves fixed-batch decode throughput by up to 2.73x, while the smaller memory footprint enables up to 2x larger batches, improving peak throughput by up to 4.15x.

Typed decision models answer a declared question without generating text: a decision head returns a probability for each of the declared options in a single forward pass. A single pass is fast, intuitive System 1 thinking. We study what lies between one pass and generated reasoning: looping, in which the same layers are recursively applied several times before one typed readout. Each loop lets the model revise its hidden state before it commits to an answer, without generating a token; we call this System 1.5 thinking. We propose SanSi, which turns a pre-trained looped language model into a typed decision model. The option probabilities are read after every loop, and every loop is trained with a proper scoring rule, so that one model serves every budget from one loop to eight in a single run. On 10,027 test decisions from 59 sources, SanSi reaches 72.0% accuracy: 13.5 points above a non-looped model of the same shape trained with the same recipe, 5.3 points above a newer non-looped model of its size, and 1.8 points below one with three times the parameters. On two depth-controlled tasks, loops extend the solvable depth beyond the depths seen in training, where the larger single-pass model fails. Used as the judge for policy optimization with reinforcement learning, without gold answers, SanSi raises the generator's F1 by 7.7 points.

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 language models (LoopLMs) improve parameter efficiency by recursively reusing Transformer blocks, enabling deeper computation under a fixed model size. However, this reuse makes LoopLMs more fragile under post-training quantization (PTQ). We present the first systematic study of quantization in LoopLMs and identify three challenges: distribution shift across roles, state reuse across loop transitions, and recursive error accumulation. To address these challenges, we propose LoopQ, a loop-aware PTQ framework that preserves a shared quantized backbone while introducing lightweight adaptations. LoopQ combines activation scaling, selective transformation, cross-loop state alignment, and trajectory-aware optimization to reduce distributional mismatch within loops and error accumulation across loops. Experiments across seven benchmarks show that, under W4A4 quantization, LoopQ improves average downstream accuracy by 68.8% and reduces average perplexity by 87.7% compared with the strongest static PTQ baseline.

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.

Recursive or looped language models have recently emerged as a new scaling axis by iteratively refining the same model computation over latent states to deepen reasoning. We extend such scaling principle from a single model to multi-agent systems, and ask: Can agent collaboration itself be scaled through recursion? To this end, we introduce RecursiveMAS, a recursive multi-agent framework that casts the entire system as a unified latent-space recursive computation. RecursiveMAS connects heterogeneous agents as a collaboration loop through the lightweight RecursiveLink module, enabling in-distribution latent thoughts generation and cross-agent latent state transfer. To optimize our framework, we develop an inner-outer loop learning algorithm for iterative whole-system co-optimization through shared gradient-based credit assignment across recursion rounds. Theoretical analyses of runtime complexity and learning dynamics establish that RecursiveMAS is more efficient than standard text-based MAS and maintains stable gradients during recursive training. Empirically, we instantiate RecursiveMAS under 4 representative agent collaboration patterns and evaluate across 9 benchmarks spanning mathematics, science, medicine, search, and code generation. In comparison with advanced single/multi-agent and recursive computation baselines, RecursiveMAS consistently delivers an average accuracy improvement of 8.3%, together with 1.2times-2.4times end-to-end inference speedup, and 34.6%-75.6% token usage reduction. Code and Data are provided in https://recursivemas.github.io.

Latent-reasoning looped language models (LoopLMs) offer a different scaling path for machine translation (MT): instead of increasing parameter count or emitting explicit chain-of-thought tokens, they spend additional recurrent computation inside hidden states. We introduce LatentMT, the first systematic study of latent-reasoning LoopLMs for machine translation. LatentMT adapts a small 2.6B-parameter backbone model with lightweight training. Across 32 translation directions spanning high-, mid-, and low-resource languages, LatentMT achieves performance comparable to models three to five times larger. It is competitive in a high-resource language and achieves state-of-the-art performance on both mid-resource and low-resource languages. Studying the behavior of scaling the number of recurrent reasoning steps, we find that recurrent computation consistently improves translation quality in early steps, then saturates quickly afterwards. Our mechanistic analysis shows that hidden-representation differences shrink along the recurrent reasoning-step axis, supporting the observed saturation in performance. Finally, our efficiency analysis shows that LatentMT requires lower training and inference compute than much larger non-latent-reasoning models with similar performance, making latent recurrent computation a promising path toward compact, efficient, and strong machine translation.

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.

We introduce Loop Flow Transformers (LiFT), a family of looped generative models that scales computation by repeatedly applying a shared Diffusion Transformer (DiT) core, with only light changes to the standard architecture. Rather than asking every recurrent step for the final prediction, LiFT trains each step with a single regression target: a point on a straight path from the model's initial estimate to the flow-matching target. Because we index these targets by a continuous depth coordinate, a trained model can loop far beyond its training depth with no retraining, early exits, or other modifications. In our experiments, these longer rollouts improve generation, so inference computation can grow without adding parameters. On ImageNet at 256x256, LiFT-L/2 achieves an FID 3.34 points lower than our dense DiT-XL/2 baseline while using approximately 60% fewer parameters, 32% fewer training FLOPs, and 52% fewer inference FLOPs.

As data propagates through a Transformer, the norm of its hidden states grows by orders of magnitude with depth, a phenomenon framed as 'curse of depth' and nearly universally treated as a pathology to be suppressed. We take the opposite view. Across 16 pre-trained LLMs from 9 families, spanning dense, mixture-of-experts and hybrid architectures and Pre-, Peri- and Post-Norm designs, we find that this growth reflects an emergent depth-positional encoding, carried by the only learned per-layer gain on the residual stream, the normalization weight γ: with depth, γ grows in magnitude and rotates in direction, jointly encoding the layer index. We make this depth-conditioned encoding explicit with LayerRoPE, an implicit analog of RoPE along the depth axis, which replaces all layerwise γ vectors with a single shared vector and depth-conditioned scalars, at a net reduction in parameters and <0.02% change in FLOPs. Across a model ladder scaled up to 100B+ tokens, LayerRoPE consistently outperforms Pre-, Post- and Peri-Norm and Layer-Norm Scaling, reaching Pre-Norm's 1.3B loss with 3.4times less compute; LayerRoPE is the only approach that shows strong convergence and improves near monotonically as depth scales to 512 layers. It improves learning-rate sensitivity by 3-10times, and transfers naively to and consistently improves looped latent models and Vision Transformers. Inspecting its learned schedule inverts the prevailing premise: LayerRoPE does not shrink the residual stream but widens it, damping what each block reads while amplifying what it writes. Depth stability, our results suggest, calls not for suppressing the residual stream, but for depth-conditioned regulation of the computational blocks it feeds.

Mixture-of-Experts (MoE) and looped architectures scale models along two orthogonal axes, namely parameter capacity and effective depth. However, mainstream looped architectures rely on dense backbones that couple parameter count with per-token FLOPs, which makes it impossible to isolate the effect of iterative computation under matched budgets. To this end, we present LoopMoE, a looped MoE language model that integrates sparse routing with iterative weight-shared computation through two designs. The first is IterAdaLN, which resolves weight-sharing symmetry via a modulation signal jointly conditioned on the iteration index and the per-token hidden state. The second is a capacity-balancing strategy that recovers the attention-to-FFN active parameter ratio of well-tuned non-looped references. Together, these designs enable the first strictly controlled, head-to-head evaluation of a looped MoE against a Vanilla MoE under identical total parameters, per-token FLOPs, and active sublayer ratios. At the 3B scale, LoopMoE outperforms the Vanilla MoE on 8 of 9 downstream benchmarks with an average improvement exceeding 1 point. At the 9B scale, LoopMoE continues to outperform the matched Vanilla MoE, indicating that the architectural gain persists at larger scale. Our work establishes a controlled synthesis of sparsity and recurrence, and suggests a promising direction for looped language models.

Masked diffusion models (MDMs) have emerged as a promising alternative to autoregressive models for language modeling, yet the effective design of transformer architectures for MDMs remains underexplored. In this paper, we show that selectively looping the early-middle transformer layers significantly improves both training efficiency and model performance in MDMs. We call this approach LoopMDM(Looped Masked Diffusion Model), which brings two key benefits: looping layers at training-time yields a depth-scaling effect without adding parameters, while varying the number of loops at inference-time enables flexible compute scaling. Despite the simplicity, the results are striking: across multiple pre-training corpora, LoopMDM matches the performance of same-size MDMs with up to 3.3 fewer training FLOPs, while its final performance outperforms them on various reasoning benchmarks, including up to 8.5 points on GSM8K. It even surpasses deeper non-looped MDMs trained with comparable per-step compute, indicating that selective looping is more effective than naive depth scaling. Furthermore, LoopMDM can scale inference-time compute by increasing the number of loops. Adaptively adjusting the number of loops throughout the sampling process further yields additional gains in compute efficiency while maintaining performance. Lastly, with attention analysis, we provide evidence that looping is effective in MDMs by promoting interactions among masked positions. Our code and weights will be publicly released.

Looped computation shows promise in improving the reasoning-oriented performance of LLMs by scaling test-time compute. However, existing approaches typically require either training recurrent models from scratch or applying disruptive retrofits, which involve substantial computational costs and may compromise pretrained capabilities. To address these limitations, we introduce Looped Depth Up-Scaling (LoopUS), a post-training framework that converts a standard pretrained LLM into a looped architecture. As a key technical contribution, LoopUS recasts the pretrained LLM into an encoder, a looped reasoning block, and a decoder. It operationalizes this latent-refinement architecture through four core components: (1) block decomposition, guided by staged representation dynamics; (2) an input-dependent selective gate to mitigate hidden-state drift; (3) random deep supervision for memory-efficient learning over long recursive horizons; and (4) a confidence head for adaptive early exiting. Collectively, these mechanisms transform a standard non-looped model into a looped form while stabilizing it against both computational bottlenecks and representation collapse. Through stable latent looping, LoopUS improves reasoning-oriented performance without extending the generated traces or requiring recurrent training from scratch. For more details, see https://thrillcrazyer.github.io/LoopUS

Diffusion language models (DLMs) enable fast generation by predicting multiple tokens in parallel, but their practical adoption remains limited by a persistent quality gap relative to comparably sized autoregressive (AR) models. We attribute this gap to a computation-difficulty mismatch: within a partially observed sequence, some unknown tokens are easy to predict, while others require substantially more computation. Existing DLMs nevertheless apply uniform computational depth to all unknown positions at each denoising step. We introduce ALoDLM, which replaces uniform computation with token-adaptive latent recurrence. At each denoising step, ALoDLM iteratively refines latent representations and allocates computation according to token difficulty. Tokens ready to commit are fed back as discrete context, while unresolved tokens retain and further refine their latent states through additional recurrent passes. To learn token prediction and computation allocation jointly, we formulate token-wise computation schedules as latent variables and derive a conditional negative evidence lower bound (NELBO). We train ALoDLM at 1.7B and 8B parameter scales. Across eleven benchmarks, ALoDLM outperforms all evaluated DLMs and the corresponding AR baselines in average benchmark score at both scales. ALoDLM also retains fast parallel decoding, yielding a strong quality-efficiency trade-off among evaluated autoregressive and diffusion models under optimized inference engines.

Uniform diffusion language models (UDLMs) generate text by iteratively denoising sequences of ordinary vocabulary tokens, any of which may be corrupted and subsequently revised. During training, UDLMs must denoise from a context that may itself be unreliable, whereas masked diffusion language models (MDLMs) always condition on a clean context. This challenging training objective makes UDLMs more FLOP-intensive to train than MDLMs, but it yields stronger downstream reasoning performance. Recently, selective looping, which recursively applies a few early Transformer layers within each denoising step, has been shown to improve training efficiency in MDLMs under an iso-FLOP budget. We investigate looping in UDLMs while keeping the training objective and sampler unchanged. We find that looping significantly improves training efficiency: looped UDLMs reach the validation NLL of a non-looped UDLM trained on about 0.5 trillion tokens with up to 2.8x fewer training FLOPs on OpenWebText and 3.3x fewer on FineWeb-Edu. We also find that, at a matched NFE budget, looped UDLMs reach up to 9.8% lower generation perplexity than the non-looped UDLM, with most gains achieved at moderate loop counts. Finally, we study different loop-count distributions and find that sampling smaller loop counts more frequently during training improves inference efficiency: on downstream tasks, this strategy uses 12% fewer loops than uniform loop-count sampling without degrading accuracy.

Transformers have demonstrated effectiveness in in-context solving data-fitting problems from various (latent) models, as reported by Garg et al. However, the absence of an inherent iterative structure in the transformer architecture presents a challenge in emulating the iterative algorithms, which are commonly employed in traditional machine learning methods. To address this, we propose the utilization of looped transformer architecture and its associated training methodology, with the aim of incorporating iterative characteristics into the transformer architectures. Experimental results suggest that the looped transformer achieves performance comparable to the standard transformer in solving various data-fitting problems, while utilizing less than 10\% of the parameter count.

Looped transformers have emerged as a parameter-efficient alternative to scaling depth for strong reasoning. By reusing one stack of layers across T iterations, they attain the effective depth and reasoning capabilities of larger models at a fixed parameter count. Yet existing approaches suffer from latent overthinking and undifferentiated computation, largely because intermediate representations receive no guidance across loops. Multi-token prediction (MTP) supplies exactly the dense, forward-looking supervision the loop is missing. We propose LoopMTP, which links the two through a structural correspondence in latent space: a model that loops T times can anticipate T future tokens. LoopMTP realizes this by softly aligning the hidden state of loop t with the embedding of the token t steps ahead, while a lightweight gate preserves useful information across iterations. LoopMTP improves average accuracy by up to 8.1\% (relative) over the non-looped baseline, with training remaining stable for up to 15 loops.

Looped Transformers provide a parameter-efficient approach to depth scaling by repeatedly applying shared Transformer blocks. Recent reasoning models have likewise highlighted the value of scaling test-time computation through longer computation trajectories. However, the principles for designing effective loop transitions remain poorly understood. We view the looped hidden state as a fast weight that is updated throughout the depth. We formulate loop transitions as local gradient-based updates, with recurrent blocks predicting implicit targets at each depth. Our framework derives loop transitions in closed form from a projection, a local objective and an optimizer update rule. Mapping representative loop transitions into this framework reveals mismatches between their transitions and projections. We first align the input maps of existing transitions. We then derive OperLoop, which combines explicit weight decay, adaptive step size and a delta objective. The aligned variants reduce training loss and improve average commonsense accuracy. OperLoop improves average generative performance over the compared looped and non-looped baselines under matched training FLOPs. These results support the framework's usefulness for loop design. We extend the analysis to additional loop models and outline a roadmap for future loop transition design.

We present Loopie, the most powerful looped Transformer to date. The Loopie series consists of two Mixture-of-Experts (MoE) models: a 20B-parameter model with 2B active parameters and a 6Bparameter model with 0.6B active parameters. Looped Transformers have long faced a challenge: given an N-fold increase in pre-training compute, increasing the parameter count by a factor of N usually outperforms looping a model N times. Loopie addresses this challenge. Extensive ablation studies, including comparisons with a vanilla 30B-A3B model, show that Loopie substantially outperforms vanilla Transformer baselines trained with the same compute budget. Our novel post-training pipeline equips Loopie with strong reasoning abilities. At the 2025 IMO and IPhO, Loopie achieves gold-medal performance without tools.

Improving text-to-image models has traditionally relied on increasing model size or the number of denoising steps. In this work, we explore an alternative way to scale computation by repeatedly running shared Transformer blocks within each denoising step, effectively increasing computational depth while keeping the parameter count fixed. This looped computation enables iterative refinement of internal representations without explicit reasoning tokens. However, naive looping fails to consistently improve image quality. We trace this problem to weak supervision across intermediate loops and unregulated attention updates that progressively erode local information. To overcome these challenges, we propose Looped Diffusion Transformer (Looped-DiT), which combines deep supervision across intermediate loops with self-modulating attention to stabilize looped feature updates. Under matched-parameter and matched-compute settings, Looped-DiT consistently outperforms non-looped baselines. Notably, a 260M-parameter looped model can surpass a model 6.5x larger across multiple text-to-image benchmarks while requiring 4.9x lower inference compute. Beyond this performance gain, we find that looped computation can offer a more effective form of iterative computation for diffusion models, with increasing loop depth yielding larger gains than adding more denoising steps under a fixed inference budget. Furthermore, deeper loops can progressively correct mistakes made in earlier loops, exhibiting behaviors suggestive of latent reasoning. Together, these results show that looped computation offers a promising way to scale visual generation models.

Looped Transformers have drawn growing attention because looping improves multi-step reasoning and increases the executed depth of a model without adding parameters. They achieve this by reusing a stack of L layers for N loops. However, increasing the loop count N not only destabilizes training but also shifts the best hyperparameters, so they must be searched again for every loop count. Inspired by Depth-μP (Yang et al., 2024), which multiplies each residual branch by L^{-1/2} for depth L so that a learning rate tuned on a shallow untied network transfers to a deep one, we study how the residual branches and the learning rate should scale with the loop count N. We find that loops cannot simply be treated as extra depth, because the loops share weights. At initialization, distinct layers add nearly uncorrelated increments to the residual stream, so their sum grows as L, whereas a shared layer adds positively correlated increments at every loop, so their sum grows linearly in N. For a single layer looped N times, we show in the infinite-width limit that multiplying the residual branch by 1/N and keeping the learning rate independent of N is the only parameterization with both a stable forward pass and maximal residual-stream updates. We then generalize the analysis to a stack of L distinct layers looped N times to obtain a parameterization we call Loop-μP, in which the loop factor 1/N combines with the depth factor L^{-1/2}. Empirically, we find in language modeling that the base learning rate tuned on the smallest model stays best under Loop-μP for loop counts up to 8 and executed depths up to 384 layers. These results make Loop-μP a practical recipe for scaling looped Transformers.

Augmenting Large Language Models (LLMs) with external tools enables them to execute complex, multi-step tasks. However, tool learning is hampered by the static synthetic data pipelines where data generation and model training are executed as two separate, non-interactive processes. This approach fails to adaptively focus on a model's specific weaknesses and allows noisy labels to persist, degrading training efficiency. We introduce LoopTool, a fully automated, model-aware data evolution framework that closes this loop by tightly integrating data synthesis and model training. LoopTool iteratively refines both the data and the model through three synergistic modules: (1) Greedy Capability Probing (GCP) diagnoses the model's mastered and failed capabilities; (2) Judgement-Guided Label Verification (JGLV) uses an open-source judge model to find and correct annotation errors, progressively purifying the dataset; and (3) Error-Driven Data Expansion (EDDE) generates new, challenging samples based on identified failures. This closed-loop process operates within a cost-effective, open-source ecosystem, eliminating dependence on expensive closed-source APIs. Experiments show that our 8B model trained with LoopTool significantly surpasses its 32B data generator and achieves new state-of-the-art results on the BFCL-v3 and ACEBench benchmarks for its scale. Our work demonstrates that closed-loop, self-refining data pipelines can dramatically enhance the tool-use capabilities of LLMs.

Existing deep learning models may encounter great challenges in handling graph structured data. In this paper, we introduce a new deep learning model for graph data specifically, namely the deep loopy neural network. Significantly different from the previous deep models, inside the deep loopy neural network, there exist a large number of loops created by the extensive connections among nodes in the input graph data, which makes model learning an infeasible task. To resolve such a problem, in this paper, we will introduce a new learning algorithm for the deep loopy neural network specifically. Instead of learning the model variables based on the original model, in the proposed learning algorithm, errors will be back-propagated through the edges in a group of extracted spanning trees. Extensive numerical experiments have been done on several real-world graph datasets, and the experimental results demonstrate the effectiveness of both the proposed model and the learning algorithm in handling graph data.

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.

We introduce training-free looped transformers, in which a lightweight inference-time wrapper loops a contiguous mid-stack block of layers of a frozen checkpoint without additional fine-tuning, continued training, or architectural changes. Unlike prior looped transformer methods that train with the looped structure end-to-end, we retrofit recurrence onto pretrained models at test time. We show that naive block reapplication usually degrades performance, highlighting the importance of the loop application strategy. Motivated by viewing a pre-norm transformer block as a forward Euler step on an ODE, we instead treat looping as a refinement of the same approximation, replacing one large update with smaller damped sub-steps. Across seven dense, sparse MoE, and MLA+MoE model families, our method improves Qwen3-4B-Instruct by +2.64 pp on MMLU-Pro, Qwen3-30B-A3B-Instruct by +1.14 pp on CommonsenseQA, and Moonlight-16B-A3B-Instruct by +1.20 pp on OpenBookQA.

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.

Large Language Models (LLMs) are powerful but often too slow and costly for real-world use during inference. Looped transformers save on parameters by reusing the same weights for multiple computational steps, or "loops." However, this approach has a major flaw: the loops run one after another, causing inference latency and memory requirements to increase with each added loop. This makes them impractical for fast applications. To solve this problem, we introduce the Parallel Loop Transformer (PLT). PLT is a new architecture that delivers the performance benefits of a deep, looped model but with the low latency of a standard, non-looped model. PLT works using two key techniques. First, Cross-Loop Parallelism (CLP) breaks the sequential dependency by computing different loops for different tokens at the same time, all within a single pass. Second, to prevent memory costs from growing, we use an Efficient Representation Enhancement strategy. This method shares the memory (KV cache) from the first loop with all other loops. It then uses a Gated Sliding-Window Attention (G-SWA) to combine this shared global information with local information, maintaining high accuracy. Our experiments show that PLT achieves the high accuracy of a traditional looped model but with almost no extra latency or memory cost compared to a standard transformer.

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.

Speculative decoding accelerates autoregressive generation in large language models. In each drafting stage, a lightweight draft model proposes tokens that the target model subsequently verifies. With increasingly capable draft models, we find that the target model frequently accepts all tokens produced in a drafting stage. A verification nevertheless follows each drafting stage, resulting in unnecessary target-model forward passes even when drafting could have continued. Adaptive draft length methods decide during decoding how many draft tokens precede a verification, but they raise the speedup only for autoregressive draft models. For a parallel draft model, drafting further requires target-model hidden states for draft tokens that have not been verified. We propose DLoop, a looped form of speculative decoding that adaptively performs multiple drafting stages before verification. DLoop continues drafting while the draft model remains confident and verifies all accumulated draft tokens together. Loop-aware training keeps the draft model reliable in the additional drafting stages by exposing it to its own hidden states for unverified draft tokens. By spending additional draft-model forward passes, DLoop reduces the number of target-model forward passes required for verification. Across diverse speculative decoding methods including EAGLE-3, DFlash, Domino, DSpark, and multi-token prediction modules, DLoop improves the wall-clock speedup by 5 to 41 percent while preserving lossless decoding. Code will be available at https://github.com/naver-ai/DLoop.

Looped Transformers have emerged as an efficient and powerful class of models for reasoning in the language domain. Recent studies show that these models achieve strong performance on algorithmic and reasoning tasks, suggesting that looped architectures possess an inductive bias toward latent reasoning. However, prior approaches fix the number of loop iterations during training and inference, leaving open the question of whether these models can flexibly adapt their computational depth under variable compute budgets. We introduce LoopFormer, a looped Transformer trained on variable-length trajectories to enable budget-conditioned reasoning. Our core contribution is a shortcut-consistency training scheme that aligns trajectories of different lengths, ensuring that shorter loops yield informative representations while longer loops continue to refine them. LoopFormer conditions each loop on the current time and step size, enabling representations to evolve consistently across trajectories of varying length rather than drifting or stagnating. Empirically, LoopFormer demonstrates robust performance on language modeling and reasoning benchmarks even under aggressive compute constraints, while scaling gracefully with additional budget. These results show that looped Transformers are inherently suited for adaptive language modeling, opening a path toward controllable and budget-aware large language models.

We introduce Elastic Looped Transformers (ELT), a highly parameter-efficient class of visual generative models based on a recurrent transformer architecture. While conventional generative models rely on deep stacks of unique transformer layers, our approach employs iterative, weight-shared transformer blocks to drastically reduce parameter counts while maintaining high synthesis quality. To effectively train these models for image and video generation, we propose the idea of Intra-Loop Self Distillation (ILSD), where student configurations (intermediate loops) are distilled from the teacher configuration (maximum training loops) to ensure consistency across the model's depth in a single training step. Our framework yields a family of elastic models from a single training run, enabling Any-Time inference capability with dynamic trade-offs between computational cost and generation quality, with the same parameter count. ELT significantly shifts the efficiency frontier for visual synthesis. With 4times reduction in parameter count under iso-inference-compute settings, ELT achieves a competitive FID of 2.0 on class-conditional ImageNet 256 times 256 and FVD of 72.8 on class-conditional UCF-101.

Large Language Models (LLMs) often exhibit a gap between their internal knowledge and their explicit linguistic outputs. In this report, we empirically investigate whether Looped Transformers (LTs)--architectures that increase computational depth by iterating shared layers--can bridge this gap by utilizing their iterative nature as a form of introspection. Our experiments reveal that while increasing loop iterations narrows the gap, it is partly driven by a degradation of their internal knowledge carried by representations. Moreover, another empirical analysis suggests that current LTs' ability to perceive representations does not improve across loops; it is only present in the final loop. These results suggest that while LTs offer a promising direction for scaling computational depth, they have yet to achieve the introspection required to truly link representation space and natural language.

Language models typically reason via explicit chain-of-thought (CoT), generating intermediate steps token-by-token. Latent CoT offers an alternative: it performs multi-step reasoning in the model's hidden states, replacing decoded tokens with continuous representations for greater efficiency. However, existing latent CoT methods underperform explicit CoT beyond 1B parameters, and the gap widens with scale. Looped, or recurrent-depth, Transformers, which reuse their weights to increase computation depth without adding parameters, are a natural fit for latent reasoning. We therefore ask whether looped Transformers can bridge this gap. We answer affirmatively with a simple recipe: a looped padded Transformer that processes K latent blocks in parallel for R iterations, with a cross-entropy loss on each latent position's gold CoT-step token, similar to explicit CoT supervision. We instantiate it as LOTUS (Looped Transformers with parallel supervision on latents). LOTUS is, to our knowledge, the first latent-CoT method to bridge the gap to explicit CoT at the 3B scale, while cutting thought-phase latency by 2.5x-6.9x from compact math expressions to natural language. Projecting LOTUS's post-loop latents through the base LM head recovers the gold reasoning steps and even surfaces alternative valid intermediate steps, evidence that its latent space is interpretable and CoT-aligned. Ablations confirm that both the looped backbone and the parallel supervision on gold CoT tokens are essential.

Classical structure-from-motion reconstruction proceeds iteratively, refining camera poses and scene geometry from a current estimate. This allows maps to easily expand without recomputing already computed views, or geometry estimates to improve as more observations are provided. Most geometric foundation models, however, predict entire geometry in a single forward pass or update a recurrent state as frames arrive sequentially, making this expansion and refinement non-trivial. We introduce Solv3R, a looped transformer that learns to continue solving multi-view geometry from its current state. Importantly, its update operator is independent of the iteration count, allowing previously refined and newly introduced views to be jointly updated regardless of how ``solved'' they are. We train the operator to preserve accurate reconstructions under continued iteration and correct inaccurate states, enabling refinement of fixed image collections and integration of new observations. Representations of solved views can also be cached once and reused as geometric context across iterations, enabling compact and efficient key-value caching. We analyze the solver dynamics and compare it with the state-of-the-art, demonstrating that a single learned solver supports offline reconstruction, incremental reconstruction, relocalization, and streaming without task-specific training or modules. The code and models will be made publicly available at https://ubc-vision.github.io/solv3r/.

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.

Scaling Large Language Models (LLMs) has been driven mainly by enlarging the Transformer backbone, but for an already-strong model this requires another round of costly pretraining. We study whether an existing backbone can keep improving by allocating more computation to each token while leaving the Transformer backbone fixed. Depth-recurrent (looped) Transformers pursue this goal but are hard to scale, because looped computation does not fit naturally with the pipeline parallelism used to train the largest models. We add computation along the sequence-length dimension, where the extra computation is simply a longer input and stays compatible with standard large-model training. We propose Hidden Decoding, a sequence-length scaling method applied during continued pretraining (CPT). It expands each token into n streams with independent embedding tables and keeps the intermediate streams' key-value cache as context, so each token performs more internal computation without adding or widening Transformer layers. To keep this affordable at scale, we introduce Stream-Factorized Attention, in which most layers attend only within each stream and only a few layers mix across streams, reducing the attention cost from quadratic to roughly linear in n. Experiments support two scaling results. At frontier scale, we train WeLM-HD4-80B and WeLM-HD4-617B at n=4 and improve their matched non-HD baselines, making Hidden Decoding the first demonstrated sequence-length scaling method at the 100B+ MoE scale. Across expansion factors, the gains grow as n increases, showing that sequence-length expansion is a practical fixed-backbone scaling path for frontier-scale LLMs.

Self-attention selects information freely across the sequence, but across depth, Transformers merely add each layer's output to the residual stream, so later layers cannot selectively reuse earlier-layer representations. Recent cross-layer methods improve this flow but operate on hidden states outside attention, adding state beyond the key-value cache at inference--a cost that becomes increasingly salient as modern LLMs compress the cache with grouped-query and multi-head latent attention. We introduce Depth-Attention, which performs this selection inside the attention module itself: before a layer attends over the sequence, its query attends over the keys of earlier layers at the same token position and mixes their values into the value that self-attention then reads. Because Depth-Attention reuses the standard attention queries, keys, and value-cache slots, storing depth-mixed values in place of the original values, it adds no parameters and introduces no persistent inference state beyond the standard key-value cache--the same cache size as a vanilla decoder and less than hidden-state-based cross-layer methods. On Qwen3-style decoders at 1.5B and 3B parameters, Depth-Attention attains the lowest perplexity and the highest average downstream accuracy, improving over the vanilla Transformer by up to 2.3 accuracy points and surpassing strong cross-layer baselines in perplexity and average accuracy, while adding under 0.01% extra arithmetic FLOPs and no additional persistent inference state. The gains hold from 360M to 3B parameters and extend to looped Transformers.

Looped architectures provide an inductive bias toward learning step-by-step procedures for tasks that require compositional reasoning. The number of effective layers reached by looping determines the quality of the solution these models find. Like deep architectures, looped architectures are prone to a signal propagation problem induced by depth as the halting decision is postponed. In this paper, we address this signal propagation issue using pre-norm layers and residual scaling. Building on these architectural modifications, we propose FPRM, a Transformer-based Fixed-Point Reasoning Model that uses fixed-point convergence as an end-to-end halting mechanism in a looped architecture. We show that fixed-point halting allows FPRM to adapt its compute to task difficulty. FPRM is effective on common reasoning benchmarks, namely Sudoku, Maze, state-tracking, and ARC-AGI.

Looped Transformers have recently demonstrated strong performance in both reasoning and language tasks by reusing a shared set of parameters across multiple iterations, achieving parameter efficiency without sacrificing representational power. Besides, looped Transformers perform inference directly in the latent space (latent reasoning) to reduce the number of tokens consumed during inference, thereby achieving improved sample efficiency. However, these models typically apply a fixed recursion depth uniformly to every token, leading to suboptimal compute allocation and leaving significant efficiency gains on the table. In this work, we propose dynamic token-choice routing for looped transformers, enabling each token to adaptively determine its own number of loop iterations based on its hidden state, which can improve the token generation accuracy. Moreover, we further introduce recursion-wise KV cache, which maintains an independent key-value cache for each recursion loop, this design ensures that tokens at different depths only attend to their corresponding cached states, effectively enabling faster autoregressive decoding. Extensive experiments show that T-LoopFormer achieves robust performance on language modeling and zero-shot reasoning tasks and our model can reach the lowest decoding latency, which validate the effectiveness of token-choice router and recursion-wise KV cache. Code is available at https://github.com/YuMingQian1234/T-LoopFormer

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.

Transformers have achieved remarkable success across a wide range of applications, and a growing body of work suggests that part of their strength comes from their ability to learn and execute algorithmic procedures. However, our understanding of how transformers learn such algorithms remains limited, especially in the presence of layer normalization (LN). In this work, we study principal component prediction as a concrete testbed for understanding the training dynamics of transformers with LN. We prove that a looped linear transformer with LN, trained by gradient descent, converges to a solution that implements the power method, with each self-attention layer performing one power iteration. Notably, the model is trained only for principal component prediction, rather than being explicitly supervised to implement the power method. Our finding thus reveals an "algorithmic implicit bias" of looped transformers with LN: principal-component prediction can in principle be achieved by many mechanisms, yet gradient descent selects one that realizes the power method. We further provide a concrete comparison between transformers with and without LN: even with layerwise guidance from power iterations, a transformer without LN cannot exactly learn the power method, whereas the corresponding transformer with LN can, leading to a provable performance gap in principal component prediction. Our results provide, to our knowledge, the first theoretical analysis of the training dynamics of looped and single-layer transformers with LN, and shed light on the role of LN in transformer models.

In-context learning has been recognized as a key factor in the success of Large Language Models (LLMs). It refers to the model's ability to learn patterns on the fly from provided in-context examples in the prompt during inference. Previous studies have demonstrated that the Transformer architecture used in LLMs can implement a single-step gradient descent update by processing in-context examples in a single forward pass. Recent work has further shown that, during in-context learning, a looped Transformer can implement multi-step gradient descent updates in forward passes. However, their theoretical results require an exponential number of in-context examples, n = exp(Ω(T)), where T is the number of loops or passes, to achieve a reasonably low error. In this paper, we study linear looped Transformers in-context learning on linear vector generation tasks. We show that linear looped Transformers can implement multi-step gradient descent efficiently for in-context learning. Our results demonstrate that as long as the input data has a constant condition number, e.g., n = O(d), the linear looped Transformers can achieve a small error by multi-step gradient descent during in-context learning. Furthermore, our preliminary experiments validate our theoretical analysis. Our findings reveal that the Transformer architecture possesses a stronger in-context learning capability than previously understood, offering new insights into the mechanisms behind LLMs and potentially guiding the better design of efficient inference algorithms for LLMs.

Looped Transformers achieve parameter efficiency by repeatedly executing a shared block across recurrent loops. Each loop yields an intermediate representation decodable for the same next token, yet standard decoding discards earlier states. Because earlier loops embody less computation, recurrence inherently supplies aligned weak-and-strong prediction pairs without auxiliary models or external training. We introduce LoopCD, a training-free contrastive decoding framework that guides token selection by contrasting the final prediction with an earlier recurrent pass, operating either in logit space with one extra output pass (LoopCD-Logits) or in hidden-state space with zero output overhead (LoopCD-Hidden). Across four looped Transformer families, LoopCD delivers substantial, consistent gains at full recurrent depth: LoopCD-Logits raises Ouro-2.6B-Thinking's AIME 2024 pass@1 from 61.88% to 73.33%, while LoopCD-Hidden lifts Huginn's HumanEval pass@1 from 22.56% to 31.71%. Crucially, these performance gains enable halving the number of recurrent loops while still matching or exceeding full-depth unguided baselines, reducing forward FLOPs by 22.5% to 48.2%. By transforming intermediate recurrent states into effective guidance signals, LoopCD achieves superior decoding quality while substantially reducing inference compute.

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.

We introduce Dory1.0-4B, a 4-billion-parameter looped (latent-recurrent) language model with elastic depth: a single checkpoint serves different inference compute budgets. Dory1.0-4B is built on a normalized Transformer (nGPT) backbone with interleaved sliding-window and full-attention layers, organized into an input block, a shared recurrent block, and an output block. At inference, the recurrent block can be applied a variable number of times. We pre-trained Dory1.0-4B on 6 trillion tokens. The first 4.5 trillion tokens were trained in NVFP4, and additional pre-training then extended the context length to 256K tokens. Next, we post-trained the model using Supervised Fine-Tuning (SFT) and Reinforcement Learning with Verifiable Rewards (RLVR), with a difficulty-aware loop curriculum that routes harder prompts to deeper loops. Increasing the loop count improves accuracy, with the largest gains on coding, agentic, and long-context benchmarks. With two or more loops, Dory1.0-4B outperforms larger models such as Qwen3.5-9B, Gemma 4 12B, and Gemma 4 26B-A4B on average across 12 benchmarks. Looping also makes Dory1.0-4B more token-efficient on agentic benchmarks: with four loops, it generates 1.3-2.5x fewer tokens than with one loop, as the extra computation is spent in latent space rather than on longer outputs. To reduce memory during looping, Dory1.0-4B caches keys and values only from the last loop, keeping its KV-cache footprint equal to that of a non-looped model at any loop count while retaining most of the accuracy gains. We open-source the BF16, MXFP8, NVFP4, and GGUF quantized checkpoints.

Looped Transformers (LT) have emerged as a powerful architecture by iterating their layers multiple times before decoding the final token. However, pairing them with full attention retains quadratic complexity, making them computationally expensive and slow. We introduce LT2 (Linear-Time Looped Transformers), a family of looped architectures that replace quadratic softmax attention with subquadratic, linear-time attention. We study two variants: LT2-linear with linear attention and LT2-sparse with sparse attention. We find that looping uniquely synergizes with these variants: it enables iterative memory refinement in linear attention and progressively expands the effective receptive field in sparse attention. We formalize these benefits theoretically and demonstrate consistent empirical gains across controlled recall, state-tracking, and language modeling tasks. We then explore LT2-hybrid, which combines different attention variants in a looped setting. Two variants are especially promising: LT2-hybrid (GDN+DSA), which interleaves linear and sparse attention to maximize efficiency and matches the standard looped transformer's quality at fully linear-time cost; and LT2-hybrid (Full+GDN), which interleaves GDN with a small fraction of full attention layers to maximize quality, surpassing the standard looped transformer in both performance and efficiency. We also show how to convert a pre-trained LT into an LT2-hybrid model. With about 1B tokens of training, our converted model, Ouro-hybrid-1.4B, outperforms industry-level 1B models and is competitive with industry-level 4B models while retaining the speed benefits of linear-time attention. Together, these results show a clear path toward making looped transformers more scalable and advancing efficient, capable small language models.

Looped Transformers achieve strong performance with compact parameter sizes by repeatedly applying a shared stack of Transformer blocks across recurrent depths. However, they incur higher decoding latency than standard Transformer models of comparable parameter size because shared weights are accessed at every recurrent depth. To improve decoding efficiency, self-speculative decoding is particularly well suited to Looped Transformers, as their intermediate recurrent states can directly provide draft predictions without an auxiliary draft model. We therefore propose LoopSpec, a training-free self-speculative decoding framework tailored for Looped Transformers. LoopSpec extracts draft tokens from early recurrent states and operates in a pipelined manner, overlapping draft generation of future tokens with target verification of the current token. To improve draft accuracy without excessive compute overhead, we introduce a selective second proposal from deeper recurrent depth while ensuring lossless decoding under both greedy and sampling regimes. Furthermore, we derive the optimal proposal depths in closed form and show the prediction matches measurement. Across reasoning and coding benchmarks, LoopSpec achieves up to 6.83times inference speedup across diverse Looped Transformers.

We investigate how looped transformers encode human preference, training lightweight evaluator heads on frozen Ouro-2.6B loop-iteration states on Anthropic HH-RLHF. v2: an erratum is prepended; the original manuscript is unchanged. A post-publication audit found the three headline results inflated by two independent evaluation errors. The 95.2% pairwise evaluator accuracy is a canonical-ordering artifact: the data were correctly split, but the evaluator learned to prefer the first-presented argument; its strict antisymmetrized accuracy on the full 8,552-pair test set is 63.9%. The 84.5% pairwise probe and the below-chance 21.75% pointwise probe were source-item leaks (orientation rows and pair partners crossing the train/test split); corrected pair-disjoint values are 56.5% and 54.2% -- above chance, so the "inverted polarity" finding is withdrawn. The central finding survives at much smaller magnitude: preference is decoded more accurately relationally than pointwise (paired +2.3 points, 95% CI [+1.3, +3.3]), and the antisymmetrized evaluator still beats the linear probe, but no corrected readout rivals end-to-end reward models. The methodological findings stand (constant-output degeneracy, flip test, swap-protocol metric deflation), with one correction: antisymmetrized accuracy, not antisymmetry correlation, certifies relational discrimination. The two errors are mutually invisible -- a split audit cannot see an ordering prior, antisymmetrization cannot see a leak -- so both checks are required. Full audit in the follow-up work (Kirin, 2026, in preparation).

Reasoning models (e.g., DeepSeek-R1) generate long chains of thought to solve harder problems, but they often loop, repeating the same text at low temperatures or with greedy decoding. We study why this happens and what role temperature plays. With open reasoning models, we find that looping is common at low temperature. Larger models tend to loop less, and distilled students loop significantly even when their teachers rarely do. This points to mismatches between the training distribution and the learned model, which we refer to as errors in learning, as a key cause. To understand how such errors cause loops, we introduce a synthetic graph reasoning task and demonstrate two mechanisms. First, risk aversion caused by hardness of learning: when the correct progress-making action is hard to learn but an easy cyclic action is available, the model puts relatively more probability on the cyclic action and gets stuck. Second, even when there is no hardness, Transformers show an inductive bias toward temporally correlated errors, so the same few actions keep being chosen and loops appear. Higher temperature reduces looping by promoting exploration, but it does not fix the errors in learning, so generations remain much longer than necessary at high temperature; in this sense, temperature is a stopgap rather than a holistic solution. We end with a discussion of training-time interventions aimed at directly reducing errors in learning.

Scaling laws predict how loss decreases with increases in computation. We show, contrary to conventional wisdom, that architectural interventions can modify scaling exponents in pre-training, leading to power-law improvements in performance as computation increases. As an anchoring point, we consider the architectural formulation of looped transformers. Although not typically used in this way, looping, also known as recursive depth, provides a mechanism for model growth, by increasing the number of loops during training. Model growth, with and without shared weights, provides the biggest changes to the scaling exponents. In particular, a 7.4B model growth architecture matches GPT-3 13B on CORE with roughly 20times less compute, and has compute efficiency gains that increase with scale. Moreover, simply using a boundary operator in a vanilla transformer, which normalizes and injects an earlier block, also provides an exponent increase, although to a lesser extent. In the data-constrained, multi-epoch setting, standard looping has a useful regularizing effect, where we find it is compute-optimal to increase the number of loops with scale. These results can be understood through the lens of computational depth: for a given computational budget, we wish to increase the usable depth of the transformer, which can lead to efficiency gains that increase with scale.

Discrete diffusion models offer a promising alternative to autoregressive generation through parallel decoding, but they suffer from a sampling wall: once categorical sampling occurs, rich distributional information collapses into one-hot vectors and cannot be propagated across steps, forcing subsequent steps to operate with limited information. To mitigate this problem, we introduce Loopholing, a novel and simple mechanism that preserves this information via a deterministic latent pathway, leading to Loopholing Discrete Diffusion Models (LDDMs). Trained efficiently with a self-conditioning strategy, LDDMs achieve substantial gains-reducing generative perplexity by up to 61% over prior baselines, closing (and in some cases surpassing) the gap with autoregressive models, and producing more coherent text. Applied to reasoning tasks, LDDMs also improve performance on arithmetic benchmarks such as Countdown and Game of 24. These results also indicate that loopholing mitigates idle steps and oscillations, providing a scalable path toward high-quality non-autoregressive text generation.

Scaling Transformer-based click-through rate (CTR) models by stacking more parameters brings growing computational and storage overhead, creating a widening gap between scaling ambitions and the stringent industrial deployment constraints. We propose LoopCTR, which introduces a loop scaling paradigm that increases training-time computation through recursive reuse of shared model layers, decoupling computation from parameter growth. LoopCTR adopts a sandwich architecture enhanced with Hyper-Connected Residuals and Mixture-of-Experts, and employs process supervision at every loop depth to encode multi-loop benefits into the shared parameters. This enables a train-multi-loop, infer-zero-loop strategy where a single forward pass without any loop already outperforms all baselines. Experiments on three public benchmarks and one industrial dataset demonstrate state-of-the-art performance. Oracle analysis further reveals 0.02--0.04 AUC of untapped headroom, with models trained with fewer loops exhibiting higher oracle ceilings, pointing to a promising frontier for adaptive inference.

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.

Looping, reusing a block of layers across depth, and depth growing, training shallow-to-deep models by duplicating middle layers, have both been linked to stronger reasoning, but their relationship remains unclear. We provide a mechanistic unification: looped and depth-grown models exhibit convergent depth-wise signatures, including increased reliance on late layers and recurring patterns aligned with the looped or grown block. These shared signatures support the view that their gains stem from a common form of iterative computation. Building on this connection, we show that the two techniques are adaptable and composable: applying inference-time looping to the middle blocks of a depth-grown model improves accuracy on some reasoning primitives by up to 2times, despite the model never being trained to loop. Both approaches also adapt better than the baseline when given more in-context examples or additional supervised fine-tuning data. Additionally, depth-grown models achieve the largest reasoning gains when using higher-quality, math-heavy cooldown mixtures, which can be further boosted by adapting a middle block to loop. Overall, our results position depth growth and looping as complementary, practical methods for inducing and scaling iterative computation to improve reasoning.

We introduce LoopVL to study whether Loop Transformers can be effectively extended to vision- language models. LoopVL combines Module-Loop and Model-Loop computation to iteratively update a unified vision-language state through shared modules. We train LoopVL from scratch through language pre-training, multimodal training, and post-training. LoopVL outperforms a range of similarly sized and larger non-recurrent models on multimodal understanding and visual reasoning benchmarks. We also observe Visual Aha Moments in LoopVL, characterized by pronounced shifts in visual attention across loops. LoopVL provides practical evidence for recurrent vision-language modeling and offers an intuitive perspective on how shared parameters can support deeper multimodal computation over continuously evolving visual-language states.

We propose Ambient Dataloops, an iterative framework for refining datasets that makes it easier for diffusion models to learn the underlying data distribution. Modern datasets contain samples of highly varying quality, and training directly on such heterogeneous data often yields suboptimal models. We propose a dataset-model co-evolution process; at each iteration of our method, the dataset becomes progressively higher quality, and the model improves accordingly. To avoid destructive self-consuming loops, at each generation, we treat the synthetically improved samples as noisy, but at a slightly lower noisy level than the previous iteration, and we use Ambient Diffusion techniques for learning under corruption. Empirically, Ambient Dataloops achieve state-of-the-art performance in unconditional and text-conditional image generation and de novo protein design. We further provide a theoretical justification for the proposed framework that captures the benefits of the data looping procedure.

The proliferation of generative models, combined with pretraining on web-scale data, raises a timely question: what happens when these models are trained on their own generated outputs? Recent investigations into model-data feedback loops proposed that such loops would lead to a phenomenon termed model collapse, under which performance progressively degrades with each model-data feedback iteration until fitted models become useless. However, those studies largely assumed that new data replace old data over time, where an arguably more realistic assumption is that data accumulate over time. In this paper, we ask: what effect does accumulating data have on model collapse? We empirically study this question by pretraining sequences of language models on text corpora. We confirm that replacing the original real data by each generation's synthetic data does indeed tend towards model collapse, then demonstrate that accumulating the successive generations of synthetic data alongside the original real data avoids model collapse; these results hold across a range of model sizes, architectures, and hyperparameters. We obtain similar results for deep generative models on other types of real data: diffusion models for molecule conformation generation and variational autoencoders for image generation. To understand why accumulating data can avoid model collapse, we use an analytically tractable framework introduced by prior work in which a sequence of linear models are fit to the previous models' outputs. Previous work used this framework to show that if data are replaced, the test error increases with the number of model-fitting iterations; we extend this argument to prove that if data instead accumulate, the test error has a finite upper bound independent of the number of iterations, meaning model collapse no longer occurs.

As synthetic data becomes higher quality and proliferates on the internet, machine learning models are increasingly trained on a mix of human- and machine-generated data. Despite the successful stories of using synthetic data for representation learning, using synthetic data for generative model training creates "self-consuming loops" which may lead to training instability or even collapse, unless certain conditions are met. Our paper aims to stabilize self-consuming generative model training. Our theoretical results demonstrate that by introducing an idealized correction function, which maps a data point to be more likely under the true data distribution, self-consuming loops can be made exponentially more stable. We then propose self-correction functions, which rely on expert knowledge (e.g. the laws of physics programmed in a simulator), and aim to approximate the idealized corrector automatically and at scale. We empirically validate the effectiveness of self-correcting self-consuming loops on the challenging human motion synthesis task, and observe that it successfully avoids model collapse, even when the ratio of synthetic data to real data is as high as 100%.

Recent advances in multimodal large language models (MLLMs) have shown impressive reasoning capabilities. However, further enhancing existing MLLMs necessitates high-quality vision-language datasets with carefully curated task complexities, which are both costly and challenging to scale. Although recent self-improving models that iteratively refine themselves offer a feasible solution, they still suffer from two core challenges: (i) most existing methods augment visual or textual data separately, resulting in discrepancies in data complexity (e.g., over-simplified diagrams paired with redundant textual descriptions); and (ii) the evolution of data and models is also separated, leading to scenarios where models are exposed to tasks with mismatched difficulty levels. To address these issues, we propose C2-Evo, an automatic, closed-loop self-improving framework that jointly evolves both training data and model capabilities. Specifically, given a base dataset and a base model, C2-Evo enhances them by a cross-modal data evolution loop and a data-model evolution loop. The former loop expands the base dataset by generating complex multimodal problems that combine structured textual sub-problems with iteratively specified geometric diagrams, while the latter loop adaptively selects the generated problems based on the performance of the base model, to conduct supervised fine-tuning and reinforcement learning alternately. Consequently, our method continuously refines its model and training data, and consistently obtains considerable performance gains across multiple mathematical reasoning benchmarks. Our code, models, and datasets will be released.

Looped Transformers have shown exceptional neural algorithmic reasoning capability in simulating traditional graph algorithms, but their application to more complex structures like hypergraphs remains underexplored. Hypergraphs generalize graphs by modeling higher-order relationships among multiple entities, enabling richer representations but introducing significant computational challenges. In this work, we extend the Loop Transformer architecture's neural algorithmic reasoning capability to simulate hypergraph algorithms, addressing the gap between neural networks and combinatorial optimization over hypergraphs. Specifically, we propose a novel degradation mechanism for reducing hypergraphs to graph representations, enabling the simulation of graph-based algorithms, such as Dijkstra's shortest path. Furthermore, we introduce a hyperedge-aware encoding scheme to simulate hypergraph-specific algorithms, exemplified by Helly's algorithm. We establish theoretical guarantees for these simulations, demonstrating the feasibility of processing high-dimensional and combinatorial data using Loop Transformers. This work highlights the potential of Transformers as general-purpose algorithmic solvers for structured data.

Language models are increasingly used in scientific discovery to generate hypotheses, propose candidate solutions, implement systems, and iteratively refine them. At the core of these trial-and-error loops lies evaluation: the process of obtaining feedback on candidate solutions via verifiers, simulators, or task-specific scoring functions. While prior work has highlighted the importance of evaluation, it has not explicitly formulated the problem of how evaluation-driven discovery loops can be scaled up in a principled and effective manner to push the boundaries of scientific discovery, a problem this paper seeks to address. We introduce Simple Test-time Evaluation-driven Scaling (SimpleTES), a general framework that strategically combines parallel exploration, feedback-driven refinement, and local selection, revealing substantial gains unlocked by scaling evaluation-driven discovery loops along the right dimensions. Across 21 scientific problems spanning six domains, SimpleTES discovers state-of-the-art solutions using gpt-oss models, consistently outperforming both frontier-model baselines and sophisticated optimization pipelines. Particularly, we sped up the widely used LASSO algorithm by over 2x, designed quantum circuit routing policies that reduce gate overhead by 24.5%, and discovered new Erdos minimum overlap constructions that surpass the best-known results. Beyond novel discoveries, SimpleTES produces trajectory-level histories that naturally supervise feedback-driven learning. When post-trained on successful trajectories, models not only improve efficiency on seen problems but also generalize to unseen problems, discovering solutions that base models fail to uncover. Together, our results establish effective evaluation-driven loop scaling as a central axis for advancing LLM-driven scientific discovery, and provide a simple yet practical framework for realizing these gains.

We reformulate the problem of supervised learning as learning to learn with two nested loops (i.e. learning problems). The inner loop learns on each individual instance with self-supervision before final prediction. The outer loop learns the self-supervised task used by the inner loop, such that its final prediction improves. Our inner loop turns out to be equivalent to linear attention when the inner-loop learner is only a linear model, and to self-attention when it is a kernel estimator. For practical comparison with linear or self-attention layers, we replace each of them in a transformer with an inner loop, so our outer loop is equivalent to training the architecture. When each inner-loop learner is a neural network, our approach vastly outperforms transformers with linear attention on ImageNet from 224 x 224 raw pixels in both accuracy and FLOPs, while (regular) transformers cannot run.

Language-model agents that run long tasks often redo work they have already done: they re-issue a tool call they made earlier and get back exactly the same observation, a step that adds no new information. We identify this failure as a loop. Loops appear natively in Qwen, Llama, and Gemini agents and become far more common once the agent's history is compressed to fit its context window. To reduce these loops, we propose DeLoop, a self-supervised training method. DeLoop builds on a simple observation: the agent's own rollout can provide the supervision needed to learn from its loops. We use a looping action as a negative example and the first later non-looping action in the same rollout as a positive example. DeLoop learns from these pairs through direct preference optimization and does not require task rewards, gold answers, or LLM judges. On CodeGym under compression, DeLoop reduces the share of Qwen3-8B tasks containing loops from 57.3% to 6.1%. It also uses about one third as many steps per task, with similar task success. On Qwen3-32B, DeLoop reduces loops and steps while raising task success from 37.7% to 44.8%.

Loop invariants are fundamental to reasoning about programs with loops. They establish properties about a given loop's behavior. When they additionally are inductive, they become useful for the task of formal verification that seeks to establish strong mathematical guarantees about program's runtime behavior. The inductiveness ensures that the invariants can be checked locally without consulting the entire program, thus are indispensable artifacts in a formal proof of correctness. Finding inductive loop invariants is an undecidable problem, and despite a long history of research towards practical solutions, it remains far from a solved problem. This paper investigates the capabilities of the Large Language Models (LLMs) in offering a new solution towards this old, yet important problem. To that end, we first curate a dataset of verification problems on programs with loops. Next, we design a prompt for exploiting LLMs, obtaining inductive loop invariants, that are checked for correctness using sound symbolic tools. Finally, we explore the effectiveness of using an efficient combination of a symbolic tool and an LLM on our dataset and compare it against a purely symbolic baseline. Our results demonstrate that LLMs can help improve the state-of-the-art in automated program verification.

Looped Transformers reuse one block of layers several times: by spending extra computation they push a model of fixed size further, and so use its parameters more fully; while sparse mixture-of-experts (MoE) models activate only a few of many experts for each token. Looped MoE bridges these two design philosophies and gives MoE models new potential for better expert usage, but it raises a question: how to loop a MoE? We answer it with Foil. With the expert parameters and the expert compute per token held fixed, Foil (1) flattens the experts, halving the expert layers, doubling the experts per layer and doubling the passes, so that every routing decision chooses from a larger pool, and (2) unties the attention, giving each pass its own attention parameters while the experts and routers stay shared. Experiments show that Foil clearly outperforms the unflattened looped baseline: at 20B tokens every Foil model has lower pretraining loss than the baseline; at 100B tokens the loss improves monotonically with the degree of flattening, the most flattened Foil ending 0.012 nat below the baseline at equal parameters and compute, with downstream accuracy on par or better; untying the attention also yields more balanced and more confident routing at equal shape. Our ablations analyse why Foil works and turn the findings into design guidance for looped MoE: the returns of looping and of widening the expert layers amplify each other, routing confidence tracks healthy expert use better than load balance, and a sparse looped MoE should therefore use more experts per layer and more passes. Code and configurations are available at https://github.com/SR-A-W/how-to-loop-moe.

Loop Engineering is emerging as a practice for organizing development work around coding agents. Instead of writing each prompt by hand, practitioners design loops that monitor progress, assign work, run checks, and decide what the agent should do next. Even with a capable coding agent, a loop may trust a stale progress note, skip needed verification, spend its budget in the wrong direction, or stop before the task is safe to submit. Yet the final outcome of one end-to-end run cannot tell whether success or failure reflects the loop's guidance or the coding agent's ability to carry out the task. We introduce LoopArena, a benchmark for evaluating how well one model can guide a separate coding agent through a long-running task. The model under evaluation is the Controller: after each coding round, it receives a structured summary of the run and instructs a separate, fixed coding agent, the Worker, on what to do or verify next, or decides whether to stop. LoopArena evaluates this ability in three complementary settings that differ in execution scope and cost. Type I scores next-step Loop Contract selection through execution-validated questions without running the Worker at evaluation time. Type II executes repeated control over a selected slice of a full task, while Type III evaluates the paired full task from its original state. On full tasks, the best observed Strict Success Rate is 24.69\%, leaving substantial room for improvement in long-horizon loop control. Across Controllers, the paired reduction in estimated inference cost averages 64.4\%, and Type II produces a similar ordering under the main Core criterion (Spearman's \(ρ=0.9747\)). We release the benchmark data and evaluation code at https://github.com/AMAP-ML/LoopArena .

Looped transformers apply a shared block multiple times and have emerged as a parameter-efficient route to scaling compute in language models. However, at fixed FLOPs a looped model has strictly less capacity than a baseline transformer. We propose a novel dual-path block that can flexibly scale compute, the number of sequential operations applied to a hidden state, and capacity, the parameters available at a single step. For this we expose both axes as parallel pathways within a single layer: a deep sublayer re-applied K times with shared parameters, and a wide sublayer with an enlarged feed-forward network applied once. Independent per-token gates combine both axes and allow detailed per-token routing analyses. We show that across two FLOP budgets, our dual-path model surpasses iso-FLOP matched models on language modeling and downstream evaluations, while using fewer parameters than the baseline at matched FLOPs. The learned gates are directly interpretable and show systematic per-token allocation with function words and lexical content trend wide, while punctuation, symbols, and arithmetic tokens trend deep.

The success of large-scale language models like GPT can be attributed to their ability to efficiently predict the next token in a sequence. However, these models rely on constant computational effort regardless of the complexity of the token they are predicting, lacking the capacity for iterative refinement. In this paper, we introduce a novel Loop-Residual Neural Network, which achieves better performance by utilizing longer computational time without increasing the model size. Our approach revisits the input multiple times, refining the prediction by iteratively looping over a subset of the model with residual connections. We demonstrate the effectiveness of this method through experiments comparing versions of GPT-2 with our Loop-Residual models, showing improved performance in language modeling tasks while maintaining similar parameter counts. Importantly, these improvements are achieved without the need for extra training data.

Looped Transformers increase effective depth by iterating a shared block of layers, but most evaluations compare at fixed model size, conflating architectural advantage with extra FLOPs. We study looping on Mixture-of-Experts Transformers while closely matching per-token FLOPs, total non-embedding parameters, and KV cache. Through a series of ablations, we arrive at a recipe we call SMELT (Sparse MoE Transformer, middle layers Loop Twice), which loops the middle half of layers twice while matching the unlooped Baseline on all three budgets. We scale SMELT across four sizes up to 54B non-embedding parameters and fit a separate Chinchilla-style scaling law for each architecture. SMELT's loss drops faster with compute, saving 6.8--18.0\% of training FLOPs on the compute-optimal frontier. The advantage transfers to downstream benchmarks beyond what validation loss predicts, is largest on Code, and grows with sample length and the number of in-context examples. Mechanistic analysis shows that the second visit reduces the attention sink and redirects mass toward content-relevant tokens, an inductive bias that may underlie the observed performance gains. These results show that looping can improve Transformers even under budget matching, offering a practical recipe that turns depth reuse into measurable gains.

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.

Looped Transformers provide advantages in parameter efficiency, computational capabilities, and generalization for reasoning tasks. However, their expressive power regarding function approximation remains underexplored. In this paper, we establish the approximation rate of Looped Transformers by defining the modulus of continuity for sequence-to-sequence functions. This reveals a limitation specific to the looped architecture. That is, the analysis prompts the incorporation of scaling parameters for each loop, conditioned on timestep encoding. Experiments validate the theoretical results, showing that increasing the number of loops enhances performance, with further gains achieved through the timestep encoding.

Diffusion language models, especially masked discrete diffusion models, have achieved great success recently. While there are some theoretical and primary empirical results showing the advantages of latent reasoning with looped transformers or continuous chain-of-thoughts, continuous diffusion models typically underperform their discrete counterparts. In this paper, we argue that diffusion language models do not necessarily need to be in the discrete space. In particular, we prove that continuous diffusion models have stronger expressivity than discrete diffusions and looped transformers. We attribute the contradiction between the theoretical expressiveness and empirical performance to their practical trainability: while continuous diffusion provides intermediate supervision that looped transformers lack, they introduce additional difficulty decoding tokens into the discrete token space from the continuous representation space. We therefore propose Coevolutionary Continuous Discrete Diffusion (CCDD), which defines a joint multimodal diffusion process on the union of a continuous representation space and a discrete token space, leveraging a single model to simultaneously denoise in the joint space. By combining two modalities, CCDD is expressive with rich semantics in the latent space, as well as good trainability and sample quality with the help of explicit discrete tokens. We also propose effective architectures and advanced training/sampling techniques for CCDD, which reveals strong empirical performance in extensive language modeling experiments on real-world tasks.

A common and effective means for improving language model capabilities involves finetuning a ``student'' language model's parameters on generations from a more proficient ``teacher'' model. Termed ``synthetic data'', these generations are often produced before any student finetuning, but some work has considered generating new synthetic samples as training progresses. This paper studies and advocates for the latter case, where data are generated in an iterative, closed-loop fashion that is guided by the current state of the student model. For a fixed budget of generated samples, or a budget in terms of compute spent querying a teacher, we show that this curation of finetuning data affords improved student performance over static generation. Further, while there have been several LLM-specific methods proposed that operate in this regime, we find that simple, inexpensive selection criteria from the active learning literature tend to be most performant. We validate these claims across four mathematical and logical reasoning datasets using four different small language models.

Industrial recommender systems face two fundamental limitations under the log-driven paradigm: (1) knowledge poverty in ID-based item representations that causes brittle interest modeling under data sparsity, and (2) systemic blindness to beyond-log user interests that constrains model performance within platform boundaries. These limitations stem from an over-reliance on shallow interaction statistics and close-looped feedback while neglecting the rich world knowledge about product semantics and cross-domain behavioral patterns that Large Language Models have learned from vast corpora. To address these challenges, we introduce ReaSeq, a reasoning-enhanced framework that leverages world knowledge in Large Language Models to address both limitations through explicit and implicit reasoning. Specifically, ReaSeq employs explicit Chain-of-Thought reasoning via multi-agent collaboration to distill structured product knowledge into semantically enriched item representations, and latent reasoning via Diffusion Large Language Models to infer plausible beyond-log behaviors. Deployed on Taobao's ranking system serving hundreds of millions of users, ReaSeq achieves substantial gains: >6.0% in IPV and CTR, >2.9% in Orders, and >2.5% in GMV, validating the effectiveness of world-knowledge-enhanced reasoning over purely log-driven approaches.

Recently, it was shown that small, looped architectures, such as Tiny Recursive Models (TRMs), can outperform Large Language Models (LLMs) on complex reasoning tasks, including the Abstraction and Reasoning Corpus (ARC). In this work, we investigate a core question: how can we further improve the efficiency of these methods with minimal changes? To address this, we frame the latent reasoning of TRMs as a form of classifier-free guidance and implicit policy improvement algorithm. Building on these insights, we propose a novel training scheme that provides a target for each loop during training. We demonstrate that our approach significantly enhances training efficiency. Our method reduces the total number of forward passes by 18x and eliminates halting mechanisms, while maintaining quality comparable to standard TRMs. Notably, we achieve 24% accuracy on ARC-1 with only 0.8M parameters, outperforming most LLMs.

Looped transformers reuse a shared block of layers to increase computational depth without increasing the number of trainable parameters. They are trained with backpropagation through time (BPTT). To save training compute, BPTT is often truncated to the last few loops, but truncation yields biased gradients that may weaken the benefit of looping. In this work, we study the value of truncation in looped transformer training in a systematic way: we jointly vary model size, training tokens, and forward and backward loop counts across nearly 600 training runs and fit a unified scaling law that captures the effects of looping and truncation through effective parameters. Despite the training-compute savings of truncation, our fitted law favors full BPTT when forward and backward loop counts are jointly optimized. In matched-compute runs, full BPTT through fewer loops consistently outperforms truncated settings of the same compute. The law further shows that at fixed model size, the compute-optimal loop count and training-token budget both grow with compute. When model size is also free, the law identifies two fully trained loops as the best looped setting. Experimentally, this setting achieves lower validation loss than the compute-optimal non-looped model with about 27% fewer non-embedding parameters and only 8% more training compute.

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.

Looped Transformers scale latent computation by repeatedly applying shared blocks, but sequential looping increases latency and KV-cache memory with the loop count. Parallel loop Transformers (PLT) alleviate this cost through cross-loop position offsets (CLP) and shared-KV gated sliding-window attention, making loop count a practical design choice. We therefore study PLT loop-count selection through a gain--cost view: an extra loop may refine representations, but CLP also introduces a positional mismatch at each loop boundary. We instantiate this study by training LoopCoder-v2, a family of 7B PLT coders with different loop counts, from scratch on 18T tokens, followed by matched instruction tuning and evaluation. Empirically, the two-loop variant delivers broad gains over the non-looped baseline across code generation, code reasoning, agentic software engineering, and tool-use benchmarks, improving SWE-bench Verified from 43.0 to 64.4 points and Multi-SWE from 14.0 to 31.0 points. In contrast, variants with three or more loops regress, revealing a strongly non-monotonic loop-count effect. Our diagnostics show that loop 2 provides the main productive refinement, while later loops yield diminishing, oscillatory updates and reduced representational diversity. Because the CLP-induced mismatch remains roughly fixed as refinement gains shrink, the offset cost increasingly dominates. This gain--cost trade-off explains PLT's saturation at two loops and provides diagnostics for loop-count selection.

Using the weights of trained Neural Network (NN) models as data modality has recently gained traction as a research field - dubbed Weight Space Learning (WSL). Multiple recent works propose WSL methods to analyze models, evaluate methods, or synthesize weights. Weight space learning methods require populations of trained models as datasets for development and evaluation. However, existing collections of models - called `model zoos' - are unstructured or follow a rudimentary definition of diversity. In parallel, work rooted in statistical physics has identified phases and phase transitions in NN models. Models are homogeneous within the same phase but qualitatively differ from one phase to another. We combine the idea of `model zoos' with phase information to create a controlled notion of diversity in populations. We introduce 12 large-scale zoos that systematically cover known phases and vary over model architecture, size, and datasets. These datasets cover different modalities, such as computer vision, natural language processing, and scientific ML. For every model, we compute loss landscape metrics and validate full coverage of the phases. With this dataset, we provide the community with a resource with a wide range of potential applications for WSL and beyond. Evidence suggests the loss landscape phase plays a role in applications such as model training, analysis, or sparsification. We demonstrate this in an exploratory study of the downstream methods like transfer learning or model weights averaging.