Recurrent Language Models

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

This manuscript presents DeepThink, a knowledge-addressable framework for recurrent language modeling. A Knowledge Bank stores typed Knowledge Units, a Knowledge Activator selects units from the current execution state, Knowledge Assembly produces an executable configuration, and a Shared Thinking Operator updates that state. An output stage then organizes generation. This organization exposes knowledge capacity, activation width, and recurrent depth as separate design variables, supporting both complete parameter packages and fine-grained expert or low-rank modules. We formalize the interfaces and their execution semantics, including state-dependent selection, parameter-space assembly, and thinking followed by generation. For compatible templates and complete runtime-state configurations, we establish an exact Fixed-Schedule Realization of an existing fixed-depth model. Exact shared-base reparameterization and pretrained output-tail reuse provide complementary initialization constructions. Together, these results define a route from pretrained models to dynamically selected and assembled recurrent computation. A synthesis of conditional computation, recurrent models, modular parameters, and knowledge interfaces positions DeepThink within the research landscape and identifies the design questions arising from their integration.

A recurrent-depth language model reuses one block of weights several times in a row. The number of repetitions, which we call loops, is chosen when the model runs, not when it is trained, and running fewer is cheaper and faster, so a deployment may choose to do it. Running fewer loops also makes these models less safe, and the loss of safety is larger than the loss of anything else we measured. We study four released Ouro checkpoints and Huginn-0125. At half of its trained depth, Ouro-1.4B-Thinking still solves 0.692 of the GSM8K maths problems, against 0.775 at full depth, keeping 89% of its accuracy. Over the same reduction, the rate at which it helps with harmful requests rises from 0.038 to 0.205, 5.3 times higher. The information is not lost: a linear classifier reads whether a request is harmful from the model's internal state after a single loop, with an ROC AUC of 0.967 against 0.986 at full depth. What the missing loops remove is the refusal that should follow. We measure the effect with four instruments, one generating no text at all, and reproduce it on a second model family. Removing a single direction from Ouro-1.4B's internal state at full depth drops its refusal rate on harmful prompts from 0.950 to 0.130, identifying what the shallow model is missing. Two run-time repairs do not help: the models' early-exit rules do not spend more loops on harmful requests, and choosing a depth per request does not recover the loss. Changing the training does. Standard safety training makes the shallow model safe only by refusing 0.865 of harmless requests, while training at randomly chosen depths reaches a separation of +0.399 between harmful and harmless refusal at one loop, against +0.118. On Huginn-0125 it keeps harmful help at or below 0.040 at every depth from 1 to 32 loops, where standard training reaches 0.194, and it costs no accuracy at full depth.

Recurrent-depth language models improve inference-time reasoning by repeatedly applying a shared network to refine internal representations, but the additional recurrent computation also increases inference cost. This overhead is particularly pronounced during long-context prefill, where the full input sequence is repeatedly processed across recurrent depth. Weight-and-activation quantization offers a natural way to accelerate this computation through lower-precision arithmetic, yet repeatedly applying quantized computation introduces numerical perturbations that accumulate along the recurrent trajectory. We systematically analyze how such errors propagate across recurrent depth and show that quantization errors introduced earlier are attenuated by more subsequent recurrent steps and therefore have a smaller effect on the final representation. Matched-budget experiments strongly support this prediction, showing that placing quantized computation earlier in recurrent depth consistently yields the best fidelity. Based on this insight, we introduce RAMP (Recurrent Adaptive Mixed Precision), which performs early recurrent refinement using low-precision weight-and-activation computation and restores high precision for terminal refinement. Across context lengths up to 4K, RAMP preserves near-full-precision prefill quality while achieving up to 1.16x end-to-end first-prefill speedup on an NVIDIA H100, corresponding to a 13.9% latency reduction.

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.

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 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.

Scaling the horizon of agentic LLMs is bottlenecked by the need to fit ever longer context traces in GPU memory. Context compaction has been the most popular mechanism to alleviate this issue, keeping GPU memory constant for a given trace. Unfortunately, most compaction strategies rely on prefilling the LLM context many times over, hindering training throughput. To alleviate this bottleneck and enable efficient trainable compaction, we propose KV-streams, a plug-and-play strategy compatible with any compaction strategy that substantially increases throughput while showing no evidence of hindering performance. KV-streams enable scalable compaction by streaming the KV cache forward rather than flushing it after each compaction. We show that KV-streams enable three different compaction strategies, achieving a 2.6 to 5x wall-clock speedup in training. Beyond efficiency, we find that the streamed KV cache can act as a recurrent state, carrying forward information that has long since disappeared from the context. Specifically, in a controlled setting we show that, contrary to prior work, RL alone is all that is needed for this behavior to emerge. Overall, we show KV-streams to be an efficient and lightweight plug-and-play addition to any post-training pipeline.

Language modeling using Transformers has become commonplace despite their fixed computational depth and quadratic runtime with respect to input tokens. Recurrent models on the other hand offer linear depth but no parallel execution. In this work, we extend balanced-tree recursive operators from sequence encoding to autoregressive prediction, enabling all prefix representations to be computed with logarithmic depth and linear runtime. Our experiments provide an initial characterization of this model class, demonstrating robust length extrapolation and performance approaching that of ALiBi-based Transformers, highlighting its potential as an alternative architecture for language modeling.

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.

In streaming tasks, recurrent models can carry latent computation across time, allowing each update to build on representations produced earlier. This raises a basic question: once temporal recurrence provides sequential computation across steps, how much depth is still needed within each step? Prior work has shown that recurrence can make shallow models competitive. We instead study this question as a compute-allocation problem, varying within-step depth, expert width, and the number of parallel experts per layer across several compute budgets. For each budget, we compare the best observed recurrent and non-recurrent allocations and the performance they achieve under approximately matched per-step computation. Across Sokoban and autoregressive FineWeb language modeling, we find that temporal recurrence shifts the best observed compute allocation toward substantially fewer layers, with comparable or better performance.

Recurrent fast-weight memories and selective state-space models compress an expanding context into a fixed-size recurrent state, making the state transition an online learning rule. We study this rule under read-after-write autoregressive semantics. For the prefix-prediction objective considered here, the local fast-memory example revealed at step $t$ is the prefix-aligned pair $(\mathbf{x}_t,\mathbf{y}_t)=(φ(\mathbf{k}_{t-1}),\mathbf{v}_t)$. The common same-step association $(φ(\mathbf{k}_t),\mathbf{v}_t)$ remains causal, but optimizes a different internal objective. We derive normalized first-order updates for squared-error regression and negative inner-product objectives. The regression family comprises Falcon-1 (a scalar NLMS update), Falcon-2 (its per-column extension), and Falcon-3 (a sliding-window mini-batch update); Falcon-1A/Falcon-2A/Falcon-3A are the corresponding inner-product variants. We provide recurrent, masked-parallel, and chunk-parallel forms, together with numerically stable positive-decay renormalization. Representative variants remain competitive in language modeling and improve length extrapolation on variable-digit addition. This framework separates temporal alignment, plasticity, forgetting, and bounded rehearsal in recurrent sequence models.

Looped language models improve reasoning and knowledge manipulation by applying shared computation repeatedly. Existing systems usually repeat an entire layer stack, although a mixer and a dense feed-forward network (FFN) perform different operations and have different costs. We ask a narrower question: what should loop? We view recurrence as repeated composition of a state update and argue that an application is valuable when it exposes a new cross-position influence direction that remains observable at the task readout. Iterative Transport Rank (ITR) describes the cumulative influence trajectory; marginal ITR describes the nonredundant influence contributed by successive applications. This view motivates MixerLoop, which repeats each Gated DeltaNet mixer while applying its dense FFN once. We compare MixerLoop with no recurrence and full-block recurrence at 15M and 110M parameters under the same data, initialization, and architecture. A finite context-off intervention tests whether later mixer applications produce distinct, non-negligible, and beneficial changes at the final language-model readout. MixerLoop surpasses FullLoop on aggregate CORE at 15M and retains 41.5% of its CORE improvement at 110M while reducing recurrent-backbone projection FLOPs by 45.9%. These results show that the benefits of recurrent depth can be retained without repeatedly executing the dense FFN.

A dense, pretrained language model can be retrofitted with recurrent depth and learn an iterative latent transition that persists after outcome-only annealing. Qwen2.5-0.5B-Instruct is split into a Prelude, a weight-tied Recurrent Block, and a Coda, with an identity-preserving one-loop path and a re-entry bridge on later loops. At loop 1 the retrofit remains non-inferior to its base on a preregistered ARC battery. Three findings. First, the mechanism is a reusable procedure rather than terminal-answer lookup, and installs at two budgets: 6M trained parameters over frozen base weights and 180M full-block. With intermediate-step supervision, the model computes one task step per loop and persists when only final answers are graded. The adapter matched the full block overall (83.8% versus 84.0%), led through depth 11, and trailed beyond. Verbal fine-tuning reached 79-86% on controlled verbal renderings (zero-shot transfer was minimal), and adapter verbal training begun from the installed mechanism outpaced matched fresh training by 18.6 points, including on a held-out test set. Second, the operation extrapolates to roughly 1.5 times its supervised depth, holding 70% accuracy through depth 18. Third, a same-size scratchpad-trained model matched the recurrent model within its learned horizon but collapsed beyond it. The recurrent model won overall, 84% versus 72%, retained 53% versus 2.5% beyond depth 10, and answered 7.6 times faster. An iterative transformer can therefore perform deeper reasoning in latent space faster than comparable or larger models fine-tuned on the same task, in a system-level comparison. A second task, running the rule in reverse, exposed the limits: the inverse was learnable in isolation, but no continuation acquired it while preserving the installed mechanism and general capability, a catastrophic-interference boundary. Learned depth selection remains open.

Training recurrent neural networks (RNNs) requires assigning credit across long sequences of computations. Standard backpropagation through time (BPTT) addresses this problem poorly: it is sequential in time, limiting parallelism, and suffers from vanishing or exploding gradients, making long-range associations difficult to learn. We propose Supervised Memory Training (SMT), a method for training nonlinear RNNs that sidesteps recurrent credit propagation entirely by reducing RNN training to supervised learning on one-step memory transition labels $(m_t, x_{t+1}) \rightarrow m_{t+1}$. SMT acquires these memory labels by training a Transformer-based encoder on a predictive state objective--retaining only information from the past necessary to predict the future. By decoupling what to remember from how to update memory, SMT enables time-parallel RNN training with a stable $O(1)$ length gradient path between any two tokens--without ever unrolling the RNN. We find that SMT outperforms BPTT when pretraining various RNN architectures on tasks like language modeling and pixel sequence modeling. SMT enables nonlinear RNNs to better capture long-range dependencies and train in parallel, potentially unlocking the scaling of models that build temporal abstractions of past experience.

What formal languages can a recurrent neural language model recognize? Formal results in the literature conflict: some authors report Turing-completeness, while others show equivalence to regular languages. The reason for this discrepancy is that the underlying arithmetic model differs. The paper develops a unified algebraic account of the expressivity of recurrent neural networks, starting with a formal account of various arithmetic models. This account reduces expressivity to an algebraic question, e.g., whether a network's syntactic monoid divides a certain wreath product. As a case study, the paper revisits diagonal state-space models: the same architecture cannot implement an even-modulus counter once floating-point recurrences are enforced, yet realizes every even-modulus counter under unsigned-integer quantization.

We introduce WriteSAE, a sparse autoencoder for the matrix updates written into recurrent language-model state. In Gated DeltaNet, Mamba-2, and RWKV-7, each token writes a matrix-shaped update to a recurrent cache; a residual-stream SAE has vector-shaped atoms and cannot replace that update directly. WriteSAE learns rank-1 matrix atoms with the same shape as the model's own write. This lets us test a direct replacement: at positions where the SAE activates an atom, we remove the model's write, insert the atom scaled by its SAE activation, and continue the forward pass. The atom gives a closer final token distribution than deleting the write on 92.4% of evaluated positions; averaged per atom, the rate is 89.8%. For Gated DeltaNet, a formula using the forget gate, read query, and output embedding predicts the resulting logit change with $R^2 = 0.98$. The same replacement test transfers to Mamba-2-370M at 88.1%. In generation, the formula chooses a write direction; writing it into three consecutive cache positions at $3\times$ the norm of the model's write makes tokens initially ranked 100--1000 by the unmodified model appear in 100% of continuations, up from 33.3%. To our knowledge this is the first cache-level steering intervention reported in a state-space or hybrid recurrent layer.

Modern language models scale depth by stacking layers, each holding its own state - a per-layer KV cache in transformers, a per-layer matrix in Mamba, Gated DeltaNet (GDN), RWKV, and xLSTM. Biological systems lean heavily on recurrence rather than on stacking. We ask how far that shape can go on language modeling. We propose Grounded Prediction Networks (GPN): one state vector revisited at every step through a single recurrent block - one FFN, one shared matrix memory. At 130M parameters, a 1-layer GPN+M reaches FineWeb-Edu perplexity 18.06, within 13% of a 12-layer Transformer++ (16.05) and 18% of a 10-layer GDN (15.34); a 2-layer variant closes the gap to 6%/11%. We do not match the deep baselines. Because the working context is a single vector, we can directly inspect its geometry: a persistent default-token direction, a content-bearing horizon of tens of tokens, and memory heads that split spontaneously into fast and slow retention pools.

Over the last two decades, language modeling has experienced a shift from the use of predominantly recurrent architectures that process tokens sequentially during training and inference to non-recurrent models that process sequence elements in parallel during training, which results in greater training efficiency and stability at the expense of lower inference throughput. Here we introduce the Structured Recurrent Mixer, an architecture that allows for algebraic conversion between a sequence parallel representation at train time and a recurrent representation at inference, notably without the need for specialized kernels or device-specific memory management. We show experimentally that this dual representation allows for greater training efficiency, higher input information capacity, and larger inference throughput and concurrency when compared to other linear complexity models. We postulate that recurrent models are poorly suited to extended sequence length scaling for information-rich inputs typical of language, but are well suited to scaling in the sample (batch) dimension due to their constant memory per sample. We provide Mojo/MAX inference implementations of SRMs exhibiting 12x the throughput and 170x the concurrency of similarly powerful Transformers inferenced on vLLM, increases characteristic of Pytorch implementations resulting in a 30% increase in compute-constant GSM8k Pass@k. We conclude by demonstrating that SRMs are effective reinforcement learning training candidates.

We introduce MinMax Recurrent Neural Cascades (MinMax RNCs), a class of recurrent neural networks built from a novel form of recurrence over the MinMax algebra. We show that MinMax RNCs enjoy key properties that are difficult to obtain simultaneously: strong formal expressivity, efficient evaluation, stable dynamics, and non-vanishing state gradients. First, their formal expressivity corresponds to the regular languages, arguably the maximal expressivity for finite-memory systems. Second, in addition to evaluation in recurrent form, they also admit parallel-scan evaluation with logarithmic depth and linear work in the input length. Third, their states and activations are uniformly bounded for all sequence lengths. Fourth, their loss gradients exist almost everywhere and are uniformly bounded for all sequence lengths. Fifth, they do not exhibit vanishing state gradients: the gradient of a state with respect to a past state can retain norm one independently of the temporal distance between the states. Empirically, we find that these theoretical properties translate into strong practical performance. MinMax RNCs solve the considered synthetic tasks perfectly, generalise to long sequences, and outperform the recurrent baselines considered in our experiments. We also train a 112M-parameter MinMax RNC for next-token prediction, obtaining competitive performance for its size and providing initial evidence that MinMax recurrence can scale to real-world sequence-modelling tasks.

Transformer-based embedding models suffer from quadratic computational and linear memory complexity, limiting their utility for long sequences. We propose recurrent architectures as an efficient alternative, introducing a vertically chunked inference strategy that enables fast embedding generation with memory usage that becomes constant in the input length once it exceeds the vertical chunk size. By fine-tuning Mamba2 models, we demonstrate their viability as general-purpose text embedders, achieving competitive performance across a range of benchmarks while maintaining a substantially smaller memory footprint compared to transformer-based counterparts. We empirically validate the applicability of our inference strategy to Mamba2, RWKV, and xLSTM models, confirming consistent runtime-memory trade-offs across architectures and establishing recurrent models as a compelling alternative to transformers for efficient embedding generation.

In this report, we introduce the IQuest-Coder-V1 series-(7B/14B/40B/40B-Loop), a new family of code large language models (LLMs). Moving beyond static code representations, we propose the code-flow multi-stage training paradigm, which captures the dynamic evolution of software logic through different phases of the pipeline. Our models are developed through the evolutionary pipeline, starting with the initial pre-training consisting of code facts, repository, and completion data. Following that, we implement a specialized mid-training stage that integrates reasoning and agentic trajectories in 32k-context and repository-scale in 128k-context to forge deep logical foundations. The models are then finalized with post-training of specialized coding capabilities, which is bifurcated into two specialized paths: the thinking path (utilizing reasoning-driven RL) and the instruct path (optimized for general assistance). IQuest-Coder-V1 achieves state-of-the-art performance among competitive models across critical dimensions of code intelligence: agentic software engineering, competitive programming, and complex tool use. To address deployment constraints, the IQuest-Coder-V1-Loop variant introduces a recurrent mechanism designed to optimize the trade-off between model capacity and deployment footprint, offering an architecturally enhanced path for efficacy-efficiency trade-off. We believe the release of the IQuest-Coder-V1 series, including the complete white-box chain of checkpoints from pre-training bases to the final thinking and instruction models, will advance research in autonomous code intelligence and real-world agentic systems.

Test-time scaling via recurrent/iterative Transformers enables large language models to spend more computation at inference, but most pretrained recurrent LMs run a fixed number of iterations, wasting compute on easy tokens and lacking token-wise adaptivity. Following the core idea of Adaptive Computation Time(ACT) and Early Exit(EE), we propose AdaPonderLM, a self-supervised recurrent language model that learns token-wise early exiting during pretraining without manually tuned per-token/per-layer pruning ratios. AdaPonderLM uses iteration-specific MLP gates with a monotonic halting mask to decide when each token stops recurring, and introduces a KV reuse mechanism that reuses cached key/value states for halted tokens, ensuring train--test consistency and practical acceleration. Across Pythia backbones from 70M to 410M (pretraining) and up to 2.8B (continued pretraining), AdaPonderLM reduces inference compute at about 10% while maintaining comparable language modeling perplexity and competitive downstream accuracy. Our analysis shows the learned gates allocate more computation to high-NLL (hard) tokens, exhibiting adaptive computation time behavior in a fully self-supervised setting. Meanwhile, under iso-FLOPs, the learned halting policy consistently outperforms fixed pruning, showing AdaPonderLM allocates compute to the right tokens rather than just reducing average depth.