LoopFold: Parallelizing Recurrent Depth in Looped Transformers
Cuts the sequential latency of deep looped models by folding several recurrent depths into one parallel batch for the same prefix. A lightweight predictor guesses the inputs of later loops so they can start early, and a corrector refines them once earlier loops finish; everything is trained with the ordinary next-token loss.
StateLoop: Persistent State for Looped Transformers
Adds one persistent state per token that survives across loop iterations. A learned gate decides how much of the old state to keep and how much of the new backbone output to absorb, and the state is projected back into the stream for later iterations and the final prediction.
Cross-Loop Query Handoff: How Later Loops Redirect Two-Hop Retrieval in the Ouro Looped Language Models
A mechanistic study of how later loops in Ouro repair two-hop retrieval. Transplanting loop-3 attention queries into loop 1 rescues most questions that loop 1 got wrong, while late keys and values help far less, pointing to queries as the carrier of the later-loop correction.
LoopGR: Parameter-Efficient Scaling for Generative Recommendation via Recurrent Depth
Brings recurrent depth to generative recommendation with a hierarchical mechanism that alternates a shared generative decoder and a lightweight context updater, plus recurrent trajectory self-distillation from a high-budget teacher to a low-budget student.
Recursive CLIP: Vision-Language Distillation with Looped Transformer Blocks
Asks how much block reuse CLIP tolerates. Recursive backbones apply N blocks M times with N·M fixed at ViT-B depth, either in the vision encoder only or in both encoders, trained contrastively or by distillation from a strong ViT-B/16 teacher.
Prefix Keys and Values Matter Late: RIPE Caching for Self-Speculative Decoding in Looped Language Models
Shows that in recurrent-depth models the KV planes of late loops matter far more than early ones and converge geometrically. RIPE therefore stores only the deepest KV plane for most positions and keeps full depth only for a recent window, which suits self-speculative decoding with few-loop drafts and full-depth verification.
ScaleRLT: Exact Log-Probabilities and Fast Deferred Rollouts for Reinforcement Learning on Looped Transformers
Makes on-policy RL practical for looped language models such as Ouro. Rollouts defer and batch the deeper loops, and the learner uses a hybrid log-probability whose value replays the rollout schedule exactly while its gradient comes from a dense pass, removing the floating-point mismatch between sampler and learner.
One Model, Two Paths: Balancing Single-Pass and Looped Performance in Language Models
Adapts a pretrained model so it can run either a single pass or a looped pass that reuses its middle layers. Joint weighted losses on both outputs, plus guidance from the frozen original model, stop looped training from degrading single-pass quality.
Accordion: Elastic-Depth Inference via Fixed-Prefix Layer Looping
A mixture-of-experts model that serves five inference budgets through fixed-prefix layer looping, with the smallest budget equal to the unlooped model. Pretraining samples one budget per micro-batch, and SFT and RL with budget-matched multi-teacher distillation preserve every budget.
Do Halting Heads Really Save Compute in Looped Language Models?
Pretrains looped models up to 1.3B parameters to test learned halting heads. Heads trained on a mix of iterations predict early stops yet tokens run almost every iteration under the usual rule, and the commonly reported average-iterations metric overstates savings in batched generation.
LSTM-Style Looped Transformers with Shared KV Caches
Studies an LSTM-style gated recurrent transformer whose recurrent core also shares a single KV cache. A consistency loss late in training aligns the cache entries seen in parallel training with those produced recursively at generation time.
BeNice! Closing Looping's Inference-Memory Gap via KV-Sharing
Makes looped models KV-shareable so earlier recurrences can reuse the final recurrence's cache. Stochastic depth during pretraining induces contractive dynamics, and the BeNice! post-training step broadcasts the final KV cache into the first recurrence while preserving quality.
Agreement rather than normalisation keeps depth reusable in looped language models
Asks what keeps extra loop steps usable beyond the trained budget. Removing Ouro's loop-boundary normalisation breaks the released models without norm blow-up, yet small models trained without it extrapolate better, suggesting that agreement between the training and inference loop is what matters.
Loop to Look Better: Learning Adaptive Depth for Question-Guided Perception in Vision-Language Models
Adds adaptive recurrent perception to pretrained vision-language models by revisiting a middle-to-late window of the vision encoder with the question injected at each loop. A learned stopper sets the depth and a token-wise Mixture-of-Looks blends features from different loops.
Applies selective looping, which repeats a few early layers inside each denoising step, to uniform diffusion language models without changing their objective or sampler.
Asymmetric Looped Transformers: Fixed Global Representations for an Iterative Decoder
Separates context building from iterative refinement: a single-pass causal encoder writes fixed per-layer global KV caches, and a weight-tied recurrent decoder attends jointly to them and to local KV from its evolving state.
PAR: Progress-Aligned Recurrence for Length Generalization in State Tracking
Progress-Aligned Recurrence makes extra iterations useful for longer state-tracking problems by supervising the prediction at iteration k to match the state after k operations and anchoring recurrent state norms to the first iteration. A learned completion readout decides when to answer.
How Much Gradient Does Truncated Backprop Lose in Looped Transformers?
Measures exactly how much weight gradient truncated backprop discards for every truncation length, across nine looped language models including Parcae and Ouro.
Internalizing Chain-of-thought into Looped Transformers
A theoretical construction of constant-bit-size looped transformers that interpolates between chain-of-thought and latent recurrence, trading generated tokens for loop iterations when simulating Turing machines.
recAMPLIFY: Large-Scale Pretraining of Recursive Protein Language Models
Adapts recursive pretraining to masked protein language models: a weight-tied core refines residue-aligned states over an adjustable number of cycles. Gated Mixture Recurrence then decodes from a probability-weighted mixture of visited states with a learned residual gate, unifying halting and gating.
Spherical Fixed-Point Reasoners: Adaptive Loop Halting on the Hypersphere
Spherical Fixed-Point Reasoners move fixed-point looped reasoning onto the unit hypersphere of normalized transformers, turning input injection, damping and halting into geodesic operations that are bounded by construction.
Truncate or Not? A Scaling Study of Looped Transformers with Truncated Backpropagation
A scaling study of truncated backpropagation through time in prelude–recurrent–coda looped transformers. Nearly 600 runs jointly vary model size, tokens and forward/backward loop counts, and a unified law expresses looping and truncation through effective parameters.
Block Recursive Transformers for Structured Parameter Sharing
Argues that how shared layers are organized matters. BRT splits the stack into contiguous blocks and recurses each block before moving to the next, an intermediate topology between per-layer repetition and looping the entire stack.
Dual-Slot: From Serial Loops to Parallel Slots in Language Models
Reorganizes serial looped computation into parallel latent and prediction slots. The prediction slot uses the previous token's deep latent state, while the latent slot prepares the state for the next step, replacing two sequential invocations with one paired call.
UpLoop: Upcycling Large Language Models with Gated Loop
A post-training recipe that upcycles a pretrained LLM into a looped one by looping selected layers locally rather than the whole stack, with hidden-wise gates on attention and MLP residuals so later loops refine instead of rewrite.
Gives a shared-depth Transformer a dedicated causal workspace: at each of four rounds, positions write gated vectors into prefix-specific latent slots and read back a content-addressed summary before the next application of the shared block, while the input embedding stays accessible.
Inside the Loop: A Mechanistic Study of Weight-Tied Transformers on Depth-Bound Algorithmic Tasks
A mechanistic study on multi-hop permutation traversal, where every intermediate hop has a known answer. Looped models advance through the hops gradually, forming a staircase that adapts to the iteration budget, and behave like iterative program learners rather than simulated deeper stacks.
Towards Looped Models Done Right, Part I: Topology, Input Injection, Recurrent-State Design
Disentangles three design axes of looped language models trained from scratch at matched parameters, logical depth and tokens: where recurrence sits, how input is injected, and how recurrent state is organized. The prelude–recurrent–coda sandwich and input injection come out ahead, favoring the Huginn-style design over Ouro.
LoopAhead: Speculating Ahead for Faster Decoding in Looped Language Models
A speculative decoding framework for looped transformers that gives different recurrent depths complementary roles: a lightweight lookahead drafter predicts several future tokens from early-loop states, deeper-loop feedback decides which draft branches to continue, and the full-depth model makes final decisions.
Retrofitting Recurrent Vision-Language Models from Pretrained Backbones
Retrofits depth recurrence into a pretrained vision-language model by pruning language-backbone layers and looping a subset of the rest, with a linear adapter merging each loop's output with earlier token representations. A three-stage recovery with distillation from the original model restores quality.
Learning to Compute Is Not Learning to Retain or Reuse in Looped Transformers
Separates three causes of overthinking in looped transformers on in-context Mealy-machine execution: failed computation, lost retention and non-reusable states. The computation itself is exact, but answers decay with extra loops unless training asks for retention.
LatentLoop: Scaling Looped MoE Models with Asymmetric Compute Allocation
Finds that in a looped MoE that applies a shared block twice, the first visit can be scaled down without harm while the second cannot. LatentLoop compresses the sequence on the first visit, keeps full resolution on the second, and spends the savings on a larger hidden size.
LEV: Thinking Fast and Deep with Looped Decision Models
A decision model that answers typed multiple-option questions in one pass but can think deeper by rerunning its layers in a loop instead of generating text. It reads the options after every loop step, so one checkpoint answers at any depth, and a small stopper sets depth per question.
Links linear attention to dynamic mode decomposition to show that looped transformers can emulate propagators of dynamical systems in context, and builds a framework in which a one-step forecaster uses gradient-descent iterations as recurrent depth.
LoopREG: Representation Supervision Enables Test-Time Recurrent Depth in Diffusion Transformers
A recurrent diffusion Transformer that repeatedly applies a shared block at a state aligned to a pretrained visual encoder after the final recurrent visit, so inference can extend the same operator to additional visits.
LoOPSD: On-Policy Self-Distillation with Looped Teachers
Builds a stronger teacher from the same frozen weights by searching which layers to repeat, then distills that looped teacher's predictions on student-generated prefixes back into the student, alternating search and distillation over several rounds.
Monitoring latent reasoning in looped language models
An interpretability-based inspection protocol with five lenses applied to eight looped language models from four families, asking whether iterations actually improve capability and whether models stay stable under prompt perturbation.
The Quotient Recurrent Reasoner adds an exact per-token phase-rotation symmetry so that slow oscillatory motion lives along directions that do not change the reasoning-relevant quotient state, avoiding an oscillation bottleneck in dynamic-depth looped reasoning.
LoopQuant: Accurate and Efficient Post-Training Quantization for Looped Transformers
A loop-aware post-training quantization framework for looped language models. It canonicalizes activation distributions across loops while keeping weights shared, allocates activation precision per loop under a fixed bit budget, and calibrates the quantized model along its own low-bit recursive trajectory.
Learning to Extrapolate in Depth with Looped Transformers
A theoretical and empirical study of how far a looped block trained at depth K can be run deeper, using all-pairs graph connectivity. Training converges to an update governed by one effective parameter, which yields a finite extrapolation window with explicit endpoints.
Running Fewer Loops Removes Refusal Behaviour in Recurrent-Depth Language Models
Shows that running a recurrent-depth model with fewer loops than trained, a tempting deployment shortcut, erodes safety much faster than capability: harmfulness is still detected internally, but the refusal that should follow is missing.
Introduces fixed-first training: train a looped model at one fixed depth for most of the run and vary depth only in a short final phase, instead of training across depths from the start.
RAMP: Recurrent Adaptive Mixed Precision for Efficient Prefill in Recurrent-Depth Language Models
Shows that quantization errors injected early in recurrent depth are attenuated by later steps. RAMP therefore runs early refinement in low-precision weights and activations and restores high precision for the terminal steps, speeding up long-context prefill.
One Model to Loop Them All: Elastic Latent Recurrence for Language Models
A training recipe that removes the one-loop tax of looped language models, so one checkpoint at one loop matches a non-recurrent model of the same size and gains accuracy with each additional loop. It uses gated state updates with learned step size, mixed loop counts per update, and loop-index and budget conditioning.
Scaling to Tens of Thousands of Test-Time Iterations with Loop-Native Attention Residuals
Identifies latent overthinking, where noisy updates overwrite correct intermediate deductions, and introduces a loop-native residual connection that keeps a constant-memory running summary of past recurrent states using content-based weighting and learned temporal decay.
From Chain-of-Thought to Loops: Non-Autoregressive Latent Reasoning via Looped Transformers
Replaces left-to-right latent thought generation with iterative refinement of a compact latent workspace by a shared transformer. A probabilistic head then samples latent tokens in parallel to condition an autoregressive answer decoder.
One Block, Multiple Depths: Recurrent Vision Transformers with Depth-Programmed Experts
Shows a single Transformer block applied recurrently can match a full-depth vision encoder at comparable FLOPs. Depth-specific behavior comes from expressing each depth's FFN as a convex combination of a small shared expert bank, programmed by a continuous normalized-depth coordinate.
Recurrent Self-Improvement: Dynamic Cross-Loop On-Policy Distillation for Looped Language Models
Uses a looped model's own terminal loop as a compute-privileged teacher for an intermediate-loop student on student-generated rollouts. The dynamic variant keeps refreshing the teacher as shared weights update, giving recurrent self-improvement without an external teacher.
ResidualQuant: KV Cache Quantization for Looped Transformers with 2-Bit Residuals
Exploits the high similarity of KV states across loops: it stores the final-loop KV as a reference and encodes other loops as low-precision residuals with least-squares scaling, rotations and loop-wise mixed precision.
Safe at One Loop, Risky at Another: Aligning Safety Across Recurrent Depths in Looped Language Models
Evaluates jailbreak robustness of looped language models at different recurrent depths and finds that safety can differ by depth and attacks transfer across depths. SafeBridge aligns safety across depths with depth-specific control of shared layers, selective state bridging and joint supervision.
Points out that Ouro's early exit does not save compute in practice because every depth needs its own KV level. A training-free best-available KV caching strategy lets exited tokens reuse the deepest cache they have, and a fix to the early-exit prior makes tokens exit at genuinely different depths.
Hybrid Latent Attention for Looped Language Models
Shrinks the T-fold KV cache of looped models by keeping exact keys and values only for a sliding window of recent tokens and storing older tokens as compact latents that every loop's query reads directly. It is uptrained on frozen Ouro models, training only the added parameters.
Splits the layers into a parallel causal encoder and a recurrent decoder that merges each token's encoder output with the previous token's final decoder state, so the computation path grows with sequence length at a fixed per-token cost.
Towards Looped Models Done Right, Part II: Rethinking at Fixed Points
Argues that states near fixed points make the path to them matter less, enabling truncated backprop, terminal KV sharing, faster distilled prefill and cheaper RL updates. It then improves the depth prior, learned from prediction feedback with an entropy term, and introduces orthogonal input injection.
Loopy: Low-Bit Quantization Framework for Looped Language Models
A post-training quantization framework that selects shared-core low-bit representations by their final prediction loss at the target deployment depth, because the ranking of quantization configurations changes with recurrent depth. Calibration windows are allocated progressively to promising candidates using forward passes only.
AdaLoop: Adaptive-Depth Latent Reasoning for Audio Language Models
A lightweight plug-in for audio language models: a shared transformer block iterates over the audio representation under question guidance, and a learned halting mechanism exits once the representation is ready, so perception-heavy questions get more steps.
On the Chain-of-Thought Monitorability of Looped Language Models
The first systematic evaluation of whether looped language models remain chain-of-thought monitorable, varying loop depth within a family and comparing against non-looped models matched by size, layer count or effective depth.
The Looped Audio Spectrogram Transformer processes all tokens once and then reuses the same blocks to refine only the class token over fixed audio features, making later passes cheap.
Looping Beyond Twice: A Scalable Recipe for Looped Mixture-of-Experts
Diagnoses why looped MoE models stop improving after two loops: variance growth across iterations and expert-selection collapse. LOOM scales residual updates, re-injects the input embedding every loop, gives each loop its own router, and adds a looping residual that carries earlier outputs forward.
Decoding Looped Transformers Better for (Almost) Free
Training-free contrastive decoding that contrasts a looped model's final prediction with an earlier recurrent pass, which serves as a built-in weak model. It works in logit space with one extra output pass or in hidden-state space with no extra output cost.
The Surprising Effectiveness of Shared Memory in Looped Transformers
Pretrains looped models in which only the first recursion writes a KV cache and later recursions read it while keeping a short local window, so memory does not grow with loops. Sharing memory improves rather than hurts quality.
Rotate the Loop: Blockwise Conformality for Looped Transformers
Treats whether a loop can track state as a geometric question about the linear map applied to the residual between steps. Blockwise conformal injections split the residual into groups and rotate within each, giving each group its own timescale while keeping exact geometry.
Closing the Loop: Practical Training Recipes for Looped Language Models
Practical recipes for looped language models: a from-scratch pipeline that cuts Ouro's 7.7T-token budget to 310B tokens, controlled comparisons against dense models, and a minimal conversion of pretrained dense models using one learned input-mixing scalar and a smoothed exit loss.
Shared Weights, Selected Computations: How Looped Transformers Route What Each Loop Does
Asks whether each loop of a looped transformer repeats the same operation. On graph walks, loops can advance by varying steps, and a learned linear map on the entering hidden state can steer a frozen loop toward different transitions, with attention routing as the causal pathway.
The first scaling law that models recurrence and MoE sparsity jointly with model size and data, through a bounded, sparsity-conditional mapping from loops to effective parameters. It recovers dense and MoE laws as special cases and guides looped MoE design.
Scales text-to-image computation by repeating shared Transformer blocks within each denoising step. Deep supervision across intermediate loops and self-modulating attention fix the weak supervision and eroding local detail that make naive looping fail.
What Limits Recursive Reasoning Models: Optimization, Architecture and Test-Time Scaling
Puts HRM, TRM and URM-style recursive reasoners into one experimental pipeline across six algorithmic domains to separate the effects of architecture, gradient propagation and training procedure. A simple recipe emerges: an intermediate gradient horizon, large physical batches and controlled recurrent-state updates, with no explicit hierarchy needed.
A Tilted Bowl Is Not a Slippery Slope: Compressing Looped Models
Tests the common claim that compressed looped models collapse because rounding error accumulates across loops. Across more than 30 models from five families, this holds only for loops that never settle; for settling loops, error shifts the fixed point instead of accumulating.
Looped Actor: Depth-Recurrent Reasoning Models for Reinforcement Learning
Brings dynamic looping to reinforcement learning: a transformer policy refines a latent toward a fixed point with a shared block and loops more in states that need planning. A complexity argument shows state-adaptive policies can need less expected computation than any fixed-runtime optimal policy.
Thinking in Depth, Speaking Directly: Recurrent Latent Reasoning for Paralinguistically Grounded Spoken Dialogue
A spoken dialogue model that reuses a decoder block for latent reasoning with acoustic grounding at every pass, trained in two stages that separate learning to reason from learning to respond so it can reply directly without chain-of-thought.
Extends looped transformers to vision-language models trained from scratch, combining Module-Loop and Model-Loop computation to iteratively update a unified vision-language state through shared modules.
Views the looped hidden state as a fast weight and derives loop transitions in closed form from a projection, a local objective and an optimizer update rule. This exposes mismatches in existing transitions and yields OperLoop, with explicit weight decay, adaptive step size and a delta objective.
Scheduling Recursive Reasoning in Looped Transformers
Adapts the scale of each recurrent update online. The Trajectory Adaptive Progress-Fluctuation Scheduler decomposes sensitivity of the terminal loss into persistent progress and fluctuation terms and adjusts the step size accordingly, at inference without retraining or also during training.
What Makes Recurrence Effective in Looped Language Models?
Controlled experiments on when recurrence helps, where to place it, and how to condition it, across inference budgets below, within and beyond the training horizon. It proposes channel-wise history-state injection with timestep conditioning as a cheap, more robust alternative to initial-state injection.
Shallow Queries, Mature Values: Depth-Asynchronous Self-Speculation for Looped Transformers
Observes that queries and keys reach their final-depth form earlier than values. DAS lets shallow draft queries read full-depth prefix values without extra recurrence, and DAS-Wave adds parallel refinement, progressive block growth and an independent full-depth verifier.
FlexLoop: Depth-Elastic Looped Policies for Adaptive Test-Time Computation in Deep RL
Finds that pretrained looped RL policies specialize to their full trained depth. FlexLoop post-trains them with the original RL objective plus adjacent-depth policy distillation, making shallower depths reliable and enabling state-wise adaptive inference.
How to Loop MoE: Flatten the Experts, Untie the Attention
Answers how to loop a mixture-of-experts model at fixed expert parameters and compute: flatten the experts by halving expert layers, doubling experts per layer and doubling passes, and untie attention so each pass has its own attention while experts and routers stay shared.
Not All Thinking is Created Equal: Latent Reasoning Discovers a Recurrent Search Algorithm for Depth Generalization
Compares vanilla, chain-of-thought, pause-token and two latent-reasoning variants of one backbone trained on extended multi-hop reasoning. Only the latent variants generalize to longer hops, and circuit analysis finds a sparse recurrent forward-search circuit propagating reachability across steps.
Loop Dropout: Regularizing Shared Updates in Looped Language Models
Finds that standard LoRA on looped models mostly adapts late loops. Loop Dropout randomly masks adapter applications per loop during training and rescales by inverse survival probability, so the shared update stays strong and effective at every loop.
LoopICL: Looping a single transformer block to solve tabular tasks
A tabular foundation model built from a single looped block with coupled cell and row streams refined by within-column and cross-column attention. Varying loop counts in pretraining and a learned exit gate let users trade inference cost for accuracy.
Improving Test-Time Scaling with Adaptive Looped Transformers
Studies whether looping improves test-time scaling as outputs grow longer and finds fixed-depth looping wastes iterations on tokens that do not benefit. TaH2 post-trains the backbone with an iteration decider supervised by online lookahead labels indicating whether another iteration improves the prediction.
Beyond the Training Horizon: Mechanisms and Limits of Length Generalization in Looped Transformers
A mechanistic comparison of two recurrence-training schemes for looped transformers on polynomial iteration, finite-state composition and knowledge-graph traversal, revealing distinct mechanisms and why both fail at greater depths.
MoR-MLLM: Mixture of Recursions for Efficient Multimodal Large Language Models
Applies Mixture-of-Recursions to multimodal LLMs so each token picks its recursion depth, spending more on visually or linguistically hard tokens. A three-stage tuning schedule and entropy-regularized routing loss stabilize recursive sparsity.
Quantizing Looped Transformers: Feedback Exposure and Calibration Blindness
Identifies two failure modes of post-training quantization in looped models: feedback exposure, where a quantized non-residual layer such as the loop-entry adapter perturbs the state and the error is fed back, and calibration blindness, where one-step calibration ignores directions used later in the recurrence.
FlashLoop: Fast and Memory-Efficient Looped Transformers via Lazy Updates
A training-free inference framework that removes cross-loop redundancy: later loops update only the few tokens whose states still change, attend to a sparse stable set of key columns, and store KV residuals between adjacent loops in low bit-width.
Depth through recurrence: Looped transformers for flow-matching TTS
Studies how to organize depth through recurrence in flow-matching text-to-speech by varying the amount, order and placement of weight reuse across seven layouts with the same number of block calls.
Attention Routing Stabilizes Early: Working-Set Inference for Recurrent Language Models
Finds that attention routing in recurrent-depth models stabilizes well before hidden states do. WISE uses full attention in early recurrent steps to discover a block-structured working set, then reuses that support in later steps while keeping attention weights and refinement dynamic; no training is needed.
LoopCD: Loop-wise Contrastive Decoding for Improving Reasoning in Looped Language Models
Links reasoning errors in looped language models to loop instability that produces uncertain hard tokens, and contrasts logits from earlier iterations with the final iteration at decoding time to refine them, with no training and negligible overhead.
WaveFront Decoding: Parallelized Self-Speculative Decoding for Looped Language Models
Training-free self-speculative decoding that arranges token states at different positions and depths into a diagonal wavefront, so drafting new positions at shallow depth and verifying earlier ones at full depth happen in the same batched call of the shared block.
Compares spending extra compute on temporal recurrence versus physical depth. Latent Recurrent Transformers feed each token's high-level hidden state into the next, and inserting one latent thought token between vocabulary tokens adds a refinement pass through the shared backbone.
Prediction Dynamics in Depth-Recurrent Language Models
Explains why intermediate answers of depth-recurrent models can already match the endpoint while scores keep changing, with a margin characterization that separates common translation, direction relative to the winner, and pairing of competitor updates with score gaps.
Beyond Depth Truncation: Controlled Evaluation of Depth Utilization in Recursive Language Models
Argues that the standard depth-truncation curve conflates fewer block applications, less distinct computation and an out-of-distribution readout. The Depth Control Protocol adds positive and negative controls, including running the full budget while executing only one distinct iteration, to measure whether a recurrent model really uses its depth.
Storing Is Not Remembering: LSTM-UT and Bounded Gated Memory for Looped Transformers
Shows on cellular-automaton tasks that keeping more history is not the same as making it usable: CoTFormer's full cache can pull corrected trajectories back toward failure. LSTM-UT adds a small, bounded, gated cell state to the shared block instead.
LoopSpec: Pipelined Self-Speculative Decoding for Looped Transformers
Training-free pipelined self-speculative decoding for looped transformers: drafts from early recurrent states overlap with verification of the current token, and a selective second proposal from a deeper depth improves draft accuracy while keeping decoding lossless.
Right Direction, Wrong Step: Geometric Analysis of Finite-Step Failure in Looped Transformers
Separates whether an extra loop's update direction is unhelpful or its full step simply moves too far. Along the model's own update direction, many failures are locally improving directions with harmful full displacement, analyzed with a pathwise curvature decomposition.
T-LoopFormer: Token-Level Elastic-Depth Looped Transformers with Dynamic Routing
Token-level elastic depth for looped transformers: a token-choice router lets each token pick its number of loop iterations from its hidden state, and a recursion-wise KV cache keeps a separate cache per loop for active tokens.
Trains the recurrence of a looped model with local denoising objectives at progressively decreasing noise levels with shared noise, so states learn to carry useful computation forward even though gradients cover only a few updates. Inference integrates a probability flow, and finer time grids spend more compute.
LoopVAE: Recurrent Depth Across Scales for Visual Tokenization
Reuses a scale- and loop-conditioned core within and across spatial scales of a hierarchical visual tokenizer, keeping only resolution-changing transitions independent.
RecurTrace: Adaptive Latent Reasoning with Loop-Time Memory
Improves middle-layer latent recurrence in two ways: Loop Memory Attention lets each looped layer attend to its own states from earlier iterations along the loop-time axis, and a halting head trained with oracle labels decides when extra depth still reduces loss.
Flow Reasoning Models: Turning Flows Into Efficient Recurrent Reasoners
Flow Reasoning Models turn a non-causal flow denoiser into a recurrent reasoner by feeding its clean-solution prediction back as a conditioning state. Fixed-Point Forcing trains the model on states produced by its own recurrent rollouts, enabling deeper iterative refinement while retaining the standard flow-matching objective.
Looped Transformers under the Jacobian Lens: Does the Global Workspace Survive Recurrence?
Extends the Jacobian lens to iterated architectures to test whether a global-workspace-like band of verbalisable, causally potent representations survives recurrence, comparing Ouro-2.6B, Huginn-0125 and an untied Qwen baseline.
SMELT: Scaling Laws for Compute-Matched MoE Looped Transformers
Repeats the middle half of a sparse MoE Transformer's layers twice while closely matching per-token FLOPs, total non-embedding parameters and KV cache to an unlooped baseline. Budget-matched scaling experiments show improved loss and downstream performance from reusing the shared middle layers.
Dissecting Hierarchical Reasoning Models: A Mechanistic Study
A mechanistic study of the Hierarchical Reasoning Model on Sudoku, Maze and ARC-AGI-2 using baselines with and without recurrence, state interventions, probes and sparse autoencoders.
Dynamical phase selection controls compute scaling in looped transformers
Shows that looped networks with the same architecture, objective and accuracy can land in different dynamical phases depending on initialization, and that the bifurcation type of each phase, such as a saddle-node fold or a Neimark–Sacker transition, sets how test-time compute scales.
Allocating Recurrent Compute in Looped Language Models
Asks what should loop. Using Iterative Transport Rank to measure the new cross-position influence each application adds, it repeats each Gated DeltaNet mixer while applying the dense FFN only once.
Shows that a trained recurrent operator's finite-time dynamical regime, estimated as settling, marginal or drifting, predicts whether extra test-time iterations help, preserve or hurt answers, and gives a sufficient condition for depth safety.
Looped Language Models Improve Compositional Tool Calling
Evaluates native and retrofitted looped language models on compositional tool use, where models must coordinate several API calls and keep dependencies, comparing looped and non-looped models under matched fine-tuning while varying recurrent depth at inference.
Gated Recurrent Transformers: Expressive Depth through Recurrent Modulation
A prelude–core–coda recurrent-depth transformer whose shared core update is modulated by a lightweight projection and an elementwise update gate conditioned on the hidden state, the prelude output and per-step noise, letting a few layers specialize across recurrences.
Mimir: A Sub-2B Hierarchical Reasoning Model Trained on Permissible Data Punches Above Its Weight Class
A 1.78B language model built on the HRM-Text architecture, whose layers form a low-level and a high-level stack applied repeatedly in an interleaved schedule, trained from scratch only on permissibly licensed data.
Depth-adaptive Inference of Looped Language Models via Continuous Depth Batching
The first efficient serving method for depth-adaptive looped models: new batches are formed between loop steps, looped and non-looped parts are scheduled separately, looped KV caches are managed, and early exits are predicted so batches can be prepared asynchronously.
Augments a frozen TinyLlama-1.1B with a small trainable recursive block that iteratively refines a latent thought before answering, aiming to combine TRM-style recursive reasoning with the semantics of a pretrained language model.
LoopMTP: A looped transformer guided by latent multi-token prediction
Links looping to multi-token prediction: a model that loops T times should anticipate T future tokens, so the hidden state at loop t is softly aligned with the embedding of the token t steps ahead, with a gate preserving useful information across iterations.
Retrofitting Recurrent Depth into a Pretrained Language Model: Installation, Extrapolation, Transfer, and Retention at Two Parameter Budgets
Retrofits Qwen2.5-0.5B-Instruct into prelude, weight-tied recurrent block and coda with an identity-preserving one-loop path and a re-entry bridge, and shows it learns a reusable iterative latent procedure, either as a 6M adapter over frozen weights or a 180M full block.
Looped Transformers with Source-Centered State Evolution
Source-Centered State Evolution reconciles input conditioning with shared recurrence: inputs enter through a learned anchor and initial deviation, zero deviation maps to zero, and the anchor is an exact fixed point, removing the forcing bias of additive input injection.
When Does Recurrence Become an Algorithm? Convergence Selection in Weight-Tied Looped Transformers
Studies when a weight-tied looped transformer implements a real algorithm on group word problems. Training installs a linear computation frontier whose speed is set by the training contract, and weight tying, not expressivity, makes the model choose a serial algorithm over a parallel scan.
Looped Latent Attention: Cross-Loop KV Compression for Looped Transformers
Shows the loop-indexed KV cache of looped models traces a short low-rank trajectory per token, layer and head. Looped Latent Attention is a post-training codec that stores compact K/V latents and reconstructs loop-specific K/V only when attention reads them.
T^2MLR: Transformer with Temporal Middle-Layer Recurrence
Temporal Middle-Layer Recurrence fuses a cached middle-layer representation from the previous token into an earlier layer at the current position, letting intermediate computation persist across decoding steps with little overhead and without looping all layers.
Per-Token Fixed-Point Convergence in Depth-Recurrent Transformers
Measures what a depth-recurrent model trained with randomized recursion counts computes per token: states converge to per-token fixed points at very different speeds, and a training-free rule that halts each token once its output stabilizes beats a learned router.
Energy-guided Recursive Model (ERM) augments recurrent reasoning with an explicit energy-based mechanism for selecting among candidate trajectories. It constructs task-structured Hopfield energies over valid local or global patterns, using them to rank recurrent outputs and to support energy-guided sampling such as parallel tempering.
CHASE: Cache-Hole-Adapted Skip Exit for Looped State-Space Language Models
Extends looping to state-space language models with Looped Mamba and looped hybrid Mamba-Transformer blocks, and adapts Ouro-style exit gates. Because early exits leave the recurrent SSM state without deeper updates, a cache-hole adaptation aligns continued training with skipped-state inference.
Adaptive Depth in Looped Transformers: Diagnosing Learned Halting Gates and Trajectory Readouts
Separates the two jobs of a learned halting distribution, choosing which state to read and weighting per-depth losses, on synthetic tasks and Ouro checkpoints. Fixed-prior depth supervision yields difficulty-aware trajectories, and simple post-hoc confidence readouts often match learned gates.
Stabilizing Extrapolation in Looped Transformers via Learned Stochastic Stopping
Traces the brittle length generalization of looped transformers to a spurious correlation between sequence length and loop count, and shows that randomizing loop counts in training, or learning a stochastic schedule with RL-Halting, stabilizes predictions across inference-time loop counts.
LoopCoder-v2: Only Loop Once for Efficient Test-Time Computation Scaling
Trains a family of 7B Parallel Loop Transformer coders with different loop counts from scratch on 18T tokens to study loop-count selection as a gain–cost trade-off between extra refinement and the positional mismatch introduced by cross-loop position offsets.
On the Residual Scaling of Looped Transformers: Stability and Transferability
Analyzes how weight sharing correlates residual updates across loop iterations and derives a 1/N residual scale for a block repeated N times. Factoring loop count from unique-layer depth improves training stability and supports learning-rate transfer across loop counts.
Dense Supervision Is Not Enough: The Readout Blind Spot in Looped Language Models
Shows that dense per-loop cross-entropy only controls state variables visible to the readout. Scale-invariant readouts such as RMSNorm hide radial scale from the loss while pre-norm residual recurrence keeps growing it, so early exits work but recurrent scale goes unchecked.
Skip a Layer or Loop It? Learning Program-of-Layers in LLMs
Finds that pretrained LLM layers can be skipped or repeated to form a per-input program-of-layers without training, and learns a lightweight network that predicts such execution programs for each input.
LoopMoE: Unifying Iterative Computation with Mixture-of-Experts for Language Modeling
A looped mixture-of-experts language model with IterAdaLN, a modulation conditioned on iteration index and per-token state to break weight-sharing symmetry, and a capacity-balancing strategy that restores the attention-to-FFN active-parameter ratio of tuned non-looped models.
Latent Recurrent Transformer: Architecture Exploration, Training Strategies, and Scaling Behavior
Reuses a high-level hidden state from the previous token as recurrent memory for the next token, adding a cross-token, cross-layer pathway while keeping standard attention, KV cache and one forward per token. Interleaved parallel training avoids sequential unrolling during pretraining.
Stabilizing Recurrent Dynamics for Test-Time Scalable Latent Reasoning in Looped Language Models
Addresses the peak-then-collapse behavior of looped language models under deeper recurrence by constraining latent states to approach asymptotically stable fixed points, using efficient Jacobian spectral-radius regularization with random loop sampling.
Shows that selectively looping the early-middle layers of masked diffusion language models improves both training efficiency and final quality, and that varying the loop count at inference gives flexible compute scaling.
Loops a contiguous mid-stack block of a frozen checkpoint at inference with no training. Viewing a pre-norm block as a forward Euler step, it replaces one large update with smaller damped sub-steps, since naive block repetition usually hurts.
Equilibrium reasoners proposes a new view of recurrence as learning stable attractors as opposed to fixed point convergence. The new framework allows for more sophisticated test time scaling, and ultimately leads to stronger performance compared to the base TRM model.
Solve the Loop: Attractor Models for Language and Reasoning
A backbone proposes output embeddings and an attractor module refines them by solving for a fixed point, with gradients from implicit differentiation, so training memory stays constant in effective depth and iterations stop at convergence.
Simply Stabilizing the Loop via Fully Looped Transformer
Traces training instability of deep looping to gradient oscillation and residual explosion, and fixes them with two parameter-free changes: distributing inter-loop signals across all layers, and attention injection that reuses the existing attention block.
LoopUS: Recasting Pretrained LLMs into Looped Latent Refinement Models
Looped Depth Up-Scaling converts a pretrained LLM into an encoder, a looped reasoning block and a decoder, using representation-guided block decomposition, an input-dependent selective gate against drift, random deep supervision, and a confidence head for early exit.
The lattice deduction transformer is a looped transformer that projects the token space through a lattice structure to perform efficient combinatorial reasoning.
Sparse Layers are Critical to Scaling Looped Language Models
Compares dense and MoE transformers with and without looping and finds that looped MoE models scale better than the standard baseline while dense looped models do not, because different experts fire on each pass through the shared layers.
A loop-aware post-training quantization framework that keeps one shared quantized backbone while adding loop-aware activation scaling, selective transformations, cross-loop state alignment and trajectory-aware calibration to limit recursive error.
Memory-Efficient Looped Transformer: Decoupling Compute from Memory in Looped Language Models
The Memory-Efficient Looped Transformer keeps one KV cache per layer shared across reasoning loops and updated by a learnable gate, decoupling reasoning depth from memory. It is obtained from Ouro by chunk-wise two-phase post-training with interpolation and attention-aligned distillation.
Organizes a looped Transformer into begin, middle and end blocks, loops only the middle block, and adds hyper-connections after each loop to expand the residual stream into matrix-valued streams at minimal extra cost.
Universal Transformers Need Memory: Depth-State Trade-offs in Adaptive Recursive Reasoning
Studies learned memory tokens as a scratchpad for a single-block Universal Transformer with ACT on Sudoku-Extreme, finding them necessary and showing that memory size and ponder depth substitute for each other at fixed accuracy.
How Much Is One Recurrence Worth? Iso-Depth Scaling Laws for Looped Language Models
Measures how much one recurrence is worth in equivalent unique parameters via a joint scaling law fitted on an iso-depth sweep over recurrence counts 1–8, yielding a recurrence-equivalence exponent φ = 0.46.
One Step Forward and K Steps Back: Better Reasoning with Denoising Recursion Models
Corrupts target solutions with noise like diffusion models but trains a looped refiner to undo the corruption over multiple recursive steps rather than one, giving a curriculum of intermediate states that better matches test-time refinement.
Stability and Generalization in Looped Transformers
A fixed-point framework that analyzes looped architectures along reachability, input-dependence and geometry, proving that looped networks without recall cannot achieve strong input dependence, while recall with outer normalization gives reachable, smooth, trainable fixed points.
A Mechanistic Analysis of Looped Reasoning Language Models
Analyzes latent states of looped reasoning language models and finds that each layer in the cycle often converges to its own fixed point, so the recurrent block follows a consistent cyclic trajectory and repeats feedforward-like stages of inference at every iteration.
ELT: Elastic Looped Transformers for Visual Generation
Elastic Looped Transformers for image and video generation use weight-shared blocks and Intra-Loop Self Distillation, distilling intermediate-loop students from the maximum-loop teacher in one training step, to obtain a family of any-time models from a single run.
Loop, Think, & Generalize: Implicit Reasoning in Recurrent-Depth Transformers
Studies implicit multi-hop reasoning in recurrent-depth transformers trained from scratch, covering systematic generalization over never-composed facts and depth extrapolation from up to 5-hop training to deeper compositions.
Combines the YOCO decoder-decoder design with recursion: a Universal Self-Decoder iterates with shared parameters, but only over shallow efficient-attention layers, while YOCO's constant global KV cache and linear prefill stay intact.
Thinking Deeper, Not Longer: Memory-Efficient Test-Time Reasoning with Depth-Recurrent Transformers for Compositional Generalization
A depth-recurrent Transformer that reasons by iterating a shared block instead of generating tokens, stabilized for 20+ steps by a final-output-only thinking objective, LayerScale initialization and an identity-biased gate.
LoopRPT: Reinforcement Pre-Training for Looped Language Models
Reinforcement pre-training for looped language models that reframes next-token prediction as next-token reasoning and assigns rewards directly to latent steps using an EMA teacher reference and noisy latent rollouts.
SCORE: Replacing Layer Stacking with Contractive Recurrent Depth
Replaces layer stacking with a single shared block applied iteratively through a contractive, ODE-inspired update h ← (1 − dt) · h + dt · F(h), using fixed discrete steps and ordinary backpropagation.
Adaptive Loops and Memory in Transformers: Think Harder or Know More?
Combines adaptive per-layer looping, where each block learns to iterate with a halting mechanism, with gated memory banks that add learned storage, to test whether models should think harder or know more.
Prioritize the Process, Not Just the Outcome: Rewarding Latent Thought Trajectories Improves Reasoning in Looped Language Models
Reward Latent Thought Trajectories distributes RL reward across the full latent reasoning trajectory of a looped model instead of only the final latent state, as a drop-in replacement for GRPO.
Looping Back to Move Forward: Recursive Transformers for Efficient and Flexible Large Multimodal Models
A recursive Transformer for large multimodal models with a Recursive Connector that fuses intermediate hidden states with modality-specific projections across steps, and a Monotonic Recursion Loss that makes quality improve with recursion depth.
Recurrent-Depth VLA: Implicit Test-Time Compute Scaling of Vision-Language-Action Models via Latent Iterative Reasoning
Recurrent-Depth VLA gives vision-language-action models a weight-tied recurrent action head that refines a latent for any number of steps at constant memory, trained with truncated BPTT and stopped adaptively when the latent converges.
LoopViT: Scaling Visual ARC with Looped Transformers
A recursive vision architecture for ARC-AGI that iterates a weight-tied hybrid block of local convolutions and global attention as a latent chain of thought, with a parameter-free dynamic exit when predictive entropy shows the state has settled.
Depth-Recurrent Attention Mixtures: Giving Latent Reasoning the Attention it Deserves
Depth-recurrent attention mixtures combine sequence attention, attention along depth and sparse expert attention, relieving the fixed hidden-size bottleneck of many-step latent reasoning and decoupling scaling dimensions in depth-recurrent models.
Teaching Pretrained Language Models to Think Deeper with Retrofitted Recurrence
Converts pretrained non-recurrent language models such as TinyLlama, OLMo-2 and Llama-3.2 into depth-recurrent prelude–core–coda models, using a curriculum that gradually increases recurrences during training.
Parallel Loop Transformer for Efficient Test-Time Computation Scaling
Removes the sequential latency of looping with Cross-Loop Parallelism, which computes different loops for different tokens in the same pass, and controls memory by sharing the first loop's KV cache with later loops combined through gated sliding-window attention.
Efficient Parallel Samplers for Recurrent-Depth Models and Their Connection to Diffusion Language Models
Connects recurrent-depth models to diffusion language models and builds a diffusion-forcing sampler that decodes a new token every forward pass while earlier tokens' latents keep being refined in parallel through recurrence.
What Makes Looped Transformers Perform Better Than Non-Recursive Ones
Explains why looped transformers beat non-recursive ones on complex reasoning through loss-landscape geometry: recursion biases training toward steep V-shaped river valleys, where valley hopping speeds convergence along the river. This motivates SHIFT, a staged progressive training schedule.
MeSH: Memory-as-State-Highways for Recursive Transformers
Memory-as-State-Highways externalizes state management of recursive transformers into an explicit memory buffer with lightweight routers, addressing undifferentiated computation across iterations and overload of a single hidden state.
Encode, Think, Decode: Scaling test-time reasoning with recursive latent thoughts
Mid-trains a base model to iterate over a small subset of reasoning-relevant layers, chosen from interpretability evidence, while keeping architecture, parameter count, hyperparameters and data unchanged.
Less is More: Recursive Reasoning with Tiny Networks
The tiny recursive model is a looped transformer architecture with an intentionally simple architecture, yet yields strong generalization compared to previous models.
Two-Scale Latent Dynamics for Recurrent-Depth Transformers
Describes recurrent-depth iterates as small refinements within a looped block and larger drift across blocks; loop steps grow smaller and more mutually orthogonal over training. This motivates an early exit based on the second-order difference of step size.
Latent Chain-of-Thought? Decoding the Depth-Recurrent Transformer
Probes Huginn-3.5B on arithmetic with the Logit Lens and Coda Lens to see whether interpretable latent chain-of-thought emerges across recurrent blocks.
A brain-inspired recurrent architecture with two interdependent modules: a slow high-level module for abstract planning and a fast low-level module for detailed computation, solving sequential reasoning tasks in one forward pass without intermediate supervision.
PonderLM: Pretraining Language Models to Ponder in Continuous Space
Teaches language models to ponder in continuous space: within one token step, the model feeds back the probability-weighted sum of token embeddings and runs another forward pass, learned through ordinary self-supervised pretraining.
To CoT or To Loop? A Formal Comparison Between Chain-of-Thought and Looped Transformers
A formal comparison of chain-of-thought and looped transformers: looped models efficiently simulate parallel deterministic computation over DAGs, while stochastic CoT excels at approximate inference for self-reducible compositional problems.
Reasoning with Latent Thoughts: On the Power of Looped Transformers
Argues that many reasoning problems need depth rather than parameters: a k-layer transformer looped L times nearly matches a kL-layer model on synthetic reasoning, backed by theory that iterative algorithms map onto looped models.
Treats layer computation as implicit thinking steps: Adaptive Token Routing sends critical tokens through extra iterations, Residual Thinking Connections refine them, and Thinking Step Encoding marks each phase.
On Expressive Power of Looped Transformers: Theoretical Analysis and Enhancement via Timestep Encoding
Establishes approximation rates of looped transformers for sequence-to-sequence functions via a modulus of continuity, revealing a looped-specific limitation that per-loop scaling parameters conditioned on timestep encoding can remove.
Shows that looped transformers with an adaptive number of steps generalize to unseen input lengths on tasks with known iterative solutions built from repeated RASP-L operations, using a dedicated training algorithm.
CoTFormer: A Chain-of-Thought Driven Architecture with Budget-Adaptive Computation Cost at Inference
Observes that chain-of-thought resembles reapplying a model several times, except that later passes attend to earlier passes' tokens. CoTFormer mimics this at the token level and pairs it with token-wise variable depth chosen by a compute-adaptive router.
Programs a fixed-weight transformer and places it in a loop so the input sequence acts as a punchcard of instructions and memory, emulating a small instruction-set computer with a 13-layer looped encoder.
This paper presents a neat yet effective recursive operation on transformers that can improve parameter utilization without involving additional parameters. This is achieved by sharing weights across the depth of transformer networks. The proposed method can obtain a substantial gain (~2%) simply using a naive recursive operation, requires no special or sophisticated knowledge for designing principles of networks, and introduces minimal computational overhead to the training procedure. To reduce the additional computation caused by recursive operation while maintaining the superior accuracy, we propose an approximation method through multiple sliced group self-attentions across recursive layers, which can reduce the cost consumption by 10~30% with minimal performance loss.
Learns end-to-end how many computational steps to take, modeling halting as a probabilistic process and trading off prediction accuracy, computational cost and generalization.
ALBERT: A Lite BERT for Self-supervised Learning of Language Representations
Reduces BERT's parameters with factorized embeddings and cross-layer parameter sharing, so every layer reuses the same weights, plus an inter-sentence coherence loss.
Directly solves for the fixed point that a weight-tied layer would reach if iterated infinitely, using root-finding forward and implicit differentiation backward, so memory is constant in effective depth.
Adaptive Computation Time for Recurrent Neural Networks
The original algorithm that lets a recurrent network learn how many steps to take per input with a deterministic, differentiable halting mechanism and a ponder cost.
Fixed loop count executed in fewer sequential stages
KV / state strategy
See paper
Reading note
With 1.4B models, deeper LoopFold settings raise the eight-task zero-shot average from 53.41% to 56.38%, where vanilla looping and PLT gain little. The 46% decoding-latency reduction versus a nine-loop model uses five sequential stages plus extra parallel compute.
[ Backbone ] × R + gated persistent state per token
Depth control
Fixed trained loop count
KV / state strategy
One extra gated state vector per token; attention cache see paper
Reading note
Lowers Pile perplexity and improves the eight-task zero-shot average over a vanilla looped baseline. The perplexity advantage grows with trained loop depth.
Late keys and values rescue far fewer cases than late queries
Reading note
On synthetic two-hop scenes, late queries rescue 90% and 97% of loop-1 failures. The effect is smaller on natural questions and replicates only in Ouro-2.6B; the paper does not locate where the bridge entity is computed.
[ N blocks ] × M (N · M = ViT-B depth), vision and optionally text encoder
Depth control
Fixed recursion
KV / state strategy
Not applicable (encoders)
Reading note
CLIP tolerates substantial recurrent reuse, though the effect depends on encoder and task. Evaluated on zero-shot classification, retrieval, few-shot learning, grounding and concept unlearning with CC3M and CC12M training.
Draft: few loops → Verify: all R loops; cache = deepest plane + recent full-depth window
Depth control
Self-speculative: draft with few loops, verify with R
KV / state strategy
Deepest recurrent KV plane everywhere except a recent full-depth window
Reading note
On Huginn-0125, a 64-position window removes 89% of tail-position error at 11.9% of the full cache memory (1024-token context). Measured self-speculative speed-ups are 2.09× with the full cache and 2.58× with the deepest plane.
Ouro-style fixed loops; mixed-depth or native readout
KV / state strategy
One-wave KV admission with CUDA graphs
Reading note
Replay is bit-exact in all reported FP32 and bf16 cells. Deferral raises single-GPU rollout throughput 1.47–1.56× and a two-node GRPO update is 2.71× faster. The GSM8K gains come from 100-update runs in the authors' setup.
Most of the benefit comes from tuning the single-pass loss weight. With validation-selected weights, single-pass quality stays within a preset tolerance of ordinary continued training while looped predictions improve.
Unlooped path → fixed-prefix layer looping for budgets 1…5
Depth control
Five predefined execution paths chosen at inference
KV / state strategy
See paper
Reading note
After 3.6T pretraining tokens, the deepest budget raises Accordion-6B-A0.5B's macro-average over 37 tasks by 3.01 points; the middle budget captures 89% of the gain with half the extra layer executions.
[ Shared layers ] → halting head after each iteration
Depth control
Learned per-token halting (evaluated critically)
KV / state strategy
See paper
Reading note
Scoring every iteration fixes the head only with a strong early-stop penalty. The authors recommend reporting iterations actually executed with threshold and batch size, and provide a pre-training test that bounds a head's possible gain.
One KV cache shared across the recurrent core; persistent cache sets drop from 15 to 4
Reading note
At width 1024 it uses 40.4% fewer parameters than an untied control at nearly matched FLOPs. The consistency loss removes 81–89% of the continuation-loss penalty on FineWeb; the cost of weight tying is smallest at low token budgets.
Pretrain with stochastic depth → post-train: final-recurrence KV broadcast to earlier recurrences
Depth control
Fixed recurrence
KV / state strategy
Final-recurrence KV cache shared across depth
Reading note
Matches unshared fine-tuning perplexity and composes with MLA, Kimi delta attention and fp8/mxfp8/nvfp4 KV quantization. Throughput and p90 TTFT gains come from simulated low-memory long-context serving.
[ Block ] × R with or without loop-boundary normalisation
Depth control
Extrapolation beyond the trained loop budget
KV / state strategy
See paper
Reading note
The damage concentrates in the readout and last block layer. At matched overshoot of the trained budget, the advantage of looped over stacked depth vanishes on one synthetic task and shrinks on the other.
Each denoising step: [ Early layers ] × k → remaining layers
Depth control
Loop count sampled during training; chosen at inference
KV / state strategy
Not an autoregressive cache setting; see paper
Reading note
Reaches a non-looped UDLM's validation NLL with up to 2.8× (OpenWebText) and 3.3× (FineWeb-Edu) fewer training FLOPs. Sampling small loop counts more often during training uses 12% fewer loops downstream without hurting accuracy.
Causal encoder (once, Global KV) → [ Weight-tied decoder, Local KV ] × R
Depth control
Fixed recurrence at matched effective depth
KV / state strategy
Fixed global KV from the encoder plus local KV per iteration, under one softmax
Reading note
From 60M to 1B on FineWeb-Edu it beats Full Loop at matched effective depth and tokens; at 1B the seven-task zero-shot average rises from 49.58% to 50.85%. Benefits extend to an OLMo-style backbone and MoE.
Iteration k ≈ state after k operations; norm anchoring; completion readout
Depth control
Iterations scale with problem length; learned stopping
KV / state strategy
See paper
Reading note
A Qwen2.5-0.5B model trained on at most 32 swap-S5 moves reaches 95.5% full-state accuracy at 512 moves. Removing norm anchoring or progress-aligned targets sharply reduces extrapolation. Results cover sequential state-tracking tasks.
Backprop through the last k of R loops → measure discarded gradient
Depth control
Analysis of truncated BPTT
KV / state strategy
See paper
Reading note
The lost share is set near the output and barely changes once R is well above k; on Ouro, keeping two of four loops still loses 81% of the output gradient. The share falls during fine-tuning, and the paper could not show it predicts training cost better than simple rules.
Looped transformer with L loops + reduced CoT tokens
Depth control
Prescribed loop count L(n)
KV / state strategy
Context window O(s(n)/L)
Reading note
Reduces both context length and generated tokens by a factor of L relative to the unlooped queue-machine simulation, including with softmax attention at constant bit size. Theory only; no trained models.
Embed → [ Weight-tied core ] × cycles (gated) → mixture of visited states → decode
Depth control
Adjustable cycles; learned halting via GMR
KV / state strategy
Not applicable (masked encoder)
Reading note
With 155M unique parameters trained on the same 2T tokens as AMPLIFY-350M, it improves contact recovery and three of four supervised tasks, raising fold accuracy by 22.5%. GMR cuts expected depth by up to 37% with task-dependent trade-offs.
nGPT block on the hypersphere: geodesic injection → damping → halting
Depth control
Adaptive halting at a fixed point
KV / state strategy
See paper
Reading note
Proves fixed-point existence and convergence conditions. Reports 96.6% on Sudoku-Extreme, 87.9% on Maze-Hard and 59.1% on Mini-ARC, ahead of FPRM; these are puzzle benchmarks, not language modeling.
Prelude → [ Recurrent block ] × T (backprop through last B) → Coda
Depth control
Fixed forward loops T with B backward loops
KV / state strategy
See paper
Reading note
When forward and backward loop counts are optimized jointly, the law favors full BPTT; in matched-compute runs, full BPTT through fewer loops beats truncation. Two fully trained loops beat the compute-optimal non-looped model with about 27% fewer non-embedding parameters and 8% more training compute.
[ Block 1 ] × r → [ Block 2 ] × r → … → [ Block n ] × r
Depth control
Static per-block recursion
KV / state strategy
See paper
Reading note
Improves validation NLL and downstream accuracy over static recursive sharing in decoder-only models up to 1.7B, with additional encoder-decoder and encoder-only studies; no routing modules or extra objectives.
Token t: prediction slot (deep latent of t−1) ‖ latent slot (state for t+1)
Depth control
One paired invocation per token
KV / state strategy
See paper
Reading note
On Qwen-style 1B dense models, Pile perplexity drops from 7.25 to 7.00 and the five-shot average rises 2.26 points. Training uses a few Jacobi iterations; decoding keeps 93.1–96.5% of vanilla throughput for the dense model.
Starting from Qwen3-4B Base with SFT, R = 3 improves the 20-benchmark average by 6.17%. The authors report breaking the common saturation at two loops.
Gated prefix-specific latent slots with content-addressed reads, alongside causal self-attention
Reading note
On three byte-level corpora at about 1M and 101.71M parameters, it has the best mean held-out perplexity in five of six settings against five controls, with roughly 1% gains at the larger budget. Small-scale study.
A four-layer block iterated eight times solves the task in every seed, while an identically trained untied model of the same effective depth never does unless given a lower learning rate or intermediate supervision. Synthetic algorithmic setting.
At 730M parameters and 336B tokens the full Huginn design beats Ouro on all ten benchmarks, and the advantage transfers to an 8B-A0.8B MoE trained on 500B tokens. Separate H/L states and random state initialization give mixed or negative returns.
Early layers → [ Looped subset + linear adapter ] × r → late layers
Depth control
Fixed recurrence
KV / state strategy
See paper
Reading note
On TinyLLaVA-1.4B, without new vision-language data, it outperforms the unpruned baseline trained on the same recovery data while storing 23.3% fewer language parameters, with gains mainly on reasoning-oriented benchmarks.
Trained to answer at loop 32; probed up to loop 64
KV / state strategy
See paper
Reading note
Models keep only 22% of answers at loop 64 by default; supervising a random position keeps 99.8% and randomized loop counts help, while gradient truncation does not. Even retained states are not reusable when the answer first appears. Synthetic tasks.
Visit 1: [ Block ] on compressed sequence → Visit 2: [ Block ] at full resolution
Depth control
Two visits with asymmetric compute
KV / state strategy
Separate attention per visit; MoE experts shared across visits
Reading note
Across 15 configurations from 1.3B to 7B parameters and beyond 10^21 FLOPs it lowers validation NLL; at a fitted NLL of 1.988 the predicted compute savings are 19.0% versus vanilla MoE and 7.9% versus the looped baseline.
Input + options → [ Looped LM layers ] × r (read options each step) → probabilities
Depth control
On-demand depth via a learned stopper
KV / state strategy
Not applicable (single-pass decision model)
Reading note
A 2.6B LEV rises from 63.2% at depth one to 76.2% at depth four on JevBench. On-demand depth saves 15–24% compute for under one point of accuracy; built on looped pretrained backbones.
On ImageNet 256×256, SiT-B/2 FID drops from 15.08 to 4.91 at 400K steps versus the REG baseline; one extra inference visit on a two-visit checkpoint improves FID from 14.71 to 5.34. Holds for SiT-L/2 and SiT-XL/2.
Teacher repeats searched layers; student runs a single pass
KV / state strategy
See paper
Reading note
On GSM8K with a 400-token thinking budget, Qwen3-1.7B rises from 42.30% to 79.61%, with further gains in later rounds. Evaluated on mathematical and logical reasoning tasks.
Carry loop (prompt only in state) ‖ Re-read loop (prompt re-supplied each iteration)
Depth control
Analysis of released models
KV / state strategy
See paper
Reading note
The key axis is whether the prompt lives only in the looped state (Ouro, LoopUS) or is re-read every iteration (Parcae, Huginn, Rec-OLMo). Findings are framed around safety and efficient use.
[ Shared block ] × R with per-loop activation precision
Depth control
Fixed loops of the base model
KV / state strategy
See paper
Reading note
On Ouro-2.6B, W4A4 LoopQuant beats all W4A8 baselines, cuts model memory by 65.3% with 3.4% extra storage over naive INT4, and decodes 1.7× faster than BF16.
[ Block ] × K at training → × K′ at test (within a certified window)
Depth control
Test depth chosen from the training distribution
KV / state strategy
See paper
Reading note
Too few test loops under-propagate and too many create false connections. The certified depth gain grows exponentially as training depth outpaces training hardness. Graph-connectivity setting.
At half depth, Ouro-1.4B-Thinking keeps 89% of its GSM8K accuracy while harmful-help rate rises 5.3×. Early-exit rules and per-request depth do not repair it; training at randomly chosen depths does, including on Huginn-0125 across 1–32 loops.
Fixed depth (most of training) → mixed depths (finishing phase)
Depth control
Adaptive inference depth learned late in training
KV / state strategy
See paper
Reading note
At 200M, fixed depth 2 for the first 67% cuts modeled training compute by 45% with mean NLL across depths within 0.5%. At 1B, starting at depth 4 cuts compute by 31–37% and wall-clock time by 32–39% with final NLL within 0.8%.
[ Shared block, gated update ] × r conditioned on (loop index, budget)
Depth control
Elastic: 1 to several loops from one checkpoint
KV / state strategy
Keeping only the last loop's KV holds memory at the one-loop level
Reading note
On 4B dense and 30B-A3B MoE hybrids, one loop is within about a point of matched baselines and four loops add 7.25 and 4.75 points on a ten-task average. Post-training yields Koi-4B. Mixed depths can be introduced late in pretraining.
[ Recursive block ] × N + loop-native residual aggregation
Depth control
Test-time looping up to tens of thousands of steps
KV / state strategy
Constant-size streaming aggregation memory over recurrent states
Reading note
A 7M-parameter model reaches 97.9% exact accuracy on Sudoku-Extreme and 13.6% pass@2 on ARC-AGI-2, and keeps improving on Sudoku-Extreme beyond 20,000 effective steps.
[ One block, FFN = mix(experts; depth coordinate) ] × L
Depth control
Elastic depth by resampling the depth coordinate
KV / state strategy
Not applicable (vision encoder)
Reading note
Trained from scratch, reViT-B/16 reaches DeiT III accuracy with about 70% fewer stored parameters; a distilled 8-expert model keeps nearly all of DINOv2's linear-probe accuracy and transfers to segmentation and depth.
On Ouro-Thinking models, LoopOPD improves mathematical reasoning and the dynamic variant adds further gains; training only on math also helps general reasoning and code. Code and checkpoints are promised upon acceptance.
KV(loop R) as reference + 2-bit residuals for loops 1…R−1
Depth control
Fixed loops of the base model
KV / state strategy
Final-loop KV reference plus quantized residuals (down to INT2)
Reading note
Retains accuracy close to BF16 while reducing theoretical KV storage by 80.7%; on an RTX 5090, decode throughput improves up to 2.73× at fixed batch and peak throughput up to 4.15×.
[ Shared layers + depth-specific control ] × R with safety supervision at every depth
Depth control
Multiple readout depths
KV / state strategy
See paper
Reading note
SFT and preference alignment alone leave depth-wise safety gaps. SafeBridge reduces matched- and cross-depth attack success with comparable over-refusal. Code and checkpoints are promised upon acceptance.
Token exits at depth d → later loops read its best-available KV
Depth control
Per-token early exit with heterogeneous depths
KV / state strategy
Best-available (deepest computed) KV per token
Reading note
Up to 30% lower FLOPs and KV memory while retaining full-depth performance; KV-aware training improves further. Validated on Ouro and smaller looped LMs trained from scratch.
Recent W tokens: exact KV ‖ older tokens: shared latent read by each loop
Depth control
Ouro T = 4 loops
KV / state strategy
Sliding-window exact KV plus compact per-token latents
Reading note
Cache shrinks 10.7× per token, fitting 4.0–8.8× more concurrent sequences; throughput improves 2.5× at 1K and up to 7.4× at 16K context. Retains over 97% of accuracy on math, knowledge and reasoning and 96–100% on retrieval up to 16K.
Causal encoder (parallel) → recurrent decoder fed by previous token's state
Depth control
Grows with sequence length; fixed per token
KV / state strategy
See paper
Reading note
Trained on at most 40 bits, two splits generalize parity to 256 bits in every seed; on swap-S5 at 8× training length it reaches 97% versus under 1% for a Transformer. Removing the feedback drops these tasks to chance.
From 100M to 1.6B, the learned prior and orthogonal injection lower perplexity relative to Huginn's prior and existing injection schemes. At 1.6B, a 3× smaller KV cache matches the downstream average of fixed-depth training with a full cache.
Deeper loops reduce monitorability on some MonitorBench tasks under stress tests, not fully explained by difficulty or length. Cross-model comparisons find no evidence that looped models are systematically less monitorable.
[ Blocks ] on all tokens → [ Same blocks ] × passes on class token only
Depth control
Fixed number of passes
KV / state strategy
Audio features fixed after the first pass
Reading note
On AudioSet, ten-pass LAST reaches 0.345 mAP, 2.1% relative above a twelve-layer transformer, with 49.4% fewer parameters, 42% fewer MACs and 9.8% higher throughput. Going from two to ten passes adds only 1.2% computation.
Across 100M–1.7B models; under near-iso-FLOP the 700M model is best at 5 loops (perplexity 18.36 → 16.54). Without FLOP matching, the 1.7B model trained on 60B tokens peaks at 9 loops (perplexity 9.62 → 7.77).
Raises Ouro-2.6B-Thinking's AIME 2024 pass@1 from 61.88% to 73.33% and Huginn's HumanEval pass@1 from 22.56% to 31.71%. With guidance, halving the loops matches or beats unguided full depth, saving 22.5–48.2% forward FLOPs.
Recursion 1 writes shared KV → recursions 2…R read it + short local window
Depth control
Fixed recursions (five in the main results)
KV / state strategy
Shared first-recursion cache plus a short per-recursion window
Reading note
From 150M to 1B, the hybrid variant with five recursions lowers FineWeb-Edu perplexity by 1.12–1.82 versus a same-size Transformer while using 76–79% less context memory. Later recursions mostly attend to the shared memory, which also acts as a gradient highway.
[ Shared block ] × R + exit gate; dense → looped conversion via input-mixing scalar
Depth control
Learned exit gate
KV / state strategy
See paper
Reading note
The 1.4B LoopLM beats a parameter-matched dense model on all 12 benchmarks with the same data (+14 GSM8K, +10 MATH, +22 DROP) and approaches a 3.9B dense model at matched inference compute. Looped Qwen3-1.7B improves over continued dense training on every benchmark.
Changing intermediate supervision during training changes which transitions can be induced, suggesting the steering selects computations learned by the backbone rather than creating new ones.
[ MoE block ] × R → effective parameters (sparsity-conditional)
Depth control
Law-derived recurrence
KV / state strategy
See paper
Reading note
Sparsity gives about 3× active-parameter efficiency and recurrence about 2× total-parameter efficiency on reasoning. At trillion-token scale and matched training compute, a looped MoE matches a roughly 2× larger non-looped MoE on reasoning benchmarks.
Each denoising step: [ Shared DiT blocks ] × r (deep supervision)
Depth control
Fixed loops per denoising step
KV / state strategy
Not applicable (diffusion)
Reading note
Beats non-looped baselines at matched parameters and compute; a 260M looped model surpasses a 6.5× larger model on several text-to-image benchmarks with 4.9× less inference compute. More loops help more than more denoising steps at a fixed budget.
Low-bit [ Block ] × r → halting head fires → 8-bit [ Block ] × few
Depth control
Halting-head controller plus final high-precision loops
KV / state strategy
See paper
Reading note
A single label-free measurement predicts which compressed models fail. The controller beats fixed-depth inference by up to 15 points on Sudoku-Extreme and Maze-Hard with under a third of the weight traffic.
State → [ Shared block ] × r(state) → fixed point → action
Depth control
State-dependent loop count
KV / state strategy
Not applicable (policy network)
Reading note
Across 22 tasks in six environments, online and offline, it matches or beats an untied baseline with 16× more parameters. On Boxoban, loop counts grow with the remaining number of pushes.
[ Shared modules (Module-Loop) ] within [ Model-Loop ] × r over a vision-language state
Depth control
Fixed loops
KV / state strategy
See paper
Reading note
Trained through language pretraining, multimodal training and post-training, it outperforms several similar-size and larger non-recurrent models on multimodal understanding and visual reasoning, with visible shifts in visual attention across loops.
[ Block ] × R where each step = local gradient update on an implicit target
Depth control
Fixed loops
KV / state strategy
See paper
Reading note
Aligning input maps of existing transitions lowers training loss and improves commonsense accuracy; OperLoop improves average generative performance over looped and non-looped baselines at matched training FLOPs.
state ← state + η_t · update, η_t from progress vs fluctuation
Depth control
Fixed loops with adaptive update scale
KV / state strategy
See paper
Reading note
Improves terminal accuracy on structured reasoning tasks across several recurrent architectures; when used in training, reaches baseline accuracy with up to 1.56× wall-clock speedup. Provides sufficient conditions for fewer loops to a target quality.
Extra recurrence can improve reasoning beyond the training horizon while degrading knowledge. Effective depth alone does not predict behavior, and non-recurrent output layers improve robustness to under-unrolling.
Across four recurrent-model checkpoints on math and code, DAS-Wave reaches 4.00–6.96× mean throughput speedup over full-depth autoregressive decoding in the same stack.
[ Shared block ] × R → distill depth d+1 into depth d
Depth control
Depth-elastic, state-wise adaptive
KV / state strategy
Not applicable (policy network)
Reading note
On 30 long-horizon goal-conditioned environments it keeps full-depth performance, cuts average recurrent depth by up to 43% and gives up to 1.34× wall-clock speedup in a stress test.
[ Untied attention + shared wide MoE ] × 2k (flattened from k layers)
Depth control
More passes over fewer, wider expert layers
KV / state strategy
Per-pass attention parameters
Reading note
At 100B tokens, loss improves monotonically with flattening; the most flattened model ends 0.012 nat below the looped baseline at equal parameters and compute. Untying attention gives more balanced, confident routing.
Strong in-distribution accuracy does not imply depth generalization: CoT and pause-token models rely on local graph features. Results are on the synthetic ProsQA-Ext task.
Improves mathematical reasoning across sizes, ranks and recipes, extending to instruction tuning and code. Inference uses standard LoRA at all loops with no extra parameters or compute.
Token-level adaptive iterations up to a maximum depth
KV / state strategy
See paper
Reading note
On AIME it improves the accuracy-compute slope by 53% over the non-looped baseline and exceeds its peak accuracy by about 3.4 points at matched compute. Gains over the baseline grow from +2.8 at depth 2 to +3.9 at depth 8.
[ Block ] × R — matched vs decoupled recurrence training
Depth control
Length generalization beyond trained depth
KV / state strategy
See paper
Reading note
Recurrent states encode both content and its computational status, and transferable residual directions causally control whether information can be used in later steps. Length generalization need not rely on faithful step-by-step reasoning.
On Huginn-3.5B, INT4 fails mainly at the loop-entry adapter. Across nine checkpoints from seven looped architectures, accumulating the GPTQ Hessian across recurrence steps beats one-step GPTQ and RTN everywhere and recovers bf16-level accuracy on Huginn.
18 block calls: e.g. each of 9 blocks × 2 consecutively, or 6 blocks cycled × 3
Depth control
Fixed executed depth
KV / state strategy
Not applicable (flow matching)
Reading note
Repeating each of nine blocks twice keeps competitive intelligibility, speaker similarity and quality at 32 steps with 47.1% fewer parameters. Layout rankings change with the sampling budget; middle placement is most consistent.
Early steps: full attention → working set; later steps: [ Core ] on reused sparse support
Depth control
Fixed recurrence in two stages
KV / state strategy
Later steps attend only to the discovered working set
Reading note
Largely preserves full-attention performance on multi-hop QA, with better trade-offs at longer contexts; a sparse implementation gives up to 1.76× late-step attention speedup over FlashAttention at 4K context.
Across six Spec-Bench categories, 2.42× speedup on Ouro-2.6B and 3.54× on Huginn-3.5B over autoregressive decoding, rising to 4.81× on Huginn with cross-recurrence KV sharing.
token t → [ Backbone ] → thought step → [ Backbone ] → token t+1 (state fed forward)
Depth control
One extra latent step per vocabulary token
KV / state strategy
See paper
Reading note
On NanoChat MoE backbones at matched training FLOPs, it beats PonderLM-2 and overtakes a two-loop Transformer as compute grows. One thought token recovers about 89% of the gain from doubling depth with about 48% fewer parameters.
Latent update per step → score changes decomposed (common vs contrast)
Depth control
Analysis of finite-depth answer preservation
KV / state strategy
See paper
Reading note
On Huginn-3.5B and Ouro-1.4B, accounting for direction and pairing lowers the earliest qualifying depth by a further 22.5–34.4% of total depth. Retrospective analysis of completed trajectories.
Truncation vs controls: same applications / fewer distinct iterations / readout shift
Depth control
Evaluation methodology
KV / state strategy
See paper
Reading note
The key control is only realizable in weight-shared architectures. Includes dense-transformer negative controls and a training intervention to check causality.
Small bounded cell state rather than a cache of every step
Reading note
Trained to depth 12 on Rule 30, it keeps 99.7% exact-row accuracy at depth 60 where a Block Universal Transformer gets no row right, and one checkpoint stays above 99.95% at depth 1,000. On OpenWebText2 it beats BUT and roughly matches CoTFormer.
Outperforms prior looped models overall on six reasoning benchmarks, with 58.8% on ARC-AGI-1 and 12.2% on ARC-AGI-2; different initial noise gives multiple valid solutions.
The 29M convolutional model reaches 0.28 rFID and 32.54 dB PSNR on ImageNet-256 with about 65% fewer parameters than 84M reference VAEs. Fewer stored weights mean more arithmetic and longer runtime in tested settings.
Each looped layer attends to its own earlier-iteration states
Reading note
On MathQA with the same looped backbone it reaches 56.9% with 2.0 loops on average, 2.2 points above the best fixed depth at matched compute, while ACT and PonderNet collapse to one loop. Gains over same-budget fine-tuning grow from 0.6B to 8B.
Virtual unrolling adapter → Jacobian lens over iterated layers
Depth control
Analysis of released models
KV / state strategy
See paper
Reading note
A workspace forms in the iterated part of both models, but access differs: Ouro rebuilds the content every loop, so writes must span all remaining loops, while Huginn carries it across all sixteen recurrences yet interventions act within about two recurrences.
Separate fitted Chinchilla-style scaling laws estimate 6.8–18.0% training-FLOP savings on the compute-optimal frontier. Experiments cover four sizes up to 54B non-embedding parameters, with the largest domain gains on Code and stronger gains for longer samples and more in-context examples.
Recurrent models beat one-pass baselines, but a single-state recurrent Transformer is comparable to HRM. HRM behaves like constraint-aware iterative refinement without a compact, causally important feature set.
Weight-tied map iterated → fold phase vs Neimark–Sacker phase
Depth control
Test-time iterations governed by critical slowing down
KV / state strategy
See paper
Reading note
In the fold phase a normal-form reduction predicts relaxation time from local derivatives and implies a workload tail P(τ > N) ~ N⁻²; in the Neimark–Sacker phase this law disappears. Theory with small-scale experiments.
At 15M, MixerLoop beats full-block looping on aggregate CORE; at 110M it keeps 41.5% of the CORE improvement while cutting recurrent-backbone projection FLOPs by 45.9%.
Settling operators do not degrade with depth, and on Sudoku accuracy rises from 0.19 to 0.34 past the training horizon. A terminal fixed-point objective controls the regime; Huginn-3.5B falls in the non-settling family.
Recurrence mostly helps compositional, dependency-aware tool use on API-Bank, BFCL and NESTful, with smaller gains on single API calls. Adaptive inference gives a better compute-performance trade-off than fixed deep recurrence.
At iso-FLOPs, a 3-layer model matches a 12-layer GPT-2 Small and leads MoR and heavy-tail depth sampling in all nine scale-by-budget cells. At iso-parameters, deeper recurrence reaches 2.76 validation loss versus 2.84.
Fixed schedule; L-stack applications adjustable at inference
KV / state strategy
See paper
Reading note
Mimir v1.5 reports the best sub-2B average across 20 English, Danish and math/code benchmarks, ahead of HRM-Text 1B and Ouro 1.4B/2.6B. Changing L-stack applications degrades gracefully while changing H-stack applications breaks the model.
Loop step k: batch of tokens still looping ‖ exited tokens → non-looped tail
Depth control
Per-token adaptive depth at serving time
KV / state strategy
Looped KV-cache management across exits
Reading note
Realizes up to 99% of the estimated maximum speedup on Ouro 1.4B and Huginn 3.5B. Fully looped architectures suit adaptive inference best, since large non-looped parts complicate scheduling.
One task step per loop; extrapolates to about 1.5× supervised depth
KV / state strategy
See paper
Reading note
Non-inferior to the base at one loop on a preregistered ARC battery. Against a same-size scratchpad model it wins overall (84% versus 72%), retains far more beyond depth 10, and answers 7.6× faster. Single-author study on synthetic tasks.
state = anchor(x) + deviation; [ Block ] × R on deviation (zero ↦ zero)
Depth control
Trained and extrapolated recurrent depth
KV / state strategy
See paper
Reading note
Improves the controlled recurrent quality frontier on WikiText-2/103, web pretraining, held-out transfer and LAMBADA; ablations credit the learned anchor and anchor invariance.
At matched cache budget it beats MLA-style head compression, cross-layer sharing, KV quantization and final-loop reuse. On one H200, Ouro-1.4B batch capacity at 4K context rises from 32 to 768 sequences at 21.3× compression; transfers to Huginn-3.5B.
token t−1: middle layer → fused into early layer of token t
Depth control
One forward per token; recurrence across time
KV / state strategy
Cached middle-layer state from the previous token
Reading note
Beats data- and parameter-matched Transformers in pretraining and multi-hop reasoning; recurrence over as little as 20% of the network often beats full-layer recurrence. Can be retrofitted into a pretrained 1.7B model with brief fine-tuning.
[ Weight-tied core ] × r until output stabilizes (per token)
Depth control
Training-free per-token halting
KV / state strategy
See paper
Reading note
On a 135M-class model trained on FineWeb-Edu, the median token converges by loop six while about 10% still update at loop eight. Halting on stability matches uniform depth-8 quality with 4.94 average loops; average depth is a FLOP proxy, not a measured speedup.
EqR-style recurrent reasoner × D → K candidate trajectories → Hopfield-energy selector → selected answer
Depth control
D=64 recurrent steps in the main experiments; test-time computation scales through recurrent depth and candidate breadth
KV / state strategy
See paper
Reading note
ERM primarily changes the inference-time selection mechanism rather than replacing the recurrent reasoning backbone. The main experiments use EqR-style recurrent rollouts with D=64 and K=128 candidates. The energy is task-structured: Sudoku uses row, column, and box permutation memories, PPBench uses puzzle-specific local memories and global distance potentials, and the reported Maze global-memory setting uses an input-conditioned shortest-path memory.
Looped Mamba beats parameter-matched baselines on controlled reasoning tasks; in pretraining it stays competitive downstream with fewer parameters, though deeper non-looped models keep a perplexity edge at strict iso-FLOPs. Adapted exits keep perplexity near full computation.
[ Block ] × R → trajectory → readout (gate or confidence)
Depth control
Learned or post-hoc halting
KV / state strategy
See paper
Reading note
Failures stem mainly from the trajectory induced by joint gate training rather than gate expressivity. Ouro's ponder gates are competitive but not uniformly Pareto-optimal; measured latency confirms that lower average exit depth saves time.
[ Shared block ] × r, r sampled (heuristic or RL-Halting) during training
Depth control
Stochastic stopping at training time
KV / state strategy
See paper
Reading note
On binary addition, Dyck-1, Unique Set and Copy, learned stochastic stopping often improves the accuracy-stability trade-off but can also stabilize a suboptimal computation.
PLT: [ Stack ] × 2 with cross-loop position offsets + shared-KV gated sliding window
Depth control
Two loops best; three or more regress
KV / state strategy
Shared KV with gated sliding-window attention (PLT)
Reading note
The two-loop model raises SWE-bench Verified from 43.0 to 64.4 and Multi-SWE from 14.0 to 31.0 over the non-looped baseline. Loop 2 provides the main refinement while later loops oscillate.
For L unique layers repeated N times, the parameterization is epsilon = lambda / (N sqrt(L)). The analysis separates correlation across loops from variance across unique layers; Transformer experiments compare loss and learning-rate transfer across loop counts at fixed unique-layer depth.
[ Block ] × R → RMSNorm readout (scale hidden) per loop
Depth control
Variable inference depth
KV / state strategy
See paper
Reading note
In 44M and 129M looped models without inter-loop normalization, hidden-state norms reach thousands or tens of thousands; scale-visible readouts, norm penalties or scale-removing recurrence keep them in the tens and lower perplexity at matched depth.
Input → program predictor → sequence of skipped / repeated pretrained layers
Depth control
Per-input layer program (skip or loop)
KV / state strategy
See paper
Reading note
On mathematical reasoning benchmarks it improves accuracy over standard inference and prior dynamic-depth methods, often with fewer executed layers, and gains persist out of distribution.
Strictly controlled against a vanilla MoE with identical total parameters, per-token FLOPs and active sublayer ratios, its nine-benchmark advantage grows from over 1 point at 3B to about 3 points at 9B.
token t−1: high-level state → input of token t (one forward per token)
Depth control
Recurrence across tokens
KV / state strategy
Standard KV cache interface
Reading note
On 1.3B and 2.1B nanochat-style backbones it improves BPB and CORE at matched effective compute, beats two-forward PonderLM-2 and matches a three-loop Transformer in BPB with 9% latency overhead.
Each denoising step: early layers → [ Early-middle layers ] × k → remaining layers
Depth control
Loop count adjustable during sampling
KV / state strategy
Not an autoregressive cache setting
Reading note
Matches same-size MDMs with up to 3.3× fewer training FLOPs and gains up to 8.5 points on GSM8K; it also beats deeper non-looped MDMs at similar per-step compute. Code and weights were announced for release.
Across seven dense, MoE and MLA+MoE families, it improves Qwen3-4B-Instruct by 2.64 points on MMLU-Pro, Qwen3-30B-A3B-Instruct by 1.14 on CommonsenseQA and Moonlight-16B-A3B by 1.20 on OpenBookQA.
Backbone → proposal → [ Attractor module ] solved to fixed point (implicit gradients)
Depth control
Adaptive iterations until convergence
KV / state strategy
See paper
Reading note
A 770M model beats a 1.3B Transformer trained on twice the tokens; a 27M model trained on about 1000 examples reaches 91.4% on Sudoku-Extreme and 93.1% on Maze-Hard. Fixed-point training can let the solver be removed at inference.
Improves reasoning-oriented performance without longer generated traces or recurrent pretraining from scratch; released conversions include Qwen3 and Phi-4 backbones.
[ Shared MoE layers ] × R (routing diverges across loops) + exits at loop boundaries
Depth control
Early exits at loop boundaries
KV / state strategy
See paper
Reading note
Loop boundaries make better exit points than intermediate layers of standard models, with earlier output convergence. Recommends looped MoE with early exits for memory and inference savings.
Shared W4A4 backbone × R + loop-aware scaling and cross-loop alignment
Depth control
Fixed loops of the base model
KV / state strategy
See paper
Reading note
Under W4A4, reports a 67.8% average relative improvement in five-task accuracy across four backbones and about 75% relative perplexity reduction versus the strongest static PTQ baseline.
Begin block → [ Middle block ] × R (hyper-connections after each loop) → End block
Depth control
Fixed recurrence
KV / state strategy
See paper
Reading note
Performs well against depth-matched Transformer and mHC baselines with about 50% fewer parameters, and the advantage persists after post-training weight quantization.
[ Single block + memory tokens ] × r (ACT halting)
Depth control
Adaptive Computation Time
KV / state strategy
Learned memory tokens
Reading note
No configuration without memory tokens works; 8–32 tokens reach 57.4% exact match. Identifies an ACT router initialization trap and fixes it with a negative bias ('deep start'); ACT then has far lower seed variance than fixed depth.
At r = 4 a 410M looped model matches a 580M non-looped model but costs as much to train as a 1B one. Truncated backprop lowers φ to 0.38 while hyper-connections raise it to 0.65, making φ a diagnostic for genuine loop improvements.
[ Single layer + recall + outer norm ] × r → fixed point
Depth control
Test-time iterations on harder problems
KV / state strategy
See paper
Reading note
Single-layer looped transformers on chess, Sudoku and prefix sums follow the framework's predictions; a new internal-recall placement is competitive and much better on Sudoku once outer normalization is used.
[ Shared layers ] × r — more iterations for more hops
Depth control
Scaling inference-time recurrence for deeper reasoning
KV / state strategy
See paper
Reading note
Systematic generalization emerges through a three-stage grokking process; depth extrapolation is unlocked by more inference-time iterations, where vanilla transformers fail at both.
On graph reachability, nested boolean logic and relational text, accuracy climbs once thinking steps meet task complexity, with extrapolation beyond the training range on the graph task.
[ Shared stack ] × R with RL signals on latent steps
Depth control
Compresses reasoning into fewer iterations
KV / state strategy
See paper
Reading note
Instantiated on Ouro at several scales; improves per-step representation quality, gives Pareto gains in accuracy versus computation, with large gains on hard tokens.
Evaluated on graph networks, MLPs and nanoGPT; generally speeds convergence and reduces parameters through sharing. Simple Euler steps give the best cost-performance trade-off.
Looping mainly helps math while memory recovers commonsense performance; together they beat an iso-FLOP baseline with three times as many layers on math. Early layers loop little and later layers loop and read memory more.
[ Shared stack ] × R → reward assigned across all latent steps
Depth control
Fixed loops of the base model
KV / state strategy
See paper
Reading note
On Ouro-1.4B/2.6B-Thinking it improves mean accuracy on MATH-500, AIME24/26 and BeyondAIME by 5.8% and 10.9% over GRPO, and transfers to non-math reasoning despite math-only training.
[ LMM backbone ] × r via Recursive Connector (vision / language projections)
Depth control
On-demand recursion depth
KV / state strategy
See paper
Reading note
Reports +3% over standard Transformers and +7% over vanilla recursive baselines, with few loops on constrained devices and more loops when compute allows.
VLM features → [ Weight-tied action head ] × r (stop on convergence) → actions
Depth control
Adaptive stopping based on latent convergence
KV / state strategy
Constant memory in recurrence depth
Reading note
Manipulation tasks that fail completely with one iteration exceed 90% success with four, while simple tasks saturate quickly. Reports up to 80× faster inference than prior reasoning-based VLAs.
Depth recurrence with attention over earlier depths
KV / state strategy
Attention over states from earlier depths
Reading note
Needs 2–8× fewer training tokens for the same accuracy as FLOP-, parameter- and memory-matched baselines, and shows 2–11× more expert selection diversity than standard MoEs.
Recurrence increased by curriculum; variable at test time
KV / state strategy
See paper
Reading note
On mathematics, converted recurrent models beat simply post-training the original model at the same compute budget. Released checkpoints include Recurrent-OLMo and Recurrent-TinyLlama.
The inspected Ouro-2.6B implementation computes all loop states before selecting an exit output. Its exit option alone does not demonstrate skipped computation. Published configuration: 48 layers, 4 passes.
Token t: loop 1 ‖ token t−1: loop 2 ‖ … in one pass; shared KV + gated SWA
Depth control
Fixed loops executed in parallel across tokens
KV / state strategy
First-loop KV shared across loops plus gated sliding-window attention
Reading note
Reports looped-model accuracy with almost no extra latency or memory over a standard transformer. LoopCoder-v2 later trains 7B PLT coders and finds two loops optimal.
Each pass: new token at depth 1 ‖ earlier tokens refined further
Depth control
Recurrence spread across positions in parallel
KV / state strategy
See paper
Reading note
Applies to existing 3.5B recurrent-depth transformers such as Huginn without tuning, giving up to 5× speedup; generation is argued to be strictly more expressive than autoregressive decoding at the same time budget.
[ Shared attention block ] × L — staged hierarchical training
Depth control
Fixed loops
KV / state strategy
See paper
Reading note
The landscape account is a conjecture supported by sample- and Hessian-level observations. SHIFT accelerates training of looped models while reaching comparable performance.
On Pythia 160M–6.9B it beats recursive baselines and, at 1.4B, outperforms the larger non-recursive model by 1.06 points with 33% fewer non-embedding parameters.
Finds limited evidence of interpretable latent CoT and inconsistent probing across blocks; more recurrence gives only marginal gains, well short of explicit CoT on these tasks.
With 27M parameters and about 1000 training examples, reports near-perfect results on hard Sudoku and large mazes and strong ARC performance without pretraining or CoT data. Later analyses (for example, Dissecting HRM and TRM) question how much the hierarchy itself contributes.
The benefit carries to language modeling: looped models compete with or beat deeper non-looped models on downstream reasoning, and scaling tracks effective depth. Also connects looping to chain-of-thought.
From 162M to 466M parameters, a 162M ITT reaches 96.5% of a 466M Transformer's performance and cuts training data by 43.2%, beating Transformer and loop variants on 11 benchmarks.
[ Model ] × r with attention to earlier passes' tokens; router picks rᵢ
Depth control
Budget-adaptive per-token depth
KV / state strategy
Keeps KV from every earlier pass
Reading note
Reaches accuracy close to much larger models and lowers computation through adaptive allocation. A later study (LSTM-UT) finds its full cache can hurt extrapolation on cellular-automaton tasks.
A constructive theory result: the looped transformer emulates a calculator, a basic linear-algebra library and in-context learning via backpropagation. Weights are hand-constructed, not trained.
[ Step function ] × n, halt with learned probability λₙ
Depth control
Learned probabilistic halting
KV / state strategy
Not applicable
Reading note
Improves over earlier adaptive computation methods on a synthetic task, succeeds at extrapolation tests, and matches state of the art on a QA dataset with less compute.
Embedding (factorized) → [ Shared encoder layer ] × L
Depth control
Fixed depth with all layers shared
KV / state strategy
Not applicable (encoder)
Reading note
Achieved state-of-the-art GLUE, RACE and SQuAD results with fewer parameters than BERT-large; a canonical example of full-stack weight sharing in pretrained Transformers.
[ RNN step ] × N(t), halt when cumulative probability ≥ 1 − ε
Depth control
Learned halting per input step
KV / state strategy
Not applicable (RNN)
Reading note
Greatly improves synthetic tasks such as parity, logic, addition and sorting. Character-level language modeling gains are small but allocate more compute to hard transitions. Basis of Universal Transformer halting.