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6415documentos

arXiv AI Papershace 24 días

Capable Teacher, Myopic Scorer: Reward-Free Teacher Repair for Agentic On-Policy Distillation

Agentic on-policy distillation (OPD) trains a student agent on its own trajectories while a teacher scores every token. A natural explanation of where it goes wrong is compounding error: the student's mistakes carry its trajectories beyond the region where the teacher is reliable, so recent methods keep training inside that region, through episode-length curricula, teacher-guided turns, prefix replay, or teacher intervention where the two disagree. We show this explanation is incomplete. From the states where the student departs most from the teacher, the teacher's own takeovers still finish the task, yet the student trained on its token scores keeps running out of turns. We attribute this to the myopia of next-token scoring. The teacher rates each token without seeing where the episode ends, so its scores let through the safe, delaying moves that outcome-based RL would punish with zero reward, a penalty agentic OPD lacks. We call this difference between what the teacher does and what it scores the rollout-scorer gap. We propose Rollout-to-Scorer Distillation (R2S), which repairs the scorer before distillation by mining high-disagreement states from the untrained student, running teacher takeovers from them, keeping the states where enough takeovers pass a reward-free check such as consensus or an LLM judge, and fine-tuning the teacher to give its takeover-informed scores on every assistant token from the divergence turn onward, without the takeover in context. The repaired teacher is frozen as the sole scorer for standard OPD, requiring no task reward during repair or training. Across three environments, two teachers and four students, R2S outperforms three state-of-the-art agentic-OPD methods in all twelve teacher-student-environment cells, by 12.8 success points on average over the best of them in each cell. Project page: https://tree-shu-zhao.github.io/r2s/.

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arXiv AI Papershace 24 días

SceneRig: An Agentic System for Simulation-Ready 3D Scene Reconstruction from Single Images

Simulation is a powerful tool for robotics, but building diverse, simulation-ready scenes at scale remains difficult. Reconstructing scenes from single images is a promising alternative, but visual similarity alone is insufficient: objects must also be recovered, accurately placed, and physically grounded. Existing methods that rely on post-hoc physics settling often misplace objects, producing scenes that look correct but behave poorly under manipulation. We present SceneRig, an agentic system that turns a single image into an editable, simulation-ready scene. SceneRig combines geometric, visual, and physical tools to reconstruct a faithful 3D scene. An agent then performs edit-and-validate loops inside a physics simulator to achieve physical stability. Across diverse inputs, SceneRig improves instance segmentation, object-centric appearance, and 3D placement while keeping the reconstructed scenes physically stable. Furthermore, SceneRig is effective for downstream robotics: its scenes predict real-robot policy outcomes more accurately than the baseline method and more reliably support replay of recorded manipulations without trajectory adaptation.

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arXiv AI Papershace 24 días

Beyond the Sequentiality Bottleneck: A Hybrid Diffusion-Autoregressive Framework for Efficient Multimodal Reasoning

Autoregressive models (ARMs) face a "Sequentiality Bottleneck" in reasoning, while diffusion language models (DLMs) offer parallelism but lack semantic coherence. To addresses both limitations, we propose Collaborative Reasoning Asymmetric Learning (CoRAL), a hybrid framework via an Asymmetric Planner-Executor (APE) architecture: a DLM Planner generates K diverse reasoning trajectories in parallel, while a large ARM Executor selects the optimal plan and synthesizes the final answer. To train the Planner effectively, we introduce Reward-Guided Denoising for CoT (RDC), which integrates outcome-based rewards into the diffusion process to steer it toward goal-directed reasoning. Experiments on vision-language benchmarks show that CoRAL significantly outperforms prior diffusion-based methods and achieves competitive performance against strong ARMs, achieving both superior efficiency and quality.

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arXiv AI Papershace 24 días

Continuous-Time Policy Gradient from Discrete Data: Convergence Analysis and Model-Free Algorithms

Continuous-time reinforcement learning seeks an optimal policy under unknown dynamics driven by stochastic differential equations using observations collected at discrete times. We develop a policy gradient framework that combines rigorous convergence analysis with learning from discrete data. We first establish a performance difference lemma and derive policy gradient formulas for randomized and parametrized policies. We then prove exponential convergence of an entropy-regularized policy flow to the regularized optimum. For discrete parametrized updates, we derive a convergence bound that explicitly separates the contributions of critic error, policy-regression error, and the learning step size. To enable learning under unknown dynamics, we integrate this framework with PhiBE, which constructs a continuous-time surrogate generator from discrete transition data. The resulting on-policy and off-policy model-free actor-critic algorithms preserve the continuous-time structure of the control problem. Numerical experiments on an entropy-regularized linear-quadratic regulator and a nonlinear wheeled robot demonstrate the performance and stability of the proposed methods with both known and data-driven dynamics.

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arXiv AI Papershace 24 días

Spatially Informed and Direction Sensitive Representation Learning for Spatial Transcriptomics

Cellular states are shaped by both intrinsic gene expression programs and the spatial microenvironment in which cells reside. Spatial transcriptomics makes this context measurable, but effective representation learning requires capturing both spatially informative molecular signals and intercellular organization. Existing approaches address these challenges only partially: conventional gene selection can overlook spatially structured signals, while methods for encoding cell-cell relationships often either reduce a cell's microenvironment to proximity alone, discarding direction, or rely on absolute coordinates tied to an arbitrary tissue frame. Consequently, cells with similar neighbor distances but distinct spatial organization may receive overly similar representations, despite experiencing different biological contexts that shape their cellular states. We introduce Tessera, a spatial representation learning framework that addresses both molecular selection and cellular context. Tessera combines spatially informed gene selection with a geometric inductive bias that contextualizes each cell within its local tissue environment. By modeling relative cell-cell relationships, Tessera is translation-invariant while preserving direction, allowing neighboring cells to contribute differently according to their spatial context and cellular state. Across held-out tissues and technologies, Tessera outperforms existing spatial foundation models in representation quality and generalizes zero-shot to contexts up to 16larger than those seen during training.

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arXiv AI Papershace 24 días

Sage-A1.5: From Long-Horizon Execution to Scientific Exploration with Capability Composition

In this report, we introduce Sage-A1.5, a 35B Mixture-of-Experts agentic model that builds on A1 to advance long-horizon problem solving through adaptive capability composition. Compared with A1, Sage-A1.5 learns not only to solve individual tasks, but also to organize work and revise its approach during execution. Sage-A1.5 highlights three improvements: agentic orchestration, verification, and open scientific discovery. First, agentic orchestration extends single-agent execution through task delegation and parallel collaboration among subagents with isolated contexts, enabling complex tasks to be decomposed and addressed concurrently and supporting efficient execution while reducing interference across concurrent subagents. Second, agentic verification strengthens long-horizon reliability through progress checking, evidence validation, and reflective strategy revision, allowing the agent to recognize unproductive search paths and pursue alternative approaches. Third, open scientific discovery extends agentic capabilities beyond answering predefined questions to conducting exploratory research, connecting scientific inquiry with interactive analysis and inspectable research artifacts. We showcase these capabilities through complicated agentic tasks and multidisciplinary scientific applications. Besides, we hope Sage-A1.5 provides the community with an accessible foundation for open scientific discovery, offering a practical path from long-horizon task execution to collaborative, verifiable, and inspectable scientific exploration.

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arXiv AI Papershace 24 días

Cache Engineering for Agents

Large Language Model (LLM) agents can handle increasingly complex tasks, navigating long trajectories of observations and actions. As these trajectories grow, key-value (KV) cache I/O and storage become the dominant cost. Trajectories can also exceed the LLM's maximum context length, requiring control over active cache growth. Prior work uses context engineering to reduce cache size by trimming or summarizing history. However, modifying the trajectory prefix invalidates subsequent cached states and triggers costly re-prefilling of retained context. Task accuracy can also suffer when too little context is retained to limit re-prefill costs. We propose cache engineering as a new scheme. It allows agents to directly control their active KV cache without modifying the trajectory or re-prefilling retained context. Specifically, agents can proactively use drop to exclude selected cache from future attention and repos to adjust retained keys' position encodings. We formalize these cache operations and identify history dependence as a serving challenge: identical token sequences can yield different cache states under different cache operation sequences, making prefix matching insufficient for correct cache reuse. Thus, we co-design the AKV serving engine. It provides a stateless interface compatible with existing serving APIs, while internally managing stateful cache reuse, eviction, and restoration. Experiments show that a tool-response rolling-drop cache policy improves average benchmark scores by 1.7 points and reduces average cost by 27.7%, compared with conventional threshold-triggered summarization. With AKV, it achieves up to 1.2x the output throughput and a 1.5x speedup in P95 request latency. Recursive self-improvement of cache policies further reduces cost by 7.1% over expert design. Our code is at https://github.com/thu-nics/AKV.

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arXiv AI Papershace 24 días

The Token-Energy Law of LLM Decode: Active Bytes and Realized Time Account for Joules per Token

Energy-efficient deployment of large language models requires understanding how model architecture, hardware, and serving settings shape energy per generated token. We introduce the Token-Energy Law, a compact, interpretable model of GPU-board energy per output token for dense, full-history decoding. It relates token energy to two core dimensions: active bytes and realized decode time. Active bytes describe logical byte demand per output, separating dense weights shared within a decode iteration from private key--value (KV) histories read and updated for each request. Realized decode time is the elapsed decode duration divided by the number of output tokens produced. We combine the two byte coordinates and time in a common affine form, with coefficients calibrated for each model and hardware/runtime configuration. This formulation explains how batching amortizes shared-weight work, longer attended histories increase private KV demand, and throughput determines the time-associated cost per output. Our experiments cover three models under BF16 decoding on NVIDIA Hopper and Blackwell GPUs, including two-GPU tensor parallelism on Hopper. Using fixed coefficients and observed decode time, held-out condition-mean energy estimates differ from measurements by at most 3.73%.

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arXiv AI Papershace 24 días

Evolving Semantic Indexes for Generative Retrieval

Generative retrieval (GR) casts search and recommendation as autoregressive generation of discrete semantic identifiers (SIDs). SID design largely determines GR quality, but its discrete, interdependent components (SID construction, collision handling, decoding, and training) cannot be optimized end-to-end with gradients and remain largely hand-designed. We introduce a framework that improves these components through LLM-driven evolutionary program search, to our knowledge the first application of this approach to GR. A locally served, open-weight 31B model mutates executable GR components, so no proprietary LLM API is used, and a 1M-parameter proxy decoder provides fast feedback in place of repeatedly training the 1B target generator. Across 21 SID programs on NQ320K, proxy and target Recall@100 show strong rank correlation (Spearman ρ= 0.90). Guided by this proxy, evolution improves the seed GR on NQ320K, MSMARCO300K, and a multilingual e-commerce dataset under a fixed training recipe. On NQ320K, jointly evolving SID construction and collision handling improves Recall@1 from 55.8% to 59.3% and Recall@100 from 85.8% to 90.6%. A component study shows that proxy reliability depends on what is evolved: improvements transfer reliably for the semantic index but not for the generator's training objective or the full pipeline. LLM-driven evolutionary search is thus a practical tool for GR design, provided proxy reliability is established for the chosen search space.

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arXiv AI Papershace 24 días

Continual Concept Personalization in Diffusion Models via Residual-Space Adaptation

Text-to-image personalization aims to equip pretrained diffusion models with the ability to learn user-specific concepts from only a few reference images. Existing methods, however, typically train each concept independently or assume that all concepts are available simultaneously. In practical applications, concepts often arrive sequentially, requiring models to continually learn new subjects, objects, or styles while minimizing interference with previously acquired concepts. We address this continual concept personalization setting by extending residual-space adaptation to the stochastic adaptation process of pretrained diffusion models. Our work achieves the highest concept fidelity among competing continual personalization methods while maintaining near-zero forgetting. Beyond these empirical gains, the learned residual-space dynamics reveal where adaptation capacity is concentrated within the diffusion model. Restricting adaptation to the identified layers by our method preserves nearly the same performance while reducing the number of trainable parameters by up to 72\%, demonstrating that residual-space dynamics can identify the layers that contribute most to continual concept personalization and directly guide more parameter-efficient adaptation.

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arXiv AI Papershace 24 días

The Dynamics of Collapse and Recovery in Clustered Federated Learning

Clustered federated learning (CFL) can suffer from cluster collapse. In particular, under unfavorable initialization, early client assignments concentrate heterogeneous groups on one cluster model, which is then trained toward a pooled compromise, while another model receives no updates. Since the current models determine client routing and routing determines subsequent model updates, we formulate clustered federated learning as control of an unknown dynamical system, with cluster models as the evolving state, client routing patterns as the control input, and routing interventions as state dependent controllers. In a two group linear model, we identify an initialization region in which greedy routing makes collapse self reinforcing. We then derive finite horizon error dynamics showing how routing interventions can reverse collapse by repeatedly training the starved model, primarily with updates from its compatible group. Motivated by this condition, each client uses blocked EXP3 to choose between greedy routing and interventions that redirect updates toward unused models. Experiments show that with simpler predictors and faster specialist formation, autonomous controller selection recovers usable specialists from unfavorable starts that trap standard CFL baselines. In transformed FEMNIST, the synchronized-entry configuration with persistent commitment recovers usable specialists in settings where the autonomous controller configuration often does not. Code and scripts for reproducing the experiments and figures are available at https://github.com/prsinha1/FedRecover.

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arXiv AI Papershace 24 días

Asymmetric Bidirectional Predictive Dynamics in Google Search and COVID-19 Time Series: A Multi-Horizon Forecast-Error Analysis

Online search activity is widely used as a digital trace of information seeking during epidemics, but its predictive value is often reduced to correlation, a selected lag, or a single forecast-error score. We examine eight pre-specified Google Trends-COVID-19 pairs in both directions using matched LightGBM forecasts over rolling 400-day windows and horizons of 1-14 days. Log-RMSE gain is combined with forecast-error distribution entropy (FEDE), which captures horizon-dependent changes in the signed error distribution. The resulting directional predictive signature comprises response class, RMSE effect, FEDE endpoint and early-slope responses, persistence across rolling regimes, and horizon profile. For screening candidate directions, we also introduce a relative persistence-weighted directional predictive response strength (DPRS) on a within-study 0-100 scale; 100 denotes the strongest observed response in the comparison set, not an absolute coupling maximum. The pairs show concordant, RMSE-only, FEDE-only, and mixed responses. "loss of smell" -> reported cases is accuracy-dominant, with mean log-RMSE gain 0.104 (about 9.9% equivalent RMSE reduction) and DPRS 100, whereas the reverse direction is FEDE-dominant (D_FEDE=0.048; DPRS=42). "covid test" -> positive rate is FEDE-dominant (D_FEDE=0.039; positive in 72% of windows; DPRS=98) despite negative RMSE gain, while "lockdown" -> stringency is the clearest concordant case (DPRS=78). World-level symptom/case pairs are less persistent. Thus, forecast accuracy and error-distribution dynamics are complementary, predictive effects can be direction- and metric-dependent, and forecast horizon is part of the relationship. Aggregated regime-level patterns should not be extrapolated to individual time points or interpreted as evidence of causality.

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arXiv AI Papershace 24 días

OptScientist: Automated Discovery of Optimization Laws

Many scientific phenomena are best understood as outcomes of underlying optimization problems, yet automated scientific discovery has largely focused on recovering equations that describe observed input-output patterns. We introduce OptScientist, a novel automated framework for discovering compact symbolic optimization laws directly from data. OptScientist comprises (i) a model that infers symbolic optimization problems by first learning to represent data as an optimization problem and then extracting its symbolic form, and (ii) an LLM agentic system embedding the model in a discovery process that integrates theoretical priors from scientific literature, discovers candidate laws, abstracts their underlying mechanisms, and suggests new data to collect. We first evaluate OptScientist on OptFeynman, a novel benchmark of 99 physics-derived optimization problems, where OptScientist recovers ground-truth laws for over 80% of the problems and remains robust to noisy contextual information, while substantially outperforming machine-learning, symbolic, and LLM-based model-discovery baselines. On human risky choice and central-bank monetary policy, OptScientist discovers compact symbolic optimization laws that achieve held-out predictive performance competitive with leading baselines, while using few parameters and substantially fewer tokens than LLM baselines. Overall, OptScientist extends automated scientific discovery to uncovering optimization laws that generate observed phenomena, enabling automated theory building across the physical, biological, behavioral, and social sciences.

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arXiv AI Papershace 24 días

Convergence of stochastic gradient descent for learning linear neural networks

Stochastic gradient descent (SGD) is the cornerstone of modern deep learning, yet its theoretical convergence properties remain poorly understood in non-convex settings. To address this gap, we study SGD dynamics in deep linear neural networks - simplified models that retain key non-convex characteristics while remaining analytically tractable. Building on recent results for gradient descent, we extend convergence guarantees to SGD under regularized squared loss. We establish almost sure boundedness of SGD iterates. Using the framework of asymptotic pseudotrajectories from stochastic approximation theory, our analysis establishes then, for both deterministic and adaptive step-size choices, convergence of SGD to the set of critical points under mild conditions (without relying on Lipschitz properties or whitened data). To our knowledge, this is the first work providing almost sure convergence guarantees for SGD in deep linear networks, bridging the gap between theoretical optimization and practical stochastic training.

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arXiv AI Papershace 24 días

Agentic Ensemble with Vision-Language Models for Visual Anomaly Detection

Visual anomaly detection aims to detect abnormal objects, while anomaly samples are often scarce during model training. Assembling multiple pretrained detectors can help by leveraging their transferable image scores and anomaly maps. However, most existing ensemble methods apply fixed fusion rules over saved predictions which struggle when pretrained detectors clearly disagree due to different training objectives, priors, and visual representations. To this end, we design AnomalyAgent, a post-training fusion framework for accurate visual anomaly detection. AnomalyAgent aligns heterogeneous scores and maps, summarizes score- and region-level disagreement, and treats fusion as a finite action-selection problem. It implements two types of actions: 1) score actions that determine how to fuse image-level detector confidence via consensus, robust aggregation, detector weighting, or score-map consistency; 2) map actions that fuse dense anomaly responses via consensus, detector weighting, geometric aggregation, or disagreement correction. A constrained vision-language model (VLM) is introduced to inspect high-conflict samples and select valid score action and map action to produce the final prediction. We evaluate AnomalyAgent on MVTec AD and VisA as primary benchmarks, together with seven additional public anomaly benchmarks and Real-IAD for large-scale score-level evaluation. Across these benchmarks, AnomalyAgent consistently improves over equal-weight fusion and achieves the best mean image AUROC and pixel AP among the compared fusion methods on MVTec AD and VisA, while demonstrating robust transfer across additional anomaly domains.

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arXiv AI Papershace 24 días

Large Flow Language Models

Continuous flow language models offer a path to parallel generation with tools for control and acceleration unavailable to discrete diffusion, yet effective techniques for building them at the billion-parameter scale remain less established. We present Lafa (Large flow language model), a fast, high-performing flow language model scaled to 8B parameters in a data-efficient manner. Lafa is a blockwise flow model with causal attention on the prefix and bidirectional, asynchronous flow within each block. This design enables efficient training by adapting pretrained autoregressive (AR) models and fast inference via prefix caching and blockwise parallel decoding. We train Lafa at 0.5B, 7B, and 8B scales with fewer than 30B tokens, more efficient than previous diffusion models, and evaluate it on instruction following, reasoning, and coding, where the learned flow matches or surpasses the quality of AR and can be distilled into flow maps for one- and few-step generation. Notably, flow training preserves or even improves pretrained AR capabilities via a multi-token prediction effect, which we leverage in a self-speculative decoding scheme that verifies flow-based drafts autoregressively within a single model. This yields 2.54x the throughput of an AR model at comparable quality, outperforming multi-token prediction and discrete diffusion methods. Our findings show that continuous flows can serve as a scalable foundation for language modeling.

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arXiv AI Papershace 24 días

When Do LLMs Help with Feature Selection? A Systematic Empirical Study

Large language models (LLMs) can leverage semantic feature names alongside summary statistics for tabular feature selection, yet it remains unclear when such information improves subset selection relative to established statistical methods. We investigate this question within an 18-dataset framework comprising 10 datasets for Group Relative Policy Optimization (GRPO) training, 2 for checkpoint validation, and 6 held-out evaluation datasets. We compare five classical baselines, frozen Qwen2.5-7B and Llama-3.3-70B selectors, and two GRPO-adapted Qwen2.5-7B models, reporting classification ROC-AUC and regression R^2 separately and using dataset-wise ranks for cross-task comparison. Across five seeded evaluation partitions, classical selectors generally achieve higher predictive performance than the tested 7B LLM/RL selectors, including under a pre-specified ~33% feature budget with strict train-only classical fitting, while the 7B selectors retain compact subsets. A cardinality-matched semantic sensitivity experiment shows that, for Zero-shot Qwen-7B, informative feature names improve regression performance over anonymized and shuffled alternatives (+0.1964 and +0.1606 mean R^2), whereas aggregate classification differences are smaller (+0.0159 and +0.0084 ROC-AUC) and dataset-dependent. These results characterize when semantic information benefits LLM-based feature selection and show that classical selectors remain strong baselines for predictive tabular feature selection.

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arXiv AI Papershace 24 días

Learning Computational Structure for Sample-Efficient Prediction

Statistical learning in high dimensions suffers from the curse of dimensionality. Fortunately, many functions of practical interest are compositionally sparse, meaning they admit a composition of local functions, each depending on only a few variables, allowing sample complexity to depend on local arity rather than ambient dimension. Yet realizing this benefit requires exploiting hidden compositional structure, which theory does not establish for practical gradient-based training. We study this gap with CSFRBench, a controlled benchmark that hides the target's compositional computation during training but retains it for diagnosis, and find that dense networks and sparsity-based baselines leave much of that benefit unrealized. Inspired by the spatial organization of biological neural circuits, we introduce Learned Wiring (LeWi): each neuron has a learned low-dimensional embedding, only nearby neurons connect, and embeddings and weights are trained jointly. LeWi improves sample efficiency on targets with hierarchical structure, intermediate reuse, or independent modules. Across 32 additional targets generated by independently sampled computation graphs, LeWi is the most sample-efficient method not given the target graph, with 27% lower mean learning-curve area than dense networks. Dense networks perform better on targets with a single high-arity function. LeWi's representations and connectivity align with the target computation. On 17 world-model prediction tasks spanning control, navigation, and manipulation, LeWi beats the dense baseline on 16 and wins 99/119 baseline comparisons. These results show that jointly learning connectivity and weights can exploit computational structure to improve sample efficiency.

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arXiv AI Papershace 24 días

Discount Yourself: Self-inclusive Freshness-weighted Gossip for Decentralised Learning under Client Outages

Decentralised federated learning (DFL) trains personalised models on devices that average with their neighbours rather than with a central server. Real devices, however, are often offline for hours or days, and a client returning from an outage can recover only by aggregating the models its neighbourhood holds. We observe that the stalest of these models is usually the client's own, a case that existing staleness-aware rules, which discount only the models a client receives, do not address. We formalise aggregation after an outage as estimating the state of a fully available network and derive three principles. First, the receiver must be discounted like any neighbour. Every rule that weights it at least as much as each neighbour has an error that grows without bound with the outage, whereas discounting all candidates by relative staleness keeps the error bounded. Second, staleness must be measured as inherited lag rather than age, since local training resets a model's age but not the lag it absorbed from stale neighbours. We introduce the effective lag, which tracks this quantity at the cost of one scalar per message. Third, the discount must be steep, because absorbed lag persists in every later round. These principles define Self-inclusive Freshness-weighted Gossip (SFG), a family of drop-in aggregation rules with the communication and computation of decentralised federated averaging, whose guarantees hold for the stale receiver itself. On three image benchmarks and eight outage processes, including real smartphone availability traces, the lag-based members SFG-L and SFG-Cut consistently outperform decentralised federated averaging and recent age-of-information rules, and ablations show that each principle is necessary. The analysis also explains why tuned limited merging from gossip learning is effective.

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arXiv AI Papershace 24 días

Binary Discrete Diffusion: Candidate-Set Diffusion for Logic and Planning Tasks

Discrete diffusion models iteratively denoise a noisy sequence of categorical data such as text, enabling any-order generation with self-correction and in-filling capabilities. This makes them promising candidates for reasoning, planning, and constrained-generation tasks like coding and math. Yet masked and uniform diffusion erase a token's identity in a single corruption event, so their noisy states cannot explicitly represent partially resolved uncertainty in which several alternatives remain plausible. We close this gap with Binary Discrete Diffusion, a family of continuous-time discrete-diffusion processes whose noisy state is a set of candidate tokens per position, represented as a binary membership vector. Binary diffusion changes candidate sets through additions and removals, allowing information to be lost and recovered gradually while the diffusion state itself represents partial uncertainty. On Sudoku-Extreme, Boxoban, and graph coloring, the symmetric formulation of binary diffusion consistently outperforms strong, well-tuned masked and uniform baselines under matched architectures and controlled training, sampling, and tuning budgets, with its largest gains on the hardest instances, while remaining competitive on language modeling (Text8, TinyStories, OpenWebText). These results support candidate sets as an effective intermediate representation for iterative reasoning and show that the corruption kernel can materially change the computation learned by the reverse model.

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