NeurIPS 2026 Papers

Reinforcement learning with verifiable rewards (RLVR) has advanced the reasoning capabilities of large language models (LLMs). However, existing RLVR methods often suffer from exploration inefficiency due to mismatches between problem difficulty and model capability: overly difficult problems hinder reasoning path discovery, while overly simple problems offer little learning signal. To address this, we first formalize the effect of problem difficulty by quantifying the relationship between loss descent magnitude and rollout accuracy. Building on this analysis, we propose SEELE, a supervision-aided RLVR framework that dynamically adjusts problem difficulty to lie within the high-performance region. SEELE augments each training sample by appending a hint (part of a full solution) for difficulty reduction. Unlike previous hint-based approaches, SEELE deliberately computes the hint length for each individual problem to achieve an optimal difficulty. The optimal hint length is determined via multi-round rollout sampling, where an item response theory model fits accuracy–hint pairs from previous rounds to predict the next-round hint. This instance-level, real-time difficulty adjustment aligns problem difficulty with the evolving model capability, thereby improving exploration efficiency. Experiments show that SEELE outperforms Group Relative Policy Optimization (GRPO) and Supervised Fine-tuning (SFT) by +10.0 and +8.4 points, respectively, and exceeds the best prior supervision-aided approach by +3.8 points on average across six math reasoning benchmarks.

Large language models (LLMs) excel at reasoning across diverse domains, from mathematical problem-solving to code generation, yet struggle with strategic reasoning that requires long-term planning, opponent modeling and integrating tactical calculation with strategic understanding. Board games, with their fully observable states, perfect-information environments and objectively verifiable decisions, offer ideal testbeds for probing these capabilities. We introduce Board Game Arena (BGA), a multi-dimensional benchmark that evaluates LLMs' strategic reasoning along three cognitive axes—deductive, inductive, and abductive reasoning—through three challenging task types across five classic board games, yielding 45 sub-datasets with over one million samples. We evaluate 28 state-of-the-art LLMs on BGA and find that current models exhibit fundamental limitations in strategic reasoning for board games. While models demonstrate competence in understanding game rules and identifying legal moves, they struggle significantly with selecting optimal moves and applying strategic principles or tactical analysis. These results position BGA as a challenging and diagnostic benchmark for tracking progress in LLMs' strategic cognition.

Large reasoning models (LRMs) generate reasoning traces before producing final answers, yielding strong performance on multi-step reasoning tasks. However, preference alignment for LRMs remains underexplored. The ideal answer-level preference objective marginalizes over reasoning traces, but this trace-marginalized objective is computationally intractable. As a result, practical methods rely on single-trace surrogates, which can induce high gradient variance due to stochastic trace sampling. We cast this challenge as a gradient-estimator design problem and propose Bias-Variance Optimized Preference Optimization (BVPO), which combines a standard trace-based gradient estimator with an empty-trace gradient estimator from a separately sampled empty-trace branch. Under our fixed-prompt, fixed-answer conditional view, the empty-trace branch is deterministic with respect to the trace randomness studied in our analysis. We show that the composite estimator contracts the conditional reasoning-trace-noise component, admits a conditional MSE-optimal mixture, and appears in the estimator-dependent term of a standard biased-SGD bound. Empirically, BVPO improves alignment over baselines on alignment benchmarks, including AlpacaEval 2 and Arena-Hard. Although trained on general conversational data, BVPO also generally preserves reasoning performance on six math reasoning benchmarks. Empirical diagnostics further show that BVPO yields lower total gradient variance during training.

Training LLMs to think and reason for longer has become a key ingredient in building state-of-the-art models that can solve complex problems previously out of reach. Recent efforts pursue this in different ways, such as RL fine-tuning to elicit long CoT or scaling latent reasoning through architectural recurrence. This makes reasoning length an important scaling knob. In this work, we identify a novel phenomenon (both theoretically and experimentally): under outcome-only supervision, out-of-distribution (OOD) performance can continue improving as training-time reasoning length (e.g., the token budget in RL, or the loop count in looped Transformers) increases, even after in-distribution (ID) performance has saturated. This suggests that robustness may require a larger budget than ID validation alone would indicate. We provide theoretical explanations via two mechanisms: (i) self-iteration can induce a stronger inductive bias in the hypothesis class, reshaping ID-optimal solutions in ways that improve OOD generalization; and (ii) when shortcut solutions that work for ID samples but not for OOD samples persist in the hypothesis class, regularization can reduce the learned solution's reliance on these shortcuts as the number of self-iterations increases. We complement the theory with empirical evidence from two realizations of scaling training-time reasoning length: increasing the number of loops in looped Transformers on a synthetic task, and increasing token budgets during RL fine-tuning of LLMs on mathematical reasoning, coding, and cross-representation tasks.

Forensic verification has converged on a binary "real vs. fake" label that conflates fully synthetic, tampered, and AI-retouched images despite their very different consequences. We argue that GenAI manipulations decompose along two complementary channels: a camera channel of sensor fingerprints (destroyed by neural processing), and a semantic channel of scene-level coherence (disrupted by content edits). Each manipulation type leaves a distinctive two-channel signature. We instantiate this taxonomy in 2CAP (2-Channel Authenticity Protocol), pairing a contrastively-trained camera encoder with a frozen semantic encoder to serve two applications on a shared backbone: (i) four-class authenticity classification via cross-attention fusion with learnable reliability weights; and (ii) evidence generation, where an authentic image serves as a reference for a query and the two encoders produce per-channel alignment scores together with a patch-level saliency map. These signals are supplied as privileged context to a frozen Vision Language Model (VLM) through an Observe-Generate-Refine loop, yielding explanations with localization without forensic fine-tuning. On a multi-source benchmark, 2CAP attains the best classification performance, including strong retouching detection where existing methods fail, and improves explanation quality.

Despite remarkable progress in image-text understanding, vision-language models (VLMs) still struggle with compositional reasoning. In particular, they often fail to distinguish relational direction and attribute-object binding, leading to similar representations for semantically different image-text pairs. This problem mainly stems from the reliance on global image-text alignment, which captures coarse correspondence but overlooks fine-grained compositional structures. Toward this end, we propose an evidence-aware framework, termed Directional Relation Bucketing with Binding Localization (DELTA), to improve compositional understanding in VLMs. The key idea of our DELTA is to improve compositionality of VLMs from two complementary perspectives: direction modeling and binding localization. More specifically, we first leverage a learnable gate to model the cumulative contextual distance between anchor terms, thereby assigning relation concepts to direction-aware discrete buckets. To enrich textual representations with fine-grained relational semantics, we calibrate global semantic representations by incorporating intermediate-layer hidden states. Furthermore, DELTA leverages textual cues to ground visual evidence for object attributes, pulling image patches closer to their matched attribute descriptions while pushing them away from incorrect augmented ones. Extensive experiments on four compositional datasets demonstrate the effectiveness of the proposed DELTA. Our implementation is available at https://anonymous.4open.science/r/DELTA-2D60.

Recent advancements in Large Vision-Language Models (LVLMs) have significantly enhanced their ability to integrate visual and linguistic information, achieving near-human proficiency in tasks like object recognition, captioning, and visual question answering. However, current benchmarks typically focus on knowledge-centric evaluations that assess domain-specific expertise, often neglecting the core ability to reason about fundamental mathematical elements and visual concepts. We identify a gap in evaluating elementary-level math problems, which rely on explicit visual dependencies-requiring models to discern, integrate, and reason across multiple images while incorporating commonsense knowledge, all o which are crucial for advancing toward broader AGI capabilities. To address this gap, we introduce VCBench, a comprehensive benchmark for multimodal mathematical reasoning with explicit visual dependencies. VCBench includes 1,720 problems across six cognitive domains, featuring 6,697 images (averaging 3.9 per question) to ensure multi-image reasoning. We evaluate 26 state-of-the-art LVLMs on VCBench, revealing substantial performance disparities, with even the top models unable to exceed 50% accuracy. Our findings highlight the ongoing challenges in visual-mathematical integration and suggest avenues for future LVLM advancements.

We introduce a mathematical framework for studying contraction of the distributional Bellman operator under Maximum Mean Discrepancy (MMD). We show that contraction reduces to conditional positive definiteness of an associated kernel constructed from the base kernel, which we call the Bellman difference kernel. Focusing on radial kernels, we use Schoenberg's theorem to connect this condition to Bernstein functions, yielding contraction criteria, valid in arbitrary dimension, that can be checked analytically or numerically. Our framework unifies known results for the Gaussian RBF, power-distance, and multiquadric kernels; using numerical search, we further identify 12 new contraction-inducing kernels. We also propose *chaining*, a general construction that produces contraction-inducing kernels at any prescribed rate. To reconcile theory with practice, we introduce conditional positive definiteness on a set, which explains why the Gaussian RBF kernel can remain a valid choice on bounded supports despite its lack of global contraction. Finally, we identify a fundamental tradeoff between contraction rate and discriminative power, where kernels that contract faster tend to discriminate less between distributions.

Geometry Problem Solving (GPS) serves as a rigorous touchstone for evaluating the complex reasoning capabilities of Multimodal Large Language Models (MLLMs), particularly when solutions necessitate auxiliary constructions absent in the visual input. However, current approaches face the following challenges: explicit diagramming methods suffer from unstable image generation quality and error cascading, while existing implicit attention mechanisms are restricted to visible elements, failing to perceive the latent auxiliary structures required for deduction. To bridge this gap, we propose AuxCA (Auxiliary-Clues-Aware GRPO), a novel framework that internalizes the capability of auxiliary construction into the model’s reasoning intuition without relying on external tools. Specifically, we introduce Auxiliary Clues Dependency (ACD) to quantify the causal influence of visual cues and integrate it into GRPO via a fine-grained advantage weighting strategy. Complementing this, we incorporate an auxiliary incentive reward to explicitly encourage the model to actively attend to and utilize these geometric elements during reasoning. Extensive experiments demonstrate that our resulting model, AuxCA-8B, significantly outperforms state-of-the-art explicit plotting methods and achieves performance competitive with top-tier proprietary models. Code is available in \url{https://anonymous.4open.science/r/AuxCA-5510}.

Reliable uncertainty estimation is essential for deploying large language models in high-stakes reasoning tasks, yet most existing approaches rely on expensive sampling strategies or shallow token-level signals that ignore the internal logical structure of model outputs. We introduce Argument Graph Uncertainty, a post-hoc framework for multiple choice question tasks, that estimates model confidence directly from the logical structure of a single chain-of-thought reasoning trace. Our method uses a lightweight instruction-tuned model to segment the reasoning chain into propositional units, then applies a simple NLI model to infer relational edges and construct two complementary argument graphs over these segments. The resulting edge scores are aggregated into a Dirichlet posterior over answer options, whose concentration yields calibrated uncertainty estimates reflecting both local step-to-step coherence and global logical consistency across the reasoning trace. Evaluated on GPQA, a challenging graduate-level multiple-choice benchmark, our method consistently surpasses token-based and consistency-based uncertainty baselines across multiple reasoning models. We further demonstrate through extensive combination analyses that argument graph uncertainty captures an orthogonal signal to existing approaches, making it a complementary component in uncertainty ensembles. Crucially, our framework requires no additional calls to the evaluated model and relies entirely on small, efficient auxiliary models, producing fully interpretable, per-node uncertainty attributions at low computational cost.

On-policy self-distillation, where a student is pulled toward a copy of itself conditioned on privileged context (e.g., a verified solution or feedback), offers a promising direction for advancing reasoning capability without a stronger external teacher. Yet in math reasoning the gains are inconsistent, even when the same approach succeeds elsewhere. A pointwise mutual information analysis traces the failure to the privileged context itself: it inflates the teacher's confidence on tokens already implied by the solution (structural connectives, verifiable claims) and deflates it on deliberation tokens (*Wait*, *Let*, *Maybe*) that drive multi-step search. We propose Anti-Self-Distillation (AntiSD), which ascends a divergence between student and teacher rather than descending it: this reverses the per-token sign and yields a naturally bounded advantage in one step. An entropy-triggered gate disables the term once the teacher entropy collapses, completing a drop-in replacement for default self-distillation. Across five models from 4B to 30B parameters on math reasoning benchmarks, AntiSD reaches the GRPO baseline's accuracy in 2 to 10× fewer training steps and improves final accuracy by up to 11.5 points. AntiSD opens a path to scalable self-improvement, where a language model bootstraps its own reasoning through its training signal.

Distilling "System 2" reasoning into compact student models is bottlenecked by the scarcity of fully-correct teacher traces: as task difficulty grows, rejection sampling discards an increasingly large fraction of teacher generations. "Wrong" (W) traces are abundant, but training on them indiscriminately causes *policy poisoning* the student internalises the teacher's hallucinated arithmetic alongside any useful reasoning structure. We propose **Atomic Reasoning Units (ARU)**, a framework that resolves this trade-off via strict syntactic constraints. ARU enforces a Backus--Naur-Form (BNF) grammar that decomposes each reasoning step into a Premise, an Operation, and a Result, algorithmically disentangling reasoning planning from arithmetic execution. *Counterfactual Logic Verification* (CLV) symbolically re-executes W-traces to recover the $68\%$ that have valid plans but faulty arithmetic, and *Syntactic Loss Masking* (SLM) trains the student on this verified structure while suppressing the gradient on hallucinated result tokens. Across the $3 \times 8 \times 6$ grid we evaluate ($\{0.6, 1.7, 8\}$\,B Qwen3 students $\times$ eight benchmarks $\times$ six baselines), ARU is the strongest method in every cell, improving GSM8K by $+11.1$ and MATH-500 by $+9.8$ over the closest baseline at $1.7$\,B; on the held-out AIME competition set ARU's Maj@$8$ is $14.8$ versus $9.5$ for PoT (the closest baseline). We further document a small-scale capability gap in which ARU at $0.5$\,B matches Gold-SFT at $\geq 1.5$\,B on GSM8K (paired-bootstrap $p<0.05$ after Bonferroni correction), with a pre-registered ProofWriter control showing the gap collapses on pure-logic reasoning---consistent with arithmetic offloading as the dominant mechanism, though we do not claim the control uniquely isolates it. The code is available at https://anonymous.4open.science/r/ARU-286D.

Studies of human reasoning have shown that people are typically stronger at evaluating reasoning than producing it from scratch. In contrast, large reasoning models (LRMs) are trained to excel at producing long chains of reasoning to solve complex problems. How then do LRMs perform at evaluating reasons? We investigate this with the Valid-Answer-Invalid-Reasoning (VAIR) dataset: math problems and solutions with trivial reasoning flaws but valid answers, designed to isolate reasoning evaluation from the confound of reasoning production. Unlike humans, who we find are only 6\% worse at grading than solving such problems, we find a substantial production-evaluation gap in LRMs: frontier models score as low as 48\% when evaluating VAIR solutions, despite near-perfect solution production. Why this enigma? Through chain-of-thought (CoT) analysis, we find evidence of an answer confirmation bias: LRMs often produce then check for the correct answer instead of carefully verifying each step, fabricating rationalizations even when noticing anomalous reasoning. Linear probes corroborate this, showing that while LRM activations encode some representation of valid reasoning, they fail to robustly represent VAIR solutions as invalid. Causal patching of the final answer's representations causes LRM verdicts and activations to flip, demonstrating that answer validity is responsible for models' confirmation biases. These findings indicate an outstanding limitation in dominant approaches to reasoning training, which incentivize LRMs to produce and confirm reasoning towards correct answers, but not to robustly evaluate the underlying reasons.

With the rising need for spatially grounded tasks such as Vision-Language Navigation/Action, allocentric perception capabilities in Vision-Language Models (VLMs) are receiving growing focus. However, VLMs remain brittle on allocentric spatial queries that require explicit perspective shifts, where the answer depends on reasoning in a target-centric frame rather than the observed camera view. Thus, we introduce $\textbf{Allocentric Perceiver}$, a training-free strategy that recovers metric 3D states from one or more images with off-the-shelf geometric experts, and then instantiates a query-conditioned allocentric reference frame aligned with the instruction’s semantic intent. By deterministically transforming reconstructed geometry into the target frame and prompting the backbone VLM with structured, geometry-grounded representations, Allocentric Perceiver offloads mental rotation from implicit reasoning to explicit computation. We evaluate Allocentric Perceiver across multiple backbone families on spatial reasoning benchmarks, observing consistent and substantial gains ($\sim$\textbf{10\%}) on allocentric tasks while maintaining strong egocentric performance, and surpassing both spatial‑perception-finetuned models and state‑of‑the‑art open‑source and proprietary models.

Recent advances in Large Language Models (LLMs) have enabled agentic systems capable of solving complex tasks through multi-turn planning, tool use, verification, and memory updates. However, learning agentic systems remains difficult due to two fundamental challenges, i.e., (1) long-horizon credit assignment, where supervision is available only at the final outcome, and (2) imbalanced data distributions, where dominant data patterns bias optimization and weaken adaptation to rare but informative reasoning behaviors. In this paper, we propose Fair Multi-Level Preference Optimization (Fair-MPO or $\Phi$-MPO), a new preference optimization framework for agentic learning. We first show that Multi-Level Preference Optimization provides a principled and more computationally efficient framework for long-horizon reasoning. Then, we introduce a Fair Multi-Level Objective that addresses imbalance in agentic learning. We provide a comprehensive theoretical analysis demonstrating that our approach addresses both long-horizon reasoning and data imbalance. Our experiments on agentic reasoning benchmarks demonstrate that our approach achieves State-of-the-Art (SOTA) performance.

Formal Geometric Problem Solving requires every step of reasoning within a strict logical framework, and has been a core challenge in artificial intelligence. This paper presents a unified neuro-symbolic reasoning framework, named Reflect Agent Verified Solve (RAVS), which deeply integrates large language models, agent architectures, and formal symbolic solvers. The core innovations of RAVS lie in two parts. First, we propose a role-separated neuro-symbolic collaboration mechanism: the LLM serves as a planner responsible for high-level semantic understanding and path reflection, while the symbolic solver acts as an executor responsible for formal verification and rigorous theorem application. The neural reasoning capability of the LLM and the logical completeness of the symbolic system complement each other, fundamentally eliminating the risk of hallucinations. Second, we construct the first bidirectional symbolic reasoning engine that fully unifies forward and backward solving. For each theorem, we define two reversible operations: forward application and backward decomposition. We further provide a theoretical analysis that bidirectional reasoning achieves an exponential complexity advantage over unidirectional reasoning. Together, these two mechanisms form an iterative closed loop between neural planning and symbolic execution. On the FormalGeo7K benchmark, RAVS achieves a 95.8% solving accuracy, substantially outperforming existing state-of-the-art methods, without requiring any additional problem-specific annotated data.

Large Reasoning Models (LRMs) rely on extensive Chain-of-Thought generation but suffer from memory bottlenecks due to linear Key-Value (KV) cache growth. While low-bit quantization offers a potential solution, we identify that current static quantization strategies suffer from severe degradation on complex reasoning tasks at extreme bit-widths. Notably, this accuracy drop is accompanied by increased generation length, which ironically limits the intended efficiency gains. We attribute this performance collapse to the fact that static quantization methods are insufficiently flexible to meed dynamic demands of precise long-range retrieval. Utilizing Average Attention Distance (AAD) to quantify retrieval patterns, we find that the attention patterns in shallow layers exhibit strong locality, whereas that in deep layers demonstrate intensive long-range dependencies. Furthermore, deep layers have fluctuating AAD values across token positions. Based on above findings, we propose AdaKVQ, an adaptive mixed-precision quantization framework that modulates the bit-width allocation guided by the dynamic demands of long-range context retrieval in each layer. Extensive experiments on complex reasoning benchmarks with Llama-3.1 and Qwen2.5 demonstrate that AdaKVQ consistently outperforms state-of-the-art baselines, achieving performance comparable to full-precision models.

Do language models make decisions under uncertainty like humans do? And if so, what role does extended reasoning play in the underlying decision process? We answer this question by introducing an active probabilistic reasoning task that cleanly separates sampling (actively acquiring evidence) from inference (integrating evidence towards a decision). Benchmarking humans and a broad set of contemporary LLMs against optimal reference policies reveals a consistent pattern: extended reasoning is the key determinant of strong performance, driving large gains in inference, while yielding only modest improvements in active sampling. To explain these differences, we fit a behavioral model that captures systematic deviations from optimal Bayesian behavior through interpretable parameter families, placing humans and models in a shared low-dimensional cognitive space. The resulting fits show how reasoning shifts models toward human-like regimes of evidence accumulation and belief-to-choice mapping, and yield testable predictions about the latent dynamics that might drive each decision. Probing residual-stream activations of an open-weight reasoning model, we find that the geometry of internal representations tracks these predicted dynamics, linking behavior to representational correlates of the fitted latent dynamics.

Self-distillation enables language models to learn in an on-policy fashion from their own trajectories by using the same model as both the teacher and student, with the teacher being conditioned on privileged information that is unavailable to the student. This information can take multiple forms or views, including solutions, demonstrations, or feedback; for example, for a math problem, one view might be a ground truth reasoning trace. Supervising the student via the privileged teacher enables fine-grained token-level feedback and removes misalignment caused by distillation from external policies with different semantic biases. However, self-distillation also creates a fundamental asymmetry: the teacher may encode view-specific information tied to the privileged information provided during training, which the student cannot access at inference time. Moreover, the best form of privileged information is often task-dependent, making it difficult to choose a single teacher view. In this work, we address both these challenges jointly by introducing AVSD (Adaptive-View Self-Distillation), a novel method of self-distillation with multiple privileged-information views, which reconstructs token-level supervision by separating stable cross-view consensus from view-specific residual support. The consensus signal provides a robust update direction, while the residual signal is selectively used to adjust the update magnitude. Together, these components form a gated aggregation mechanism that balances a conservative target (geometric mean) with a more permissive target (arithmetic mean) for aggregating teacher signals at the token level. Experiments on math competition benchmarks (AIME24, AIME25, and HMMT25) show that AVSD consistently outperforms single-view self-distillation baselines and GRPO, achieving 3.1\% Avg@8 gain over the strongest baseline on Qwen3-8B on average. Moreover, on code-generation benchmarks (Codeforces, LiveCodeBench) using Qwen3-8B, AVSD outperforms single-view self-distillation baselines by 2.9\% on average. More broadly, our analysis shows that AVSD provides better token-level learning signals than any single-view method by reconstructing supervision from multiple privileged teachers.