NeurIPS 2026 Papers

We consider two-layer neural networks trained in the feature-learning regime using gradient descent, and relate the output of the finite-width network $f_{\hat{\rho}}$ to its infinite-width counterpart $f_{\rho^{MF}}$, which evolves in the mean-field dynamics. While constant-time horizon bounds for $\|f_{\rho^{MF}} -f_{\hat{\rho}}\|$ may be obtained via standard Grönwall estimates, the long-time behavior of the fluctuation is a more delicate matter. Uniform-in-time bounds often rely on (local) strong convexity in the landscape or Logarithmic Sobolev inequalities present in noisy gradient dynamics. % In this work, we study a noiseless setting and do not make assumptions on the geometry of the landscape near the optimum. Instead, we impose a weaker condition on the convergence rate of the mean-field deterministic Wasserstein-gradient-flow dynamics. In this work, we establish non-asymptotic weak propagation-of-chaos that holds uniformly in time, obtained by exploiting instead the convergence rate of the mean-field deterministic Wasserstein-gradient-flow dynamics. Specifically, denoting by $L_t$ the mean-field loss at time $t$ and $m$ the number of neurons, under standard regularity assumptions and the condition $\int_0^\infty L_t^{1/2} dt =O_d(1)$, we obtain the uniform in time bound $\|f_{\rho^{MF}} -f_{\hat{\rho}}\|^2 \lesssim \text{poly}(d) m^{-\min(1,c/6)}$ whenever $L_t \lesssim t^{-c}$. Our result holds in a noiseless setting and does not make any assumptions on the geometry of the landscape near the optimum. A key implication of our result is that whenever the convergence rate of the mean-field, population-loss dynamics is faster than $1/t^2$, we can attain a loss of $\epsilon$ with only $\text{poly}(d/\epsilon)$ neurons, training samples, and GD steps.

Pairwise constrained clustering typically relies on hard must-link/cannot-link labels, whereas realistic pairwise supervision may be real-valued and entangle intrinsic ambiguity, expert judgment, and stochastic corruption. Existing deep constrained clustering (DCC) methods mainly target hard, expert-agnostic constraints, treating soft labels mostly numerically rather than semantically. We formalize this setting as uncertainty-aware probabilistic constrained clustering (UPCC), defining a canonical aleatoric target through a heterogeneous observation process and analyzing its conditional identifiability. We introduce ProbPair, an angular pairwise objective for probabilistic relations, and build ECI-PP, an estimator--corrector--integrator framework that refines imperfect supervision via belief estimation, correction, and reliability-aware integration. Across challenging probabilistic supervision settings, experiments on diverse benchmarks show that ECI-PP outperforms state-of-the-art DCC methods and remains robust with a shared default configuration.

Geometric foundation models show promise in 3D reconstruction, yet their progress is severely constrained by the scarcity of diverse, large-scale 3D annotations. While Internet videos offer virtually unlimited raw data, utilizing them as a scaling source for geometric learning is challenging due to the absence of ground-truth geometry and the presence of observational noise. To address this, we propose SAGE, a framework for Scalable Adaptation of GEometric foundation models from raw video streams. SAGE leverages a hierarchical mining pipeline to transform videos into training trajectories and hybrid supervision: (1) Informative training trajectory selection; (2) Sparse Geometric Anchoring via SfM point clouds for global structural guidance; and (3) Dense Differentiable Consistency via 3D Gaussian rendering for multi-view constraints. To prevent catastrophic forgetting, we introduce a regularization strategy using anchor data. Experiments on applying SAGE to two architecturally distinct backbones (including both MV-DUSt3R and VGGT) show that Chamfer Distance is consistently reduced by 20-42% on unseen benchmarks (7Scenes, TUM-RGBD, Matterport3D). SAGE establishes Internet video as a viable and scalable adaptation source for geometric foundation models.

Neurosymbolic learning can use symbolic rules to provide supervision for latent concepts from weak labels, but it commonly assumes that the entities referenced by these rules are already specified. Object-centric models decompose images into slot-like representations; however, such slots are not necessarily aligned with the predicates required for symbolic reasoning. We investigate object-centric neurosymbolic learning under distant supervision, where the object-level arguments of a logic program are learned directly from images using only global task labels. We introduce DeepObjectLog, a probabilistic neurosymbolic model that integrates a slot-based perceptual encoder with a probabilistic logic layer. The encoder predicts objectness and class probabilities for candidate object representations, while the logic layer marginalizes over latent objectness and class assignments to compute the likelihood of the observed label. This formulation provides a differentiable task-level learning signal for object-centric perception without requiring per-object labels, masks, bounding boxes, or heuristic set matching. Evaluations across diverse visual reasoning tasks demonstrate that DeepObjectLog achieves superior out-of-distribution generalization to compositional, object-count, and rule shifts compared to neural object-centric and standard neurosymbolic baselines.

Language models are increasingly trained to "reason" before answering users' queries, outputting hundreds of intermediate tokens before a final answer. While reasoning is designed with direct model capabilities in mind, we argue that reasoning language models (RLMs) should also be assessed by interaction effects between their reasoning traces and weaker models. In contexts like safety monitoring and distillation, the performance of weak models increasingly depends on legibility---how thoroughly and precisely a reasoning trace externalizes an RLM's actions. Existing efficiency-based metrics for legibility fail to capture "thoroughness", instead focusing on conciseness. Thus, we introduce *transfer utility*, a method for measuring trace legibility derived from interactions where weak models finish tasks given incomplete traces. Evaluating 85k traces from 12 RLMs across three tasks, we find that reasoning traces generated by the highest-performing and the most efficient models rank the *lowest* for transfer utility. We also find evidence that transfer utility predicts the ability of weak monitors to verify procedurally dense traces for math or logical problem-solving, though the effect moderates when verifying fact-driven traces. Finally, we show that open-source reward models decouple from transfer utility signals on correctness. Together, these findings surface status quo trade-offs in weak-to-strong legibility, motivating an interaction-first framework for future applications.

Most explanations of weak-to-strong (W2S) generalization focus on why a strong student may fail to exactly imitate weak-label errors. We study a complementary mechanism: before student training, weak-teacher predictions may be inconsistent along semantic constraints specified independently of the weak model. When such constraints are valid for the target, these inconsistencies give label-free lower bounds on the risk reduction achievable by correcting weak labels, rather than merely indicating unstructured prediction error. We formalize this mechanism using Markov and Laplacian operators. Under exact target validity, Markov trajectory inconsistency equals the squared-risk gain from smoothing; under approximate validity, it remains a conservative lower bound after an explicit validity penalty. We then solve the associated minimax correction problem, obtaining a resolvent correction that continuously attenuates constraint-violating directions and reduces to the identity when validity uncertainty is too large. These results yield validity-adjusted correction (VAC), a split-sample pipeline that selects corrections from weak-teacher queries and predeclared validity budgets before training the strong student. Controlled graph-signal experiments and a CIFAR-10 cats-versus-dogs W2S experiment are consistent with the proposed mechanism: validity-adjusted inconsistency predicts useful correction and transfers to student gain, whereas high raw inconsistency under invalid controls does not provide such evidence.

Reinforcement learning (RL) with verifiable rewards has driven recent gains in large language model (LLM) reasoning, but whether it creates new capabilities or merely sharpens existing ones remains debated, especially for compositional generalization, where models must combine learned primitives into novel multi-step functions. Using a controlled program synthesis domain over a typed domain-specific language (DSL) of 119 primitives with verifiable compositional depth and irreducibility, where the base model has zero capability prior to fine-tuning, we show on Qwen3-8B (and replicate within-family at 4B and 32B, and cross-family on Llama-3.1-8B) that compositional capability is established during supervised fine-tuning (SFT) and depends on irreducibility at two levels of SFT data: structural irreducibility of compositional exemplars and observational irreducibility, where prompt input-output pairs are chosen adversarially to rule out shallower programs. Neither alone is sufficient; even moderate contamination with structurally reducible exemplars substantially degrades performance, and full contamination yields zero pass@64. Given this SFT foundation, standard outcome-reward RL (GRPO, PPO) maintains but does not extend compositional performance, contrasting with prior results in settings with simpler nested function execution. Compositional capability is bottlenecked by structural properties of SFT data, before RL begins. Code release pending institutional IP review.

Supervised fine-tuning (SFT) on long teacher trajectories is the dominant method for instilling investigation and reasoning capabilities into open software-engineering (SWE) agents. Under SFT, every retained response is an imitation target, so the student inherits not only the trajectory's outcome but also any flaw in its intermediate steps, including ungrounded leaps and redundant loops. High-quality training data must therefore be jointly effective (each step is grounded and narrows the agent's epistemic gap to the correct fix) and efficient (each step is information-bearing rather than redundant or looping). Existing recipes filter or relabel teacher rollouts using only a binary terminal verifier, which does not directly target these axes and provides no supervision on instances where the teacher fails. Every real issue ships with a developer-authored reference patch $p^\star$ that implicitly testifies to the file paths, runtime behaviors, and conventions a fix presupposes, but the standard pipeline discards it. We propose P2T (Patches-to-Trajectories), which uses $p^\star$ as privileged information during curation, and frames trajectory construction as a bi-objective program over per-step effectiveness and trajectory length. A reverse phase distills $p^\star$ into a latent process graph $G^\star$ of contextual facts and solution milestones, encoding dense intermediate anchors in constructive ordering. A forward phase curate trajectories from blinded teacher continuations, scoring per-step progress against $G^\star$ under a leakage-blocking groundedness check and committing the shortest segments that retain effectiveness. Using only 1.8k curated SWE-Gym instances, P2T improves both axes simultaneously over outcome-filtered SFT and its tool-error-masking variant: on SWE-bench Verified, it lifts Pass@1 by up to +10.8 points while cutting per-instance inference cost by 15%, with consistent gains on SWE-bench Lite and across two teachers. A size-matched ablation and qualitative analysis further isolate per-trajectory quality from data scale.

Clustering is a fundamental task in unsupervised learning, serving as a core tool for data mining and exploratory data analysis. Classical $k$-clustering objectives such as $k$-median and $k$-means formalize this task, but typically assume access to exact pairwise distances. This assumption is increasingly unrealistic in modern applications where distances are estimated through proxy embedding models, learned similarity functions, human feedback, or other low-cost procedures that may be noisy. We study $k$-clustering in an unknown metric space where the algorithm has access only to a weak distance oracle: for each pair of objects, the oracle returns the true distance with probability of $1/2+\varepsilon$ , where $\varepsilon$ is a constant, and otherwise may return an arbitrary corrupted value. We present randomized algorithms that, given a metric space with $n$ vertices and a parameter $k$, use only weak distance-oracle queries to compute a set of representative centers together with a mapping from every input object to one of these centers, such that the total mapping cost is within a constant factor of the optimal $k$-clustering cost. Our first algorithm returns $O(k\cdot\mathsf{polylog}(n))$ centers using $O(n\cdot k\cdot \mathsf{polylog}(n))$ weak-oracle queries and runs in quasi-polynomial time. We also give a polynomial-time algorithm that returns $O(k^2\cdot\mathsf{polylog}(n))$ centers using $O(n\cdot k\cdot \mathsf{polylog}(n))$ weak-oracle queries. Preliminary experiments for $k$-means clustering show that our approach remains close to the optimum under substantial oracle noise, while noise-oblivious baselines degrade sharply.

Text-to-SQL systems are typically evaluated by query-level execution correctness, but this terminal signal provides little guidance about which intermediate SQL decision caused success or failure. Token-level dense supervision is also ill-suited: SQL tokens do not align with complete semantic decisions, can penalize execution-equivalent queries, and are difficult to label reliably at scale. We therefore propose CAPER, which automatically derives clause-level supervision via counterfactual intervention on the SQL abstract syntax tree (AST), enabling root-cause error localization for reward modeling; the resulting data is used to train CAPER-9B, a lightweight Clause-PRM that provides clause-boundary feedback for policy optimization and candidate verification. Experiments on BIRD and Spider show that clause-aligned supervision not only improves execution accuracy, achieving up to a 15.3\% relative EX improvement over GPT-5.4, but also strengthens failure-localization capability, reaching 84.53\% accuracy and 90.60\% MRR on held-out failures.

Modern gradient boosting algorithms are based on Newton's method, which excels empirically but lacks global convergence guarantees. While recent work on Gradient-Regularized Newton (GRN) boosting achieved a global $\mathcal{O}(1/\gamma^{12}k^2)$ rate, its analysis was not able to exploit the natural Hessian geometry and suffers from a $\gamma^{12}$ dependence on a worst-case `weak gradient edge' $\gamma$. In this paper, we tackle these theoretical drawbacks by generalizing the Affine-Invariant Cubic Newton (AICN) scheme to convex optimization with weak learners. Our analysis is based on average cosine values $\overline{\Theta}_k$ along the optimization path and directional improvements $\delta_k$ in the Hessian-induced norm. We show that the weak-AICN update reduces to a damped weak-Newton step with an explicit stepsize determined by $\delta_k$. Under semi-strong self-concordance, we establish a global $\mathcal{O}(1/\overline{\Theta}_k^4 k^2)$ rate, significantly improving upon the weak-GRN bound. We further prove a local linear rate, global linear convergence for Hessian-dominated losses, and cosine angle lower bounds. Finally, we demonstrate the broader applicability of this framework beyond gradient boosting by showing that Newton coordinate descent emerges naturally as a special case of optimization with weak learners.