NeurIPS 2026 论文列表

Setting the learning rate for a deep learning model is a critical part of successful training, yet choosing this hyperparameter is often done empirically with trial and error. In this work, we explore a solvable model of optimal learning rate schedules for a powerlaw random feature model trained with stochastic gradient descent (SGD). We consider the optimal schedule $\eta_T^\star(t)$ where $t$ is the current iterate and $T$ is the total training horizon. This schedule is computed both as a numerical optimization problem and also analytically (when possible) using optimal control theory. Our analysis reveals two regimes which we term the easy phase and hard phase. In the easy phase the optimal schedule is a polynomial decay $\eta_T^\star(t) \simeq T^{-\xi} (1-t/T)^{\delta}$ where $\xi$ and $\delta$ depend on the properties of the features and task. In the hard phase, the optimal schedule resembles warmup-stable-decay with constant (in $T$) initial learning rate and annealing performed over a vanishing (in $T$) fraction of training steps. We investigate joint optimization of learning rate and batch size, identifying a degenerate optimality condition and showing that batch ramps can improve the scaling in wall-clock time in the easy phase. Beyond SGD, we derive optimal schedules for the momentum parameter $\beta(t)$ and show that momentum achieves a better loss-scaling exponent in the hard phase. We compare our optimal schedule to various benchmarks in our task including (1) optimal constant learning rates $\eta_T(t) \sim T^{-\xi}$ (2) optimal power laws $\eta_T(t) \sim T^{-\xi} t^{-\chi}$, finding that our schedule achieves better rates than either of these. Our theory suggests that learning rate transfer across training horizon depends on the structure of the model and task. For ResNet image classification on CIFAR-5M, the learning curves exhibit hard-phase behavior where optimal base learning rates are constant under sufficient annealing. GPT-2 style transformers trained in language modeling exhibit easy-phase behavior where optimal learning rates shift even under annealing.

Theory of Mind (ToM) benchmarks for Large Language Models (LLMs) typically rely on passive question-answering formats, but the deployment of LLMs in increasingly agentic and autonomous forms demands new evaluations. In this paper we evaluate an agent's ability to induce specific belief states in other agents by taking actions rather than using conversational persuasion, a capability we call Non-Conversational Planning ToM (NCP-ToM). NCP-ToM is likely to be essential for many agent use-cases, including within user-assistant interactions and pedagogical contexts, but may also present manipulation or misinformation risks. Using a novel framework, NCP-ExploreToM, we subvert the conventional task structure by providing models with a set of belief state goals and requiring them to move objects or direct characters into rooms to achieve their goals. We evaluated six frontier models, including GPT-5, Gemini 2.5 Pro and the Claude 4 series, and a cohort of human participants, across 600 task instances. GPT-5 was successful on approximately 80\% of tasks in the agentic setting, and was the only model to outperform human participants on our task, but was still less robust than humans across contexts. We additionally found that all models, like humans, performed better on tasks inducing true belief states than false belief states, which is a positive signal for alignment efforts. These findings highlight emerging social-reasoning capabilities in LLMs for non-conversational task completion and underscore the necessity of agentic evaluations for understanding the safety and alignment of autonomous social agents.

Accurate numerical solutions of partial differential equations (PDEs) are crucial in numerous science and engineering applications. In this work, we introduce a novel neural PDE solver named AFDONet, which incorporates neural operator learning and adaptive Fourier decomposition (AFD) theory for the first time into a specifically designed variational autoencoder (VAE) structure, to solve a general class of nonlinear PDEs on smooth manifolds. AFDONet is the first neural PDE solver whose architectural and component design is fully guided by an established mathematical framework (in this case, AFD theory), turning neural operator design from an art to a science. Thus, AFDONet also exhibits exceptional mathematical explainability and groundness, and enjoys several desired properties. Furthermore, AFDONet achieves outstanding solution accuracy and competitive computational efficiency in several benchmark problems. In particular, thanks to its deep connections with AFD theory, AFDONet shows superior performance in solving PDEs on i) arbitrary (Riemannian) manifolds, and ii) datasets with sharp gradients. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

Compositional score-based approaches to simulation-based inference (SBI) approximate the posterior over a shared parameter given $n$ independent observations by aggregating individually learned posterior scores: currently, there are two main propositions of such methods (Geffner et al., 2023; Linhart et al., 2026). As the resulting composite score does not correspond to the score of any distribution along the forward diffusion path of the true multi-observation posterior, sampling from it via a reverse SDE leads to an irreducible bias. Annealed Langevin dynamics provides a principled alternative: it treats the composite score as the genuine score of a sequence of tractable bridging densities and samples from them in succession. When properly tuned, it could lead to a controllable bias. However, its hyperparameters, namely step sizes, the number of steps per level, and the number of annealing levels, have so far been chosen empirically. We derive Wasserstein bounds for annealed Langevin with approximate scores and translate them into explicit decision rules for these hyperparameters that guarantee a prescribed sampling accuracy, while highlighting different theoretical aspects of each composite score formulation. In the Gaussian setting, we obtain closed-form expressions for all relevant quantities and prove that the bridging densities of Linhart et al. (2026) consistently admit larger step sizes and require fewer total Langevin steps than those of Geffner et al. (2023). Furthermore, we show empirically that the tuning obtained in the Gaussian setting generalizes to more complex problems, thus providing a well-understood and theoretically grounded starting point for practitioners using compositional score-based approaches.

Language model (LM) alignment improves model outputs to reflect human preferences while preserving the capabilities of the base model. The most common alignment approaches are (i) reinforcement learning, which maximizes the expected reward under a KL-divergence constraint, and (ii) best-of-$N$ alignment, which selects the highest-reward output among $N$ independent samples. Despite their widespread use, the fundamental limits of reward improvement under a KL budget remain poorly understood. We characterize the information-theoretic limits of KL-regularized alignment by deriving the maximum achievable expected reward gain for a fixed KL-divergence budget. Our first result provides a closed-form expression for the optimal reward improvement, governed by a Jeffreys divergence term rather than the $\sqrt{\texttt{KL}}$ used in prior analyses. We further reformulate this expression as a covariance under the base model, yielding a practical estimator that predicts achievable alignment gains from base model samples alone. We extend our analysis to the proxy reward setting, showing that the gap between ideal and proxy alignment (reward hacking) grows with the KL penalty factor and magnitude of reward error. We then prove that reward ensembling mitigates reward hacking, providing a theoretical justification for this technique used in practice. Empirically, we compute the KL–reward Pareto frontier for two alignment tasks for LMs, safety and summarization, and show that best-of-$N$ closely approaches the theoretical limit, while PPO and GRPO remain substantially suboptimal. Our theoretical results shed light on several empirically observed phenomena in the alignment literature and suggest that algorithmic improvements are needed to achieve optimal alignment without high inference costs.

Multi-parameter persistent homology is a rapidly developing branch of topological data analysis that improves the robustness of single-parameter persistent homology to outliers, while still capturing the metric characteristics of the data. However, a notable limitation is its lack of scalability. In this paper, we introduce a novel approach for efficiently computing 2-parameter persistent homology on large point sets. Our work extends the Flood filtration, originally developed for single-parameter persistence. Our construction, called the sublevel Flood bifiltration, offers a scalable approximation of the sublevel Flood bifiltration. We show that it benefits from theoretical stability properties and describe how to compute it efficiently. We demonstrate the performance of our approach in classification tasks on low-dimensional synthetic datasets, where density awareness is critical, as well as on real-world time series datasets.

With the rapid advancement of generative models, AI-generated images (AIGIs) have become increasingly realistic, posing new challenges for reliable detection. In this work, we revisit the root causes of poor generalization in AIGI detectors from the perspective of shortcut learning. We find that detectors often rely on spurious semantic cues or a single prominent artifact, causing the feature space to collapse toward low diversity and severely limiting generalization. To characterize this issue, we propose the Generative Artifact Manifold Clustering (GAMC) hypothesis, which shows that although generative artifacts are multi-dimensional, different generators occupy distinct distributions along these dimensions, making detectors that rely on local artifact cues inherently fragile. To address this, we shift AIGI detection from “single-cue reliance” to “multi-dimensional discrimination” and introduce the Semantic–Artifact Self-Adversarial Feature Learning (SA-AFL) framework. SA-AFL progressively disentangles semantics from artifacts and employs a self-adversarial mechanism to balance learning across artifact dimensions, enabling the model to retain richer and more shared generative features. As a result, SA-AFL reduces shortcut reliance and learns more transferable evidence for cross-generator detection. Comprehensive experiments on images from 23 generative models show that SA-AFL improves ACC by 6.84\% and mAP by 8.25\% over state-of-the-art methods.

Understanding what individual components of a neural network do is a central goal of mechanistic interpretability, and a valuable tool for auditing or editing trained models. The dominant approach picks a behavior and reverse-engineers the circuit that implements it. This gives a local picture, since we learn what a weight does in service of one task on one distribution, but not what it does in general. We therefore ask the reverse question. Can we pick any single nonzero parameter in a trained transformer and say what it does, without first choosing a behavior? Until recently, this was out of reach, because the weight basis of a standard transformer is not aligned with the computations the network performs (Elhage, et al. 2022). However, weight-sparse transformers (Gao et al. 2025) offer an alternative path forward. By training most weights to zero, the surviving weights align more cleanly with computation, which opens the door to per-weight interpretation. In this work, we introduce an automated LLM-pipeline that produces a short, human-readable account of when a given weight matters, verifies the account on held-out data, and applies it at scale to compare two weight-sparse transformers against two dense controls. Empirically, we find that a significant percentage of nonzero weights (17-35%) on sparse transformers are interpretable, compared to 5-9% on the equivalent dense model and Pythia-70m. Together these results suggest that sparse training produces transformers whose individual weights are interpretable to a degree not achievable in conventional transformers, providing a potential starting point for per-weight interpretation of larger models

Path-integration-trained recurrent neural networks (RNNs) can reproduce grid-like and head-direction-like representations, but most existing models rely on low-dimensional velocity inputs and do not capture the hierarchical organisation of the brain's self-localisation circuit. We introduce a hierarchical recurrent neural network (HRNN) that separately processes angular velocity and linear speed to perform simultaneous angular and positional path integration. Trained on these tasks, the HRNN generalises to longer timescales than seen during training and develops neural response properties resembling all the key cell types: angular velocity cells, head direction cells, speed cells, conjunctive and pure grid cells. The learned circuit also recovers key motifs of the navigation system, including attractor dynamics and asymmetric connectivity patterns. Perturbation analyses reveal functional information flow across the hierarchy, consistent with experimental observations. Finally, introducing theta-paced mutual inhibition between two learned angular-velocity subpopulations enables the HRNN to reproduce the bidirectional theta sweeps observed in empirical data. These results establish hierarchical task-driven RNNs as a biologically interpretable framework for linking neural representations, circuit motifs, and population dynamics in spatial navigation, while enabling targeted in silico perturbations and the generation of experimentally testable predictions.

Over the past decade, a growing body of work has established that implicit biases in learning dynamics fundamentally shape the solutions found by neural networks, governing their learned features and generalisation performance. However, despite these strides in deep learning theory, far less is known about representation learning in generative models based on dynamical systems, from normalising flows to diffusion. Importantly, continuous-time generative models can be prone to mode collapse, yet the learning dynamics underpinning such biases remain elusive. To address this, we use tools from dynamical systems and operator theory to show that bounds on the rank of the gradient of the model weights can explain how collapse occurs in generative models. We show that, in both variational inference and flow matching tasks, optimisation induces low-rank biases that encourage parsimonious representation learning but may also cause the learned distribution to collapse. Together these findings point to a trade-off between promoting feature learning while avoiding both memorisation and collapse in dynamical systems-based generative models.

Understanding how deep neural networks learn representations remains a central challenge in machine learning theory. In this work, we propose a feature-centric framework for analyzing neural network training by relating weight updates to feature evolution. We introduce a simple identity, the *Feature Learning Equation*, which identifies the weight Gram matrix as the key object capturing feature dynamics. This enables us to interpret gradient descent as implicitly inducing a hypothetical evolution of features, whose covariance structure — termed the *Virtual Covariance* — characterizes how representations evolve during training. Building on this perspective, we introduce *Target Linearity*, a measure quantifying the linear alignment between features and targets. By analyzing the training and layer-wise dynamics, we show that deep networks learn to *sequentially* transform representations toward target-linear structure. This linearization perspective provides a unified interpretation of several empirical phenomena, including Neural Collapse and linear interpolation in generative models.

Large Language Models (LLMs) have shown promise for empowering search agents to tackle complex information-seeking tasks. However, since acquiring external information necessitates web search, these agents are highly susceptible to various adversarial attacks (e.g., malicious information injection or stealthy context manipulation). Existing attack strategies inevitably rely on static and human-crafted templates, failing to emulate the dynamic and complex nature of real-world threats. In this paper, we propose an adversarial self-evolution framework that integrates bi-level optimization between the attacker's strategy evolution and the defender's adaptive mitigation, to expose the vulnerabilities of search agents. Specifically, we introduce the Contrastive Rollout Evolutionary Optimization (CREO) method to drive the directional evolution of the attack strategy population via contrastive evolutionary signals. To provide a stable environment for this continuous evolution, we further construct a controlled sandbox and design fine-grained metrics to quantify internal vulnerabilities. Extensive experiments across various multi-hop QA benchmarks and frontier LLMs demonstrate that the proposed framework yields attack efficacy superior to static baselines, underscoring the urgent need to develop more robust defense frameworks. The anonymized code repository is available at https://anonymous.4open.science/r/adversarial_rag-3EFA.

A central error measure in Gaussian DDPMs is the path-space KL divergence between the exact reverse chain and the learned Gaussian reverse process. This quantity is especially relevant for procedures such as classifier guidance, which perturb the entire reverse trajectory rather than only the terminal sample. Prior analyses show that standard isotropic reverse covariances suffer an unavoidable $\Omega(1/T)$ path-KL error as the number of denoising steps $T$ grows. We show that matching the full posterior covariance breaks this barrier, yielding an order-wise improvement that reduces the path KL to $O(1/T^2)$. To make full covariance matching practical, we introduce the Lanczos Gaussian sampler, a training-free, matrix-free method for sampling from the optimal reverse covariance using only covariance-vector products, which are available through Jacobian-vector products of the posterior mean. The sampler avoids dense covariance storage and auxiliary covariance models. We prove that its approximation error decays exponentially in the number of Lanczos steps, where each Lanczos step requires a single Jacobian-vector product. Empirically, using only just three such steps improves sample quality over strong diagonal-covariance baselines, including OCM-DDPM, across standard image benchmarks. This identifies full covariance matching as both theoretically valuable and practically accessible for fast DDPM sampling.

What if the best way to search a video with a vision-language model (VLM) is to stop asking the model to search? Recent approaches give the VLM control over temporal navigation, tying search quality to the model's reasoning ability. We propose the opposite: BeliefSearch treats the VLM as a noisy sensor and delegates navigation to an external Bayesian controller. A single belief over video segments anchors the system: the question sets the prior, each VLM observation updates it, and the same belief decides whether to search, which segment to examine next, and when to stop. Because the belief is explicit, we can trace why each choice was made. The same signal also drives training: each turn earns credit for the uncertainty it reduces, and the final reward is gated by how much the search lowered entropy. This blocks a reward-hacking shortcut where the model otherwise learns to skip search and guess from the preview frames. On long-video benchmarks, our method outperforms all prior search-based methods by up to 8.8 points while using $5$ to $13\times$ fewer frames than recent RL search baselines.

Large language models consistently fail at elementary counting tasks despite strong performance on complex reasoning benchmarks. Recent work has characterized counting failures behaviorally or identified circuits through which counting succeeds, but a mechanistic account of where and why counting breaks down remains absent. In this work, we provide such an account by tracing the count signal from embedding to output and identifying the geometric bottleneck that prevents correct readout for single digit counting problems. Using a dual-metric framework combining Signal-to-Noise Ratio (SNR) with linear probing, we show that counting information is computed correctly and persists in a linearly decodable subspace through the final layer. The failure in counting is not representational but geometric i.e the digit-token columns of the unembedding matrix $W_U$ are orthogonal to the count subspace in the residual stream, rendering correctly computed counts unreadable at the output. We term this as the $\textbf{Unembedding Bottleneck}$ and to causally validate it, we apply a $\textbf{Probe-Informed LoRA}$ correction targeting $W_U$, which restores single-digit counting accuracy by up to 80-95% across six models while updating only ${\sim}$0.001% of parameters and preserving general capabilities. Our findings reveal a broader class of failure in LLMs where information is internally present but geometrically inaccessible to the output projection.

The method of types, pioneered by Csiszar (1998), uses combinatorial arguments to approximate the probabilities of empirical measures and has become a standard tool for deriving error rates in hypothesis testing, source coding, and related information-theoretic problems. Although the technique is widely used across machine learning, computer science, and econometrics, rigorous type-theoretic guarantees for stationary Markov decision processes and reinforcement learning remain limited. This paper addresses that gap by developing a method-of-types framework for the empirical pair measure of the state--action process of a stationary Markov decision process. We establish finite-sample type bounds for the state--action process and characterize the associated entropy cost through the conditional Kullback--Leibler divergence relative to the true pair kernel. These results yield large deviation principles for learned MDP dynamics and variational rate functions for plug-in value- and Q-function estimators in offline reinforcement learning. We further develop finite-sample uncertainty-quantification and hypothesis-testing guarantees based on KL uncertainty sets, including applications to pessimistic value and Q-function estimation under model misspecification. Finally, we apply the same framework to universal source coding and obtain explicit coding error exponents for controlled Markov trajectories. Together, these results provide a unified type-theoretic approach to rare-event analysis, uncertainty quantification, offline reinforcement learning, and information-theoretic coding for finite controlled Markov systems.

Existing Large Language Model (LLM) benchmarks primarily focus on syntactically correct inputs, leaving a significant gap in evaluation on imperfect text. In this work, we study how word-boundary corruption affects how LLMs detect targeted information. By inserting whitespace characters within words to break them into fragments, LLMs' detection accuracy follows a U-shaped curve with the increase in insertion rate. We refer to this curve as the Text Uncanny Valley. To explain such observation, we propose a mode transition hypothesis: LLMs operate in a word-level mode for near-normal text and a character-level mode for heavily fragmented text, with the valley marking the disordered transition where neither mode is effective. Four experiments and one analysis are consistent with this account: in-context learning fails to rescue valley-bottom performance; regularizing the perturbation substantially reduces the U-shape; a math reasoning task replicates the U-shape for Gemini 3.0 Flash but not for stronger models, suggesting the effect is attenuated when tasks rely less on exact lexical alignment; and tokenization entropy peaks before the F1 minimum, consistent with a regime-conflict interpretation. These findings reveal a failure mode invisible to clean-text benchmarks yet directly relevant to any deployment scenario involving noisy or uncurated text inputs.

We develop a framework for analyzing parameter symmetries in deep ReLU networks and obtain a complete characterization of the generic parameter fibers for three-layer bottleneck architectures. Our approach provides explicit semi-algebraic descriptions of these fibers and yields a polynomial time algorithm for deciding functional equivalence of two parameters. The symmetries include discrete and continuous transformations arising from layer composition, and depend on whether deeper layers hide or preserve geometric structure from preceding layers. Finally, we show that some of these symmetries induce local conservation laws along gradient flow, while others do not.

How little of a pretrained large language model has to change for it to acquire new capabilities? We find that zeroing fewer than 0.05% of its weights, with no other modification, matches full-parameter fine-tuning (FPT) at scale across the three dominant LLM fine-tuning paradigms: on-policy reinforcement learning with verifiable rewards (GRPO), supervised fine-tuning (SFT), and on-policy distillation (SDFT). We call the method Bit-Mask Tuning (BMT): a learnable binary keep/zero mask over the gate projection of transformer blocks. BMT reaches FPT accuracy at our largest backbones, with the GRPO gap closing monotonically across model sizes from 0.5B to 8B. Two practical advantages follow: the trained mask is several 1000x smaller than the full-parameter checkpoint, and BMT forgets substantially less than FPT, retaining prior-task accuracy throughout training where FPT measurably drifts. To understand why masking alone suffices, we analyze the update geometry on GRPO and find that BMT's weight delta lands closer to full-parameter updates than other adapters on three measures: effective rank, spectrum drift, and principal-weight overlap. In sum, deletion alone produces a tiny yet high-performing adapter, reduced forgetting, and an update geometry that tracks FPT's more closely than any baseline we evaluate.