NeurIPS 2025 Papers

Extant studies predominantly address catastrophic forgetting within a simplified continual learning paradigm, typically confined to a singular data domain. Conversely, real-world applications frequently encompass multiple, evolving data domains, wherein models often struggle to retain many critical past information, thereby leading to performance degradation. This paper addresses this complex scenario by introducing a novel dynamic expansion approach called Learning Expandable and Adaptable Representations (LEAR). This framework orchestrates a collaborative backbone structure, comprising global and local backbones, designed to capture both general and task-specific representations. Leveraging this collaborative backbone, the proposed framework dynamically create a lightweight expert to delineate decision boundaries for each novel task, thereby facilitating the prediction process. To enhance new task learning, we introduce a novel Mutual Information-Based Prediction Alignment approach, which incrementally optimizes the global backbone via a mutual information metric, ensuring consistency in the prediction patterns of historical experts throughout the optimization phase. To mitigate network forgetting, we propose a Kullback–Leibler (KL) Divergence-Based Feature Alignment approach, which employs a probabilistic distance measure to prevent significant shifts in critical local representations. Furthermore, we introduce a novel Hilbert-Schmidt Independence Criterion (HSIC)-Based Collaborative Optimization approach, which encourages the local and global backbones to capture distinct semantic information in a collaborative manner, thereby mitigating information redundancy and enhancing model performance. Moreover, to accelerate new task learning, we propose a novel Expert Selection Mechanism that automatically identifies the most relevant expert based on data characteristics. This selected expert is then utilized to initialize a new expert, thereby fostering positive knowledge transfer. This approach also enables expert selection during the testing phase without requring any task information. Empirical results demonstrate that the proposed framework achieves state-of-the-art performance.

We study the problem of non-stationary Lipschitz bandits, where the number of actions is infinite and the reward function, satisfying a Lipschitz assumption, can change arbitrarily over time. We design an algorithm that adaptively tracks the recently introduced notion of significant shifts, defined by large deviations of the cumulative reward function. To detect such reward changes, our algorithm leverages a hierarchical discretization of the action space. Without requiring any prior knowledge of the non-stationarity, our algorithm achieves a minimax-optimal dynamic regret bound of $\mathcal{\widetilde{O}}(\tilde{L}^{1/3}T^{2/3})$, where $\tilde{L}$ is the number of significant shifts and $T$ the horizon. This result provides the first optimal guarantee in this setting.

A major problem of kernel-based methods (e.g., least squares support vector machines, LS-SVMs) for solving linear/nonlinear ordinary differential equations (ODEs) is the prohibitive $O(an^3)$ ($a=1$ for linear ODEs and 27 for nonlinear ODEs) part of their computational complexity with increasing temporal discretization points $n$. We propose a novel Nyström-accelerated LS-SVMs framework that breaks this bottleneck by reformulating ODEs as primal-space constraints. Specifically, we derive for the first time an explicit Nyström-based mapping and its derivatives from one-dimensional temporal discretization points to a higher $m$-dimensional feature space ($1< m\le n$), enabling the learning process to solve linear/nonlinear equation systems with $m$-dependent complexity. Numerical experiments on sixteen benchmark ODEs demonstrate: 1) $10-6000$ times faster computation than classical LS-SVMs and physics-informed neural networks (PINNs), 2) comparable accuracy to LS-SVMs ($<0.13\%$ relative MAE, RMSE, and $\left \| y-\hat{y} \right \| _{\infty } $difference) while maximum surpassing PINNs by 72\% in RMSE, and 3) scalability to $n=10^4$ time steps with $m=50$ features. This work establishes a new paradigm for efficient kernel-based ODEs learning without significantly sacrificing the accuracy of the solution.

Local SGD, or Federated Averaging, is one of the most widely used algorithms for distributed optimization. Although it often outperforms alternatives such as mini-batch SGD, existing theory has not fully explained this advantage under realistic assumptions about data heterogeneity. Recent work has suggested that a second-order heterogeneity assumption may suffice to justify the empirical gains of local SGD. We confirm this conjecture by establishing new upper and lower bounds on the convergence of local SGD. These bounds demonstrate how a low second-order heterogeneity, combined with third-order smoothness, enables local SGD to interpolate between heterogeneous and homogeneous regimes while maintaining communication efficiency. Our main technical contribution is a refined analysis of the consensus error, a central quantity in such results. We validate our theory with experiments on a distributed linear regression task.

This paper presents a novel theoretical framework for understanding how diffusion models can learn disentangled representations with commonly used weak supervision such as partial labels and multiple views. Within this framework, we establish identifiability conditions for diffusion models to disentangle latent variable models with \emph{stochastic}, \emph{non-invertible} mixing processes. We also prove \emph{finite-sample global convergence} for diffusion models to disentangle independent subspace models. To validate our theory, we conduct extensive disentanglement experiments on subspace recovery in latent subspace Gaussian mixture models, image colorization, denoising, and voice conversion for speech classification. Our experiments show that training strategies inspired by our theory, such as style guidance regularization, consistently enhance disentanglement performance.

We study the noise-free Gaussian Process (GP) bandit problem, in which a learner seeks to minimize regret through noise-free observations of a black-box objective function that lies in a known reproducing kernel Hilbert space (RKHS). The Gaussian Process Upper Confidence Bound (GP-UCB) algorithm is a well-known approach for GP bandits, where query points are adaptively selected based on the GP-based upper confidence bound score. While several existing works have reported the practical success of GP-UCB, its theoretical performance remains suboptimal. However, GP-UCB often empirically outperforms other nearly-optimal noise-free algorithms that use non-adaptive sampling schemes. This paper resolves the gap between theoretical and empirical performance by establishing a nearly-optimal regret upper bound for noise-free GP-UCB. Specifically, our analysis provides the first constant cumulative regret bounds in the noise-free setting for both the squared exponential kernel and the Matérn kernel with some degree of smoothness.

Finding low-rank matrix weights is a key technique for addressing the high memory usage and computational demands of large models. Most existing algorithms rely on the factorization of the low-rank matrix weights, which is non-unique and redundant. Their convergence is slow especially when the target low-rank matrices are ill-conditioned, because the convergence rate depends on the condition number of the Jacobian operator for the factorization and the Hessian of the loss function with respect to the weight matrix. To address this challenge, we adopt the Riemannian gradient descent (RGD) algorithm on the Riemannian manifold of fixed-rank matrices to update the entire low-rank weight matrix. This algorithm completely avoids the factorization, thereby eliminating the negative impact of the Jacobian condition number. Furthermore, by leveraging the geometric structure of the Riemannian manifold and selecting an appropriate metric, it mitigates the negative impact of the Hessian condition number. Ultimately, this results in our two plug-and-play optimizers: RAdaGrad and RAdamW, which are RGD with metrics adapted from AdaGrad and AdamW and restricted to the manifold. Our algorithms can be seamlessly integrated with various deep neural network architectures without any modifications. We evaluate the effectiveness of our algorithms through fine-tuning experiments on large language models and diffusion models. Experimental results consistently demonstrate that our algorithms provide superior performance compared to state-of-the-art methods. Additionally, our algorithm is not only effective for fine-tuning large models but is also applicable to deep neural network (DNN) compression.

We consider the problem of computing persistent homology (PH) for large-scale Euclidean point cloud data, aimed at downstream machine learning tasks, where the exponential growth of the most widely-used Vietoris-Rips complex imposes serious computational limitations. Although more scalable alternatives such as the Alpha complex or sparse Rips approximations exist, they often still result in a prohibitively large number of simplices. This poses challenges in the complex construction and in the subsequent PH computation, prohibiting their use on large-scale point clouds. To mitigate these issues, we introduce the Flood complex, inspired by the advantages of the Alpha and Witness complex constructions. Informally, at a given filtration value $r\geq 0$, the Flood complex contains all simplices from a Delaunay triangulation of a small subset of a point cloud $X$ that are fully covered by the union of balls of radius $r$ emanating from $X$, a process we call flooding. Our construction allows for efficient PH computation, possesses several desirable theoretical properties, and is amenable to GPU parallelization. Scaling experiments on 3D point cloud data show that we can compute PH of up to dimension 2 on several millions of points. Importantly, when evaluating object classification performance on real-world and synthetic data, we provide evidence that this scaling capability is needed, especially if objects are geometrically or topologically complex, yielding performance superior to other PH-based methods and neural networks for point cloud data. Source code is available on GitHub: https://github.com/plus-rkwitt/flooder

Direct Preference Optimization (DPO) has emerged as an effective approach for mitigating hallucination in Multimodal Large Language Models (MLLMs). Although existing methods have achieved significant progress by utilizing vision-oriented contrastive objectives for enhancing MLLMs' attention to visual inputs and hence reducing hallucination, they suffer from non-rigorous optimization objective function and indirect preference supervision. To address these limitations, we propose a Symmetric Multimodal Preference Optimization (SymMPO), which conducts symmetric preference learning with direct preference supervision (i.e., response pairs) for visual understanding enhancement, while maintaining rigorous theoretical alignment with standard DPO. In addition to conventional ordinal preference learning, SymMPO introduces a preference margin consistency loss to quantitatively regulate the preference gap between symmetric preference pairs. Comprehensive evaluation across five benchmarks demonstrate SymMPO's superior performance, validating its effectiveness in hallucination mitigation of MLLMs.

What features neural networks learn, and how, remains an open question. In this paper, we introduce Alternating Gradient Flows (AGF), an algorithmic framework that describes the dynamics of feature learning in two-layer networks trained from small initialization. Prior works have shown that gradient flow in this regime exhibits a staircase-like loss curve, alternating between plateaus where neurons slowly align to useful directions and sharp drops where neurons rapidly grow in norm. AGF approximates this behavior as an alternating two-step process: maximizing a utility function over dormant neurons and minimizing a cost function over active ones. AGF begins with all neurons dormant, corresponding to an initialization at the origin. At each iteration, a dormant neuron activates, triggering the acquisition of a feature and a drop in the loss. AGF quantifies the order, timing, and magnitude of these drops, matching experiments across several commonly studied architectures. We show that AGF unifies and extends existing saddle-to-saddle analyses in fully connected linear networks and attention-only linear transformers, where the learned features are singular modes and principal components, respectively. In diagonal linear networks, we prove AGF converges to gradient flow in the limit of vanishing initialization. Applying AGF to quadratic networks trained to perform modular addition, we give the first complete characterization of the training dynamics, revealing that networks learn Fourier features in decreasing order of coefficient magnitude. Altogether, AGF offers a promising step towards understanding feature learning in neural networks.

We investigate scaling laws for stochastic momentum algorithms on the power law random features model, parameterized by data complexity, target complexity, and model size. When trained with a stochastic momentum algorithm, our analysis reveals four distinct loss curve shapes determined by varying data-target complexities. While traditional stochastic gradient descent with momentum (SGD-M) yields identical scaling law exponents to SGD, dimension-adapted Nesterov acceleration (DANA) improves these exponents by scaling momentum hyperparameters based on model size and data complexity. This outscaling phenomenon, which also improves compute-optimal scaling behavior, is achieved by DANA across a broad range of data and target complexities, while traditional methods fall short. Extensive experiments on high-dimensional synthetic quadratics validate our theoretical predictions and large-scale text experiments with LSTMs show DANA's improved loss exponents over SGD hold in a practical setting.

In this paper, we consider a block coordinate descent (BCD) algorithm for training deep neural networks and provide a new global convergence guarantee under strictly monotonically increasing activation functions. While existing works demonstrate convergence to stationary points for BCD in neural networks, our contribution is the first to prove convergence to global minima, ensuring arbitrarily small loss. We show that the loss with respect to the output layer decreases exponentially while the loss with respect to the hidden layers remains well-controlled. Additionally, we derive generalization bounds using the Rademacher complexity framework, demonstrating that BCD not only achieves strong optimization guarantees but also provides favorable generalization performance. Moreover, we propose a modified BCD algorithm with skip connections and non-negative projection, extending our convergence guarantees to ReLU activation, which are not strictly monotonic. Empirical experiments confirm our theoretical findings, showing that the BCD algorithm achieves a small loss for strictly monotonic and ReLU activations.

Decentralized bilevel optimization has garnered significant attention due to its critical role in solving large-scale machine learning problems. However, existing methods often rely on prior knowledge of problem parameters—such as smoothness, convexity, or communication network topologies—to determine appropriate stepsizes. In practice, these problem parameters are typically unavailable, leading to substantial manual effort for hyperparameter tuning. In this paper, we propose \textbf{AdaSDBO}, a fully problem-parameter-free algorithm for decentralized bilevel optimization with a single-loop structure. AdaSDBO leverages adaptive stepsizes based on cumulative gradient norms to update all variables simultaneously, dynamically adjusting its progress and eliminating the need for problem-specific hyperparameter tuning. Through rigorous theoretical analysis, we establish that AdaSDBO achieves a convergence rate of $\widetilde{\mathcal{O}}\left(\frac{1}{T}\right)$, matching the performance of well-tuned state-of-the-art methods up to polylogarithmic factors. Extensive numerical experiments demonstrate that AdaSDBO delivers competitive performance compared to existing decentralized bilevel optimization methods while exhibiting remarkable robustness across diverse stepsize configurations.

We consider a common case of the combinatorial semi-bandit problem, the $m$-set semi-bandit, where the learner exactly selects $m$ arms from the total $d$ arms. In the adversarial setting, the best regret bound, known to be $\mathcal{O}(\sqrt{nmd})$ for time horizon $n$, is achieved by the well-known Follow-the-Regularized-Leader (FTRL) policy. However, this requires to explicitly compute the arm-selection probabilities via optimizing problems at each time step and sample according to them. This problem can be avoided by the Follow-the-Perturbed-Leader (FTPL) policy, which simply pulls the $m$ arms that rank among the $m$ smallest (estimated) loss with random perturbation. In this paper, we show that FTPL with a Fr\'echet perturbation also enjoys the near optimal regret bound $\mathcal{O}(\sqrt{nm}(\sqrt{d\log(d)}+m^{5/6}))$ in the adversarial setting and approaches best-of-both-world regret bounds, i.e., achieves a logarithmic regret for the stochastic setting. Moreover, our lower bounds show that the extra factors are unavoidable with our approach; any improvement would require a fundamentally different and more challenging method.

In this paper, we study the problem of solving a simple bilevel optimization problem, where the upper-level objective is minimized over the solution set of the lower-level problem. We focus on the general setting in which both the upper- and lower-level objectives are smooth but potentially nonconvex. Due to the absence of additional structural assumptions for the lower-level objective—such as convexity or the Polyak–Łojasiewicz (PL) condition—guaranteeing global optimality is generally intractable. Instead, we introduce a suitable notion of stationarity for this class of problems and aim to design a first-order algorithm that finds such stationary points in polynomial time. Intuitively, stationarity in this setting means the upper-level objective cannot be substantially improved locally without causing a larger deterioration in the lower-level objective. To this end, we show that a simple and implementable variant of the dynamic barrier gradient descent (DBGD) framework can effectively solve the considered nonconvex simple bilevel problems up to stationarity. Specifically, to reach an $(\epsilon_f, \epsilon_g)$-stationary point—where $\epsilon_f$ and $\epsilon_g$ denote the target stationarity accuracies for the upper- and lower-level objectives, respectively—the considered method achieves a complexity of $\mathcal{O}(\max(\epsilon_f^{-\frac{3+p}{1+p}}, \epsilon_g^{-\frac{3+p}{2}}))$, where $p \geq 0$ is an arbitrary constant balancing the terms. To the best of our knowledge, this is the first complexity result for a discrete-time algorithm that guarantees joint stationarity for both levels in general nonconvex simple bilevel problems.

The rise of smart manufacturing under Industry 4.0 introduces mass customization and dynamic production, demanding more advanced and flexible scheduling techniques. The flexible job-shop scheduling problem (FJSP) has attracted significant attention due to its complex constraints and strong alignment with real-world production scenarios. Current deep reinforcement learning (DRL)-based approaches to FJSP predominantly employ constructive methods. While effective, they often fall short of reaching (near-)optimal solutions. In contrast, improvement-based methods iteratively explore the neighborhood of initial solutions and are more effective in approaching optimality. However, the flexible machine allocation in FJSP poses significant challenges to the application of this framework, including accurate state representation, effective policy learning, and efficient search strategies. To address these challenges, this paper proposes a $\textbf{M}$emory-enhanced $\textbf{I}$mprovement $\textbf{S}$earch framework with he$\textbf{t}$erogeneous gr$\textbf{a}$ph $\textbf{r}$epresentation—$\textit{MIStar}$. It employs a novel heterogeneous disjunctive graph that explicitly models the operation sequences on machines to accurately represent scheduling solutions. Moreover, a memory-enhanced heterogeneous graph neural network (MHGNN) is designed for feature extraction, leveraging historical trajectories to enhance the decision-making capability of the policy network. Finally, a parallel greedy search strategy is adopted to explore the solution space, enabling superior solutions with fewer iterations. Extensive experiments on synthetic data and public benchmarks demonstrate that $\textit{MIStar}$ significantly outperforms both traditional handcrafted improvement heuristics and state-of-the-art DRL-based constructive methods.

We study the problem of minimizing non-convex functionals on the space of probability measures, regularized by the relative entropy (KL divergence) with respect to a fixed reference measure, as well as the corresponding problem of solving entropy-regularized non-convex-non-concave min-max problems. We utilize the Best Response flow (also known in the literature as the fictitious play flow) and study how its convergence is influenced by the relation between the degree of non-convexity of the functional under consideration, the regularization parameter and the tail behaviour of the reference measure. In particular, we demonstrate how to choose the regularizer, given the non-convex functional, so that the Best Response operator becomes a contraction with respect to the $L^1$-Wasserstein distance, which ensures the existence of its unique fixed point that is then shown to be the unique global minimizer for our optimization problem. This extends recent results where the Best Response flow was applied to solve convex optimization problems regularized by the relative entropy with respect to arbitrary reference measures, and with arbitrary values of the regularization parameter. Our results explain precisely how the assumption of convexity can be relaxed, at the expense of making a specific choice of the regularizer. Additionally, we demonstrate how these results can be applied in reinforcement learning in the context of policy optimization for Markov Decision Processes and Markov games with softmax parametrized policies in the mean-field regime.

A primary impediment to scaling reinforcement learning (RL) for large language model (LLM) training is the substantial computational cost, predominantly arising from the necessity of multi-sampling for policy optimization and evaluation. This underscores the critical yet challenging nature of efficient training data selection. Drawing inspiration from the Zone of Proximal Development (ZPD) theory, which posits that learners acquire knowledge more effectively from tasks of intermediate difficulty, we hypothesize that LLMs exhibit optimal learning from data they have not yet mastered but demonstrate the potential to comprehend. Conventional methodologies for assessing data difficulty or informativeness typically rely on computationally intensive multi-sampling or iterative procedures. To address this limitation, we introduce UFO-RL (**U**ncertainty-**F**ocused **O**ptimization for **R**einforcement **L**earning), a novel framework that employs a computationally efficient single-pass uncertainty estimation technique to identify informative training instances. This method, requiring only a single forward pass and obviating the need for iterative next-token computation, achieves a significant acceleration (up to 185$\times$) in data evaluation compared to multi-sampling approaches. UFO-RL leverages this efficient metric to select data within the model's estimated ZPD for training. Extensive experimentation across diverse LLMs and mathematical benchmarks demonstrates that training with a mere 10\% of the data, carefully selected by UFO-RL, yields performance comparable to or even surpassing that of full-data training. Furthermore, this targeted data selection results in up to a 16$\times$ reduction in overall training time, concurrently enhancing training stability and improving generalization capabilities. Thus, UFO-RL presents a practical and highly efficient strategy for scaling RL fine-tuning of LLMs by focusing learning efforts on the most informative and valuable data, thereby mitigating the computational bottlenecks associated with traditional RL training.

$1$-Lipschitz neural networks are fundamental for generative modelling, inverse problems, and robust classifiers. In this paper, we focus on $1$-Lipschitz residual networks (ResNets) based on explicit Euler steps of negative gradient flows and study their approximation capabilities. Leveraging the Restricted Stone–Weierstrass Theorem, we first show that these $1$-Lipschitz ResNets are dense in the set of scalar $1$-Lipschitz functions on any compact domain when width and depth are allowed to grow. We also show that these networks can exactly represent scalar piecewise affine $1$-Lipschitz functions. We then prove a stronger statement: by inserting norm-constrained linear maps between the residual blocks, the same density holds when the hidden width is fixed. Because every layer obeys simple norm constraints, the resulting models can be trained with off-the-shelf optimisers. This paper provides the first universal approximation guarantees for $1$-Lipschitz ResNets, laying a rigorous foundation for their practical use.