NeurIPS 2025 论文列表

We provide the first proof of convergence for normalized error feedback algorithms across a wide range of machine learning problems. Despite their popularity and efficiency in training deep neural networks, traditional analyses of error feedback algorithms rely on the smoothness assumption that does not capture the properties of objective functions in these problems. Rather, these problems have recently been shown to satisfy generalized smoothness assumptions, and the theoretical understanding of error feedback algorithms under these assumptions remains largely unexplored. Moreover, to the best of our knowledge, all existing analyses under generalized smoothness either i) focus on single-node settings or ii) make unrealistically strong assumptions for distributed settings, such as requiring data heterogeneity, and almost surely bounded stochastic gradient noise variance. In this paper, we propose distributed error feedback algorithms that utilize normalization to achieve the $\mathcal{O}(1/\sqrt{K})$ convergence rate for nonconvex problems under generalized smoothness. Our analyses apply for distributed settings without data heterogeneity conditions, and enable stepsize tuning that is independent of problem parameters. Additionally, we provide strong convergence guarantees of normalized error feedback algorithms for stochastic settings. Finally, we show that due to their larger allowable stepsizes, our new normalized error feedback algorithms outperform their non-normalized counterparts on various tasks, including the minimization of polynomial functions, logistic regression, and ResNet-20 training.

We study the problem of Online Convex Optimization (OCO) over a convex set $\mathcal{K} \subset \mathbb{R}^d$, accessed via a separation oracle. While classical projection-based algorithms such as projected Online Gradient Descent (OGD) achieve the optimal $O(\sqrt{T})$ regret, they require computing Euclidean projections onto $\mathcal{K}$ whenever an iterate falls outside the feasible set. These projections can be computationally expensive, especially for complex or high-dimensional sets. Projection-free algorithms address this by replacing projections with alternative oracle-based procedures, such as separation or linear optimization oracles. However, the regret bounds of existing separation-based methods scale poorly with the set's \emph{asphericity} $\kappa$, defined as the ratio between the radii of the smallest enclosing ball and the largest inscribed ball in $\mathcal{K}$; for ill-conditioned sets, $\kappa$ can be arbitrarily large. We introduce a new separation-based algorithm for OCO that achieves a regret bound of $\tilde{O}(\sqrt{dT} + d^2)$, with only logarithmic dependence on $\kappa$. This removes a key limitation of prior work and eliminates the need for costly geometric pre-processing, such as transforming $\mathcal{K}$ into isotropic position. Our algorithm is based on a novel reduction to online optimization over a sequence of dynamically updated ellipsoids, inspired by the classical ellipsoid method for convex optimization. It requires only $\tilde{O}(1)$ separation oracle calls per round, on par with existing separation-based approaches. These advances make our method particularly well suited for online optimization over geometrically complex feasible sets.

A common approach for accelerating optimization algorithms is to minimize the loss achieved in a fixed time, which enables a differentiable framework with respect to the algorithm's hyperparameters. In contrast, the complementary objective of minimizing the time to reach a target loss is traditionally considered non-differentiable. To address this limitation, we propose a differentiable discrete stopping time and theoretically justify it based on its connection to continuous differential equations. We design an efficient algorithm to compute its sensitivities, thereby enabling a new differentiable formulation for directly accelerating algorithms. We demonstrate its effectiveness in applications such as online hyperparameter tuning and learning to optimize. Our proposed methods show superior performance in comprehensive experiments across various problems, which confirms their effectiveness.

We develop a scalable second-order algorithm for a recently proposed $\ell_1$-regularized pseudolikelihood-based partial correlation network estimation framework. While the latter method admits statistical guarantees and is inherently scalable compared to likelihood-based methods such as graphical lasso, the currently available implementations rely only on first-order information and require thousands of iterations to obtain reliable estimates even on high-performance supercomputers. In this paper, we further investigate the inherent scalability of the framework and propose locally and globally convergent semismooth Newton methods. Despite the nonsmoothness of the problem, these second-order algorithms converge at a locally quadratic rate, and require only a few tens of iterations in practice. Each iteration reduces to solving linear systems of small dimensions or linear complementary problems of smaller dimensions, making the computation also suitable for less powerful computing environments. Experiments on both simulated and real-world genomic datasets demonstrate the superior convergence behavior and computational efficiency of the proposed algorithm, which position our method as a promising tool for massive-scale network analysis sought for in, e.g., modern multi-omics research.

The loss landscapes of neural networks contain minima and saddle points that may be connected in flat regions or appear in isolation. We identify and characterize a special structure in the loss landscape: channels along which the loss decreases extremely slowly, while the output weights of at least two neurons, $a_i$ and $a_j$, diverge to $\pm$infinity, and their input weight vectors, $\mathbf{w_i}$ and $\mathbf{w_j}$, become equal to each other. At convergence, the two neurons implement a gated linear unit: $a_i\sigma(\mathbf{w_i} \cdot \mathbf{x}) + a_j\sigma(\mathbf{w_j} \cdot \mathbf{x}) \rightarrow c \sigma(\mathbf{w} \cdot \mathbf{x}) + (\mathbf{v} \cdot \mathbf{x}) \sigma'(\mathbf{w} \cdot \mathbf{x})$. Geometrically, these channels to infinity are asymptotically parallel to symmetry-induced lines of critical points. Gradient flow solvers, and related optimization methods like SGD or ADAM, reach the channels with high probability in diverse regression settings, but without careful inspection they look like flat local minima with finite parameter values. Our characterization provides a comprehensive picture of this quasi-flat region in terms of gradient dynamics, geometry, and functional interpretation. The emergence of gated linear units at the end of the channels highlights a surprising aspect of the computational capabilities of fully connected layers.

What property of the data distribution determines the excess risk of principal component analysis? In this paper, we provide a precise answer to this question. We establish a central limit theorem for the error of the principal subspace estimated by PCA, and derive the asymptotic distribution of its excess risk under the reconstruction loss. We obtain a non-asymptotic upper bound on the excess risk of PCA that recovers, in the large sample limit, our asymptotic characterization. Underlying our contributions is the following result: we prove that the negative block Rayleigh quotient, defined on the Grassmannian, is generalized self-concordant along geodesics emanating from its minimizer of maximum rotation less than $\pi/4$.

In recent years, many studies proposed to generate solutions for Directed Acyclic Graph (DAG) scheduling problem in one shot by combining reinforcement learning and list scheduling heuristic. However, these existing methods suffer from biased estimation of sampling probabilities and inefficient guidance in training, due to redundant comparisons among node priorities and the sparse reward challenge. To address these issues, we analyze of the limitations of these existing methods, and propose a novel one-shot DAG scheduling method with comparability identification and dense reward signal, based on the policy gradient framework. In our method, a comparable antichain identification mechanism is proposed to eliminate the problem of redundant nodewise priority comparison. We also propose a dense reward signal for node level decision-making optimization in training, effectively addressing the sparse reward challenge. The experimental results show that the proposed method can yield superior results of scheduling objectives compared to other learning-based DAG scheduling methods.

This paper investigates convex-concave minimax optimization problems where only the function value access is allowed. We introduce a class of Hessian-aware quantum zeroth-order methods that can find the $\epsilon$-saddle point within $\tilde{\mathcal{O}}(d^{2/3}\epsilon^{-2/3})$ function value oracle calls. This represents an improvement of $d^{1/3}\epsilon^{-1/3}$ over the $\mathcal{O}(d\epsilon^{-1})$ upper bound of classical zeroth-order methods, where $d$ denotes the problem dimension. We extend these results to $\mu$-strongly-convex $\mu$-strongly-concave minimax problems using a restart strategy, and show a speedup of $d^{1/3}\mu^{-1/3}$ compared to classical zeroth-order methods. The acceleration achieved by our methods stems from the construction of efficient quantum estimators for the Hessian and the subsequent design of efficient Hessian-aware algorithms. In addition, we apply such ideas to non-convex optimization, leading to a reduction in the query complexity compared to classical methods.

In the context of smooth stochastic optimization with first order methods, we introduce the stability ratio of gradient estimates, as a measure of local relative noise level, from zero for pure noise to one for negligible noise. We show that a schedule-free variant (Stab-SGD) of stochastic gradient descent obtained by just shrinking the learning rate by the stability ratio achieves real adaptivity to noise levels (i.e. without tuning hyperparameters to the gradient's variance), with all key properties of a good schedule-free algorithm: neither plateau nor explosion at intialization, and no saturation of the loss. We believe this theoretical development reveals the importance of estimating the local stability ratio in the construction of well-behaved (last-iterate) schedule-free algorithms, particularly when hyperparameter-tuning budgets are a small fraction of the total budget since noise-adaptivity and cheaper horizon-free tuning are most crucial in this regime.

Adding entropic regularization to Optimal Transport (OT) problems has become a standard approach for designing efficient and scalable solvers. However, regularization introduces a bias from the true solution. To mitigate this bias while still benefiting from the acceleration provided by regularization, a natural solver would adaptively decrease the regularization as it approaches the solution. Although some algorithms heuristically implement this idea, their theoretical guarantees and the extent of their acceleration compared to using a fixed regularization remain largely open. In the setting of semi-discrete OT, where the source measure is continuous and the target is discrete, we prove that decreasing the regularization can indeed accelerate convergence. To this end, we introduce DRAG: Decreasing (entropic) Regularization Averaged Gradient, a stochastic gradient descent algorithm where the regularization decreases with the number of optimization steps. We provide a theoretical analysis showing that DRAG benefits from decreasing regularization compared to a fixed scheme, achieving an unbiased $\mathcal{O}(1/t)$ sample and iteration complexity for both the OT cost and the potential estimation, and a $\mathcal{O}(1/\sqrt{t})$ rate for the OT map. Our theoretical findings are supported by numerical experiments that validate the effectiveness of DRAG and highlight its practical advantages.

Learning pseudo-contractive denoisers is a fundamental challenge in the theoretical analysis of Plug-and-Play (PnP) methods and the Regularization by Denoising (RED) framework. While spectral methods attempt to address this challenge using the power iteration method, they fail to guarantee the truly pseudo-contractive property and suffer from high computational complexity. In this work, we rethink gradient step (GS) denoisers and establish a theoretical connection between GS denoisers and pseudo-contractive operators. We show that GS denoisers, with the gradients of convex potential functions parameterized by input convex neural networks (ICNNs), can achieve truly pseudo-contractive properties. Furthermore, we integrate the learned truly pseudo-contractive denoiser into the RED-PRO (RED via fixed-point projection) model, definitely ensuring convergence in terms of both iterative sequences and objective functions. Extensive numerical experiments confirm that the learned GS denoiser satisfies the truly pseudo-contractive property and, when integrated into RED-PRO, provides a favorable trade-off between interpretability and empirical performance on inverse problems.

The capacity of deep learning models is often large enough to both learn the underlying statistical signal and overfit to noise in the training set. This noise memorization can be harmful especially for data with a low signal-to-noise ratio (SNR), leading to poor generalization. Inspired by prior observations that label noise provides implicit regularization that improves generalization, in this work, we investigate whether introducing label noise to the gradient updates can enhance the test performance of neural network (NN) in the low SNR regime. Specifically, we consider training a two-layer NN with a simple label noise gradient descent (GD) algorithm, in an idealized signal-noise data setting. We prove that adding label noise during training suppresses noise memorization, preventing it from dominating the learning process; consequently, label noise GD enjoys rapid signal growth while the overfitting remains controlled, thereby achieving good generalization despite the low SNR. In contrast, we also show that NN trained with standard GD tends to overfit to noise in the same low SNR setting and establish a non-vanishing lower bound on its test error, thus demonstrating the benefit of introducing label noise in gradient-based training.

Federated Learning (FL) enables multiple clients to collaboratively train models without sharing raw data, but is vulnerable to Byzantine attacks and data heterogeneity, which can severely degrade performance. Existing Byzantine-robust approaches tackle data heterogeneity, but incur high computational overhead during gradient aggregation, thereby slowing down the training process. To address this issue, we propose a simple yet effective Federated Normalized Gradients Algorithm (Fed-NGA), which performs aggregation by merely computing the weighted mean of the normalized gradients from each client. This approach yields a favorable time complexity of $\mathcal{O}(pM)$, where $p$ is the model dimension and $M$ is the number of clients. We rigorously prove that Fed-NGA is robust to both Byzantine faults and data heterogeneity. For non-convex loss functions, Fed-NGA achieves convergence to a neighborhood of stationary points under general assumptions, and further attains zero optimality gap under some mild conditions, which is an outcome rarely achieved in existing literature. In both cases, the convergence rate is $\mathcal{O}(1/T^{\frac{1}{2} - \delta})$, where $T$ denotes the number of iterations and $\delta \in (0, 1/2)$. Experimental results on benchmark datasets confirm the superior time efficiency and convergence performance of Fed-NGA over existing methods.

In multi-objective learning (MOL), several possibly competing prediction tasks must be solved jointly by a single model. Achieving good trade-offs may require a model class $\mathcal{G}$ with larger capacity than what is necessary for solving the individual tasks. This, in turn, increases the statistical cost, as reflected in known MOL bounds that depend on the complexity of $\mathcal{G}$. We show that this cost is unavoidable for some losses, even in an idealized semi-supervised setting, where the learner has access to the Bayes-optimal solutions for the individual tasks as well as the marginal distributions over the covariates. On the other hand, for objectives defined with Bregman losses, we prove that the complexity of $\mathcal{G}$ may come into play only in terms of unlabeled data. Concretely, we establish sample complexity upper bounds, showing precisely when and how unlabeled data can significantly alleviate the need for labeled data. This is achieved by a simple pseudo-labeling algorithm.

Machine learning solutions are rapidly adopted to enable a variety of key use cases, from conversational AI assistants to scientific discovery. As the adoption of machine learning models becomes increasingly prevalent, the associated lifecycle carbon footprint is expected to increase, including both *operational carbon* from training and inference and *embodied carbon* from AI hardware manufacturing. We introduce CATransformers, the first carbon-aware co-optimization framework for Transformer-based models and hardware accelerators. By integrating both operational and embodied carbon into early-stage design space exploration, CATransformers enables sustainability-driven model architecture and hardware accelerator co-design that reveals fundamentally different trade-offs than latency- or energy-centric approaches. Evaluated across a range of Transformer models, CATransformers consistently demonstrates the potential to reduce total carbon emissions --by up to 30\% -- while maintaining accuracy and latency. We further highlight its extensibility through a focused case study on multi-modal models. Our results emphasize the need for holistic optimization methods that prioritize carbon efficiency without compromising model capability and execution time performance. The source code of CATransformers is available at https://github.com/facebookresearch/CATransformers.

Many modern learning tasks require models that can take inputs of varying sizes. Consequently, dimension-independent architectures have been proposed for domains where the inputs are graphs, sets, and point clouds. Recent work on graph neural networks has explored whether a model trained on low-dimensional data can transfer its performance to higher-dimensional inputs. We extend this body of work by introducing a general framework for transferability across dimensions. We show that transferability corresponds precisely to continuity in a limit space formed by identifying small problem instances with equivalent large ones. This identification is driven by the data and the learning task. We instantiate our framework on existing architectures, and implement the necessary changes to ensure their transferability. Finally, we provide design principles for designing new transferable models. Numerical experiments support our findings.

Large Language Models (LLMs) have shown remarkable capabilities, with optimizing their input prompts playing a pivotal role in maximizing their performance. However, while LLM prompts consist of both the task-agnostic system prompts and task-specific user prompts, existing work on prompt optimization has focused on user prompts specific to individual queries or tasks, and largely overlooked the system prompt that is, once optimized, applicable across different tasks and domains. Motivated by this, we introduce the novel problem of bilevel system prompt optimization, whose objective is to design system prompts that are robust to diverse user prompts and transferable to unseen tasks. To tackle this problem, we then propose a meta-learning framework, which meta-learns the system prompt by optimizing it over various user prompts across multiple datasets, while simultaneously updating the user prompts in an iterative manner to ensure synergy between them. We conduct experiments on 14 unseen datasets spanning 5 different domains, on which we show that our approach produces system prompts that generalize effectively to diverse user prompts. Also, our findings reveal that the optimized system prompt enables rapid adaptation even to unseen tasks, requiring fewer optimization steps for test-time user prompts while achieving improved performance.

We study bilevel optimization problems where the lower-level problems are strongly convex and have coupled linear constraints. To overcome the potential non-smoothness of the hyper-objective and the computational challenges associated with the Hessian matrix, we utilize penalty and augmented Lagrangian methods to reformulate the original problem as a single-level one. Especially, we establish a strong theoretical connection between the reformulated function and the original hyper-objective by characterizing the closeness of their values and derivatives. Based on this reformulation, we propose a single-loop, first-order algorithm for linearly constrained bilevel optimization (SFLCB). We provide rigorous analyses of its non-asymptotic convergence rates, showing an improvement over prior double-loop algorithms -- form $O(\epsilon^{-3}\log(\epsilon^{-1}))$ to $O(\epsilon^{-3})$. The experiments corroborate our theoretical findings and demonstrate the practical efficiency of the proposed SFLCB algorithm. Simulation code is provided at https://github.com/ShenGroup/SFLCB.

Stochastic gradient optimization is the dominant learning paradigm for a variety of scenarios, from classical supervised learning to modern self-supervised learning. We consider stochastic gradient algorithms for learning problems whose objectives rely on unknown nuisance parameters, and establish non-asymptotic convergence guarantees. Our results show that, while the presence of a nuisance can alter the optimum and upset the optimization trajectory, the classical stochastic gradient algorithm may still converge under appropriate conditions, such as Neyman orthogonality. Moreover, even when Neyman orthogonality is not satisfied, we also show that an algorithm variant with approximately orthogonalized updates (with an approximately orthogonalized gradient oracle) may achieve similar convergence rates. Examples from orthogonal statistical learning/double machine learning and causal inference are discussed.