ICLR 2026 论文列表

While normalization techniques are widely used in deep learning, their theoretical understanding remains relatively limited. In this work, we establish the benefits of (generalized) weight normalization (WN) applied to the overparameterized matrix sensing problem. We prove that WN with Riemannian optimization achieves linear convergence, yielding an $\textit{exponential}$ speedup over standard methods that do not use WN. Our analysis further demonstrates that both iteration and sample complexity improve polynomially as the level of overparameterization increases. To the best of our knowledge, this work provides the first characterization of how WN leverages overparameterization for faster convergence in matrix sensing.

This paper introduces a new constraint-free concave dual formulation for the Wasserstein barycenter. Tailoring the vanilla dual gradient ascent algorithm to the Sobolev geometry, we derive a scalable Sobolev gradient ascent (SGA) algorithm to compute the barycenter for input distributions supported on a regular grid. Despite the algorithmic simplicity, we provide a global convergence analysis that achieves the same rate as the classical subgradient descent methods for minimizing nonsmooth convex functions in the Euclidean space. A central feature of our SGA algorithm is that the computationally expensive $c$-concavity projection operator enforced on the Kantorovich dual potentials is unnecessary to guarantee convergence, leading to significant algorithmic and theoretical simplifications over all existing primal and dual methods for computing the exact barycenter. Our numerical experiments demonstrate the superior empirical performance of SGA over the existing optimal transport barycenter solvers.

This paper focuses on decentralized composite optimization over networks without a central coordinator. We propose a novel decentralized Symmetric ADMM algorithm that incorporates multiple communication rounds within each iteration, derived from a new constraint formulation that enables information exchange beyond immediate neighbors. While increasing per-iteration communication, our approach significantly reduces the total number of iterations and overall communication cost. We further design optimal communication rules that minimize the number of rounds and variables transmitted per iteration. The proposed algorithms are shown to achieve linear convergence under standard assumptions. Extensive experiments on regression and classification tasks validate the theoretical results and demonstrate superior performance compared to existing decentralized optimization methods.

We propose an always-feasible ``flexible'' quadratic programming (QP) optimizer, FlexQP, which is based on an exact relaxation of the QP constraints. If the original constraints are feasible, then the optimizer finds the optimal solution to the original QP. On the other hand, if the constraints are infeasible, the optimizer identifies a solution that minimizes the constraint violation in a sparse manner. FlexQP scales favorably with respect to the problem dimension, is robust to both feasible and infeasible QPs with minimal assumptions on the problem data, and can be effectively warm-started. We subsequently apply deep unfolding to improve our optimizer through data-driven techniques, leading to an accelerated version called Deep FlexQP. By learning dimension-agnostic feedback policies for the parameters from a small number of training examples, Deep FlexQP generalizes to problems with larger dimensions and can optimize for many more iterations than it was initially trained for. Our approach outperforms two recently proposed state-of-the-art accelerated QP approaches on a suite of benchmark systems including portfolio optimization, classification, and regression problems. We provide guarantees on the expected performance of our deep QP optimizer through probably approximately correct (PAC) Bayes generalization bounds. These certificates are used to design an accelerated sequential quadratic programming solver that solves nonlinear optimal control and predictive safety filter problems faster than traditional approaches. Overall, our approach is very robust and greatly outperforms existing non-learning and learning-based optimizers in terms of both runtime and convergence to the optimal solution across multiple classes of NLPs.

Submodular maximization has broad applications in machine learning, network design, and data mining. However, classical algorithms often suffer from prohibitively high computational costs, which severely limit their scalability in practice. In this work, we focus on maximizing Sums of Concave over Modular functions (SCMs), an important subclass of submodular functions, under three fundamental constraints: cardinality, knapsack, and partition matroids. Our method integrates three components: continuous relaxation, Accelerated Approximate Projected Gradient Ascent (AAPGA), and randomized rounding, to efficiently compute near-optimal solutions. We establish a $(1 - \varepsilon - \eta - e^{-\Omega(\eta^2)})$ approximation guarantee for both cardinality and partition matroid constraints, with query complexity $O\left(n^{1/2}\varepsilon^{-1/2} (T_1 + T_2)\right)$. For the knapsack constraint, the approximation ratio degrades by a factor of $1/2$, with query complexity $O\left(n T_1 + n^{1/2}\varepsilon^{-1/2} T_2\right)$, where $T_1$ denotes the computational cost of evaluating the concave extension, and $T_2$ denotes the computational cost of backpropagation. By leveraging efficient convex optimization techniques, our approach substantially accelerates convergence toward high-quality solutions. In empirical evaluations, we demonstrate that AAPGA consistently outperforms standard PGA. On small-scale experiments, AAPGA achieves superior results in significantly less time, being up to $32.3\times$ faster than traditional methods. On large-scale experiments, our parallel multi-GPU implementation further enhances performance, demonstrating the scalability of our approach.

We introduce novel techniques to enhance Frank-Wolfe algorithms by leveraging function smoothness beyond traditional short steps. Our study focuses on Frank-Wolfe algorithms with step sizes that incorporate primal-dual guarantees, offering practical stopping criteria. We present a new Frank-Wolfe algorithm utilizing an optimistic framework and provide a primal-dual convergence proof. Additionally, we propose a generalized short-step strategy aimed at optimizing a computable primal-dual gap. Interestingly, this new generalized short-step strategy is also applicable to gradient descent algorithms beyond Frank-Wolfe methods. Empirical results demonstrate that our optimistic algorithm outperforms existing methods, highlighting its practical advantages.

We introduce Cautious Weight Decay (CWD), a one-line, optimizer-agnostic modification that applies weight decay only to parameter coordinates whose signs align with the optimizer update. Unlike standard decoupled decay, which implicitly optimizes a regularized or constrained objective, CWD preserves the original loss and admits a bilevel interpretation: it induces sliding-mode behavior upon reaching the stationary manifold, allowing it to search for locally Pareto-optimal stationary points of the unmodified objective. In practice, CWD is a drop-in change for optimizers such as AdamW, Lion, and Muon, requiring no new hyperparameters or additional tuning. For language model pre-training and ImageNet classification, CWD consistently improves final loss and accuracy at million- to billion-parameter scales.

We present a general strategy for turning generative models into candidate solution samplers for batch Bayesian optimization (BO). The use of generative models for BO enables: large batch scaling as generative sampling, optimization of non-continuous design spaces, and high-dimensional and combinatorial design by using generative priors over feasible regions. Inspired by the success of direct preference optimization (DPO) and its variants, we show that its approach of directly training generative models using preferential rewards without the need for an intermediate reward model is extensible to the BO case. Furthermore, this framework is generalizable beyond preference-based feedback to general types of reward signals and loss functions. In essence, one can train a generative model with noisy, simple utility values directly computed from observations to then form proposal distributions whose densities are proportional to the expected utility, i.e., BO's acquisition function values. This perspective unifies recent progress in using generative models for black-box optimization and connects it with batch Bayesian optimization under a general framework. As theoretical results, we show that the generative models within the BO process approximately follow a sequence of distributions which asymptotically concentrate at the global optima under certain conditions. We also demonstrate this effect through experiments on challenging optimization problems involving large batches in high dimensions.

Direct Preference Optimization (DPO) has emerged as a cornerstone of reinforcement learning from human feedback (RLHF) due to its simplicity and efficiency. However, existing DPO-based methods typically treat all preference pairs equally, overlooking substantial variations in data quality and learning difficulty, which leads to inefficient data utilization and suboptimal performance. To address this limitation, we propose **Uni-DPO**, a unified dynamic preference optimization framework that jointly considers (a) the inherent quality of preference pairs and (b) the model's evolving performance during training. By adaptively reweighting samples based on both factors, Uni-DPO enables more effective use of preference data and achieves superior performance. Extensive experiments across models and benchmarks demonstrate the effectiveness and generalization of Uni-DPO. On textual tasks, Gemma-2-9B-IT fine-tuned with Uni-DPO surpasses the leading LLM, Claude 3 Opus, by 6.7 points on Arena-Hard. On mathematical and multimodal tasks, Uni-DPO consistently outperforms baseline methods across all benchmarks, providing strong empirical evidence of its effectiveness and robustness.

Different federated optimization algorithms typically employ distinct client-selection strategies: some methods communicate only with a randomly sampled subset of clients at each round, while others need to periodically communicate with all clients or use a hybrid scheme that combines both strategies. However, existing metrics for comparing optimization methods typically do not distinguish between these strategies, which often incur different communication costs in practice. To address this disparity, we introduce a simple and natural model of federated optimization that quantifies communication and local computation complexities. This new model allows for several commonly used client-selection strategies and explicitly associates each with a distinct cost. Within this setting, we propose a new algorithm that achieves the best-known communication and local complexities among existing federated optimization methods for non-convex optimization. This algorithm is based on the inexact composite gradient method with a carefully constructed gradient estimator and a special procedure for solving the auxiliary subproblem at each iteration. The gradient estimator is based on SAGA, a popular variance-reduced gradient estimator. We first derive a new variance bound for it, showing that SAGA can exploit functional similarity. We then introduce the Recursive-Gradient technique as a general way to potentially improve the error bound of a given conditionally unbiased gradient estimator, including both SAGA and SVRG. By applying this technique to SAGA, we obtain a new estimator, RG-SAGA, which has an improved error bound compared to the original one.

Quality-Diversity (QD) algorithms have shown remarkable success in discovering diverse, high-performing solutions, but rely heavily on hand-crafted behavioral descriptors that constrain exploration to predefined notions of diversity. Leveraging the equivalence between policies and occupancy measures, we present a theoretically grounded approach to automatically generate behavioral descriptors by embedding the occupancy measures of policies in Markov Decision Processes. Our method, AutoQD, leverages random Fourier features to approximate the Maximum Mean Discrepancy (MMD) between policy occupancy measures, creating embeddings whose distances reflect meaningful behavioral differences. A low-dimensional projection of these embeddings that captures the most behaviorally significant dimensions can then be used as behavioral descriptors for CMA-MAE, a state of the art blackbox QD method, to discover diverse policies. We prove that our embeddings converge to true MMD distances between occupancy measures as the number of sampled trajectories and embedding dimensions increase. Through experiments in multiple continuous control tasks we demonstrate AutoQD's ability in discovering diverse policies without predefined behavioral descriptors, presenting a well-motivated alternative to prior methods in unsupervised Reinforcement Learning and QD optimization. Our approach opens new possibilities for open-ended learning and automated behavior discovery in sequential decision making settings without requiring domain-specific knowledge.

Clustering is a hard discrete optimization problem. Nonconvex approaches such as low-rank semidefinite programming (SDP) have recently demonstrated promising statistical and local algorithmic guarantees for cluster recovery. Due to the combinatorial structure of the $K$-means clustering problem, current relaxation algorithms struggle to balance their constraint feasibility and objective optimality, presenting tremendous challenges in computing the second-order critical points with rigorous guarantees. In this paper, we provide a new formulation of the $K$-means problem as a smooth unconstrained optimization over a submanifold and characterize its Riemannian structures to allow it to be solved using a second-order cubic-regularized Riemannian Newton algorithm. By factorizing the $K$-means manifold into a product manifold, we show how each Newton subproblem can be solved in linear time. Our numerical experiments show that the proposed method converges significantly faster than the state-of-the-art first-order nonnegative low-rank factorization method, while achieving similarly optimal statistical accuracy.

In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The key innovation of our analysis is a novel notion called Gradient-Normalized Smoothness, which characterizes the maximum radius of a ball around the current point that yields a good relative approximation of the gradient field. Our theory establishes a natural intrinsic connection between Hessian approximation and the linearization of the gradient. Importantly, Gradient-Normalized Smoothness does not depend on the specific problem class of the objective functions, while effectively translating local information about the gradient field and Hessian approximation into the global behavior of the method. This new concept equips approximate second-order algorithms with universal global convergence guarantees, recovering state-of-the-art rates for functions with Hölder-continuous Hessians and third derivatives, quasi-self-concordant functions, as well as smooth classes in first-order optimization. These rates are achieved automatically and extend to broader classes, such as generalized self-concordant functions. We demonstrate direct applications of our results for global linear rates in logistic regression and softmax problems with approximate Hessians, as well as in non-convex optimization using Fisher and Gauss-Newton approximations.

The indicator matrix plays an important role in machine learning, but optimizing it is an NP-hard problem. We propose a new relaxation of the indicator matrix and compared with other existing relaxations, it can flexibly incorporate class information. We prove that this relaxation forms a manifold, which we call the Relaxed Indicator Matrix Manifold (RIM manifold). Based on Riemannian geometry, we develop a Riemannian toolbox for optimization on the RIM manifold. Specifically, we provide several methods of Retraction, including a fast Retraction method to obtain geodesics. We point out that the RIM manifold is a generalization of the double stochastic manifold, and it is much faster than existing methods on the double stochastic manifold, which has a complexity of \( \mathcal{O}(n^3) \), while RIM manifold optimization is \( \mathcal{O}(n) \) and often yields better results. We conducted extensive experiments, including image denoising, with millions of variables to support our conclusion, and applied the RIM manifold to Ratio Cut, we provide a rigorous convergence proof and achieve clustering results that outperform the state-of-the-art methods. Our Code is presented in Appendix H.

Regret matching (RM)---and its modern variants---is a foundational online algorithm that has been at the heart of many AI breakthrough results in solving benchmark zero-sum games, such as poker. Yet, surprisingly little is known so far in theory about its convergence beyond two-player zero-sum games. For example, whether regret matching converges to Nash equilibria in potential games has been an open problem for two decades. Even beyond games, one could try to use RM variants for general constrained optimization problems. Recent empirical evidence suggests that they---particularly regret matching$^+$ (RM$^+$)---attain strong performance on benchmark constrained optimization problems, outperforming traditional gradient descent-type algorithms. We show that alternating RM$^+$ converges to an $\epsilon$-KKT point after $O_\epsilon(1/\epsilon^4)$ iterations, establishing for the first time that it is a sound and fast first-order optimizer. Our argument relates the KKT gap to the accumulated regret, two quantities that are entirely disparate in general but interact in an intriguing way in our setting, so much so that when regrets are bounded, our complexity bound improves all the way to $O_\epsilon(1/\epsilon^2)$. From a technical standpoint, while RM$^+$ does not have the usual one-step improvement property in general, we show that it does in a certain region that the algorithm will quickly reach and remain in thereafter. In sharp contrast, our second main result establishes a lower bound: RM, with or without alternation, can take an exponential number of iterations to reach a crude approximate solution even in two-player potential games. This represents the first worst-case separation between RM and RM$^+$. Our lower bound shows that convergence to coarse correlated equilibria in potential games is exponentially faster than convergence to Nash equilibria.

We introduce an output layer for neural networks that ensures satisfaction of convex constraints. Our approach, $\Pi$net, leverages operator splitting for rapid and reliable projections in the forward pass, and the implicit function theorem for backpropagation. We deploy $\Pi$net as a feasible-by-design optimization proxy for parametric constrained optimization problems and obtain modest-accuracy solutions faster than traditional solvers when solving a single problem, and significantly faster for a batch of problems. We surpass state-of-the-art learning approaches by orders of magnitude in terms of training time, solution quality, and robustness to hyperparameter tuning, while maintaining similar inference times. Finally, we tackle multi-vehicle motion planning with non-convex trajectory preferences and provide $\Pi$net as a GPU-ready package implemented in JAX.

Image identity customization aims to synthesize realistic and diverse portraits of a specified identity, given a reference image and a text prompt. This task presents two key challenges: (1) generating realistic portraits that preserve fine-grained facial details of the reference identity, and (2) maintaining identity consistency while achieving strong alignment with the text prompt. Our findings suggest that existing single-stream methods fail to capture and guide fine-grained identity details. To address these challenges, we introduce \textit{OmniPortrait}, a novel diffusion-based framework for fine-grained identity fidelity and high editability in portrait synthesis. Our core idea is pivotal optimization, which leverages dual-stream identity guidance in a coarse-to-fine manner. First, a Pivot ID Encoder is proposed and trained with a face localization loss while avoiding the degradation of editability typically caused by fine-tuning the denoiser. Although this encoder primarily guides coarse-level identity synthesis, it provides a good initialization that serves as the identity pivot for optimization during inference. Second, we propose Reference-Based Guidance, which performs on-the-fly feature matching and optimization over diffusion intermediate features conditioned on the identity pivot. In addition, our approach is able to generalize naturally to multi-identity customized image generation scenarios. Extensive experiments demonstrate significant improvements in both identity preservation and text alignment, establishing a new benchmark for image identity customization.

As large language models (LLMs) continue to scale up, their performance on various downstream tasks has significantly improved. However, evaluating their capabilities has become increasingly expensive, as performing inference on a large number of benchmark samples incurs high computational costs. In this paper, we revisit the model-item performance matrix and show that it exhibits sparsity, that representative items can be selected as anchors, and that the task of efficient benchmarking can be formulated as a sparse optimization problem. Based on these insights, we propose SparseEval, a method that, for the first time, adopts gradient descent to optimize anchor weights and employs an iterative refinement strategy for anchor selection. We utilize the representation capacity of MLP to handle sparse optimization and propose the Anchor Importance Score and Candidate Importance Score to evaluate the value of each item for task-aware refinement. Extensive experiments demonstrate the low estimation error and high Kendall’s~$\tau$ of our method across a variety of benchmarks, showcasing its superior robustness and practicality in real-world scenarios.

Spectral bias, the tendency of neural networks to learn low frequencies first, can be both a blessing and a curse. While it enhances the generalization capabilities by suppressing high-frequency noise, it can be a limitation in scientific tasks that require capturing fine-scale structures. The delayed generalization phenomenon known as grokking is another barrier to rapid training of neural networks. Grokking has been hypothesized to arise as learning transitions from the NTK to the feature-rich regime. This paper explores the impact of preconditioned gradient descent (PGD), such as Gauss-Newton, on spectral bias and grokking phenomena. We demonstrate through theoretical and empirical results how PGD can mitigate issues associated with spectral bias. Additionally, building on the rich learning grokking hypothesis, we study how PGD can be used to reduce delays associated with grokking. Our conjecture is that PGD, without the impediment of spectral bias, enables uniform exploration of the parameter space in the NTK regime. Our experimental results confirm this prediction, providing strong evidence that grokking represents a transitional behavior between the lazy regime characterized by the NTK and the rich regime. These findings deepen our understanding of the interplay between optimization dynamics, spectral bias, and the phases of neural network learning.