ICML 2026 论文列表

Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process. To manage this realistic and challenging setting, new mechanisms, such as gradient clipping and gradient normalization, have been introduced to ensure the convergence of first-order algorithms. However, adaptive gradient methods, a famous class of modern optimizers that includes popular $\mathtt{Adam}$ and $\mathtt{AdamW}$, often perform well even without any extra operations mentioned above. It is therefore natural to ask whether adaptive gradient methods can converge under heavy-tailed noise without any algorithmic changes. In this work, we take the first step toward answering this question by investigating a special case, $\mathtt{AdaGrad}$, the origin of adaptive gradient methods. We provide the first provable convergence rate for $\mathtt{AdaGrad}$ in non-convex optimization when the tail index $p$ satisfies $4/3<p\leq2$. Notably, this result is achieved without requiring any prior knowledge of $p$ and is hence adaptive to the tail index. In addition, we develop an algorithm-dependent lower bound, suggesting that the existing minimax rate for heavy-tailed optimization is not attainable by $\mathtt{AdaGrad}$. Lastly, we consider $\mathtt{AdaGrad}\text{-}\mathtt{Norm}$, a popular variant of $\mathtt{AdaGrad}$ in theoretical studies, and show an improved rate that holds for any $1<p\leq2$ under an extra mild assumption.

Communication costs are a major bottleneck in distributed learning and first-order optimization. A common approach to alleviate this issue is to compress the gradient information exchanged between agents. However, such compression typically degrades the convergence guarantees of gradient-based methods. Error feedback mechanisms provide a simple and computationally cheap remedy for this issue, but numerous variants have been proposed, and their relative performance remains poorly understood. This paper provides tight convergence analyses for two of the main error-feedback algorithms from the literature, the classic Error Feedback method (EF) and Error Feedback 21 (EF21), by identifying optimal step-size choices and constructing optimal Lyapunov functions tailored to each method. The results hold independently of the number of agents and recover the known best guarantees possible in the single-agent regime.

The Gromov-Wasserstein (GW) distance provides a powerful framework for aligning structured data by comparing the intrinsic geometries of metric measure spaces, and has become a fundamental tool in machine learning. Most existing methods leverage entropy regularization to reduce the computational complexity to $\boldsymbol{\mathrm{O}}(n^3)$, where $n$ is the number of samples. However, this cubic time complexity remains a major bottleneck in large-scale applications, severely limiting the scalability. To address this challenge, we propose LoBCD-GW, an efficient GW optimization algorithm. Specifically, we reveal the data-dependent sparsity of large-magnitude updates to the coupling matrix and introduce a localized block coordinate selection strategy. This confines the optimization to a "selected set" of size $r$ (which is a parameter that depends on the given data set, and usually is much less than $n$), thereby reducing the complexity to $\boldsymbol{\mathrm{O}}(r^3)$. In addition, unlike prior acceleration methods often based on constraint relaxation, our method can guarantee the strict feasibility through a novel "marginal compensation mechanism" to synchronize local mass redistribution with global constraints. Finally, we conduct a set of experiments on various datasets, and the results demonstrate that our method achieves a $5.8\times$ speedup on large-scale graph alignment benchmarks, while maintaining state-of-the-art accuracy.

Oversmoothing has long been identified as a major limitation of Graph Neural Networks (GNNs): input node features are smoothed at each layer and converge to a constant non-informative representation, *if the weights of the GNN are sufficiently bounded*. This assumption is crucial: if, on the contrary, the weights are sufficiently large, then oversmoothing may be compensated. Theoretically, GNN could thus *learn* to not oversmooth. However, this does not really happen in practice, which prompts us to examine oversmoothing from an *optimization* point of view. In this paper, we analyze *backward oversmoothing*, that is, the notion that backpropagated errors are also subject to oversmoothing from output to input. With non-linearities, we outline the key role of the *interaction* between forward and backward smoothing. Moreover, we show that, due to backward oversmoothing, GNNs provably exhibit many *spurious stationary points*: as soon as the *last* layer is trained, the *whole* GNN is at a stationary point. As a result, we can exhibit regions where gradients are near-zero while the loss stays high. Additionally, we prove that this is *specific* to GNNs, and does not necessarily hold for Multi-Layer Perceptrons. This paper is a step toward a more complete comprehension of the optimization landscape of GNNs.

Item response theory (IRT) is widely used in areas such as recommender systems, education, psychology, and other fields. A popular model for IRT is the Rasch model. Under this model, if a user with ability $\theta$ performs a task with difficulty $\beta$ then its label $X \sim \text{Bernoulli} (1 / (1 + \exp(-(\theta - \beta)))$. Existing joint maximum likelihood estimation approaches for this problem do not perform well on small datasets and also lack theoretical guarantees. Recently, Nguyen and Zhang proposed a two step approach: (1) spectral method for estimation of task parameters, (2) likelihood optimization for estimation of user parameters. While this approach is theoretically sound, it is not computationally efficient. In this work, we propose an EM-based algorithm for joint estimation of item and user parameters by introducing Pólya-Gamma latent variables, which simplify the logistic log-likelihood. We show that our algorithm is both theoretically sound and consistently outperforms existing methods on synthetic and real-world datasets.

We present the first comprehensive Lean 4 formalization of statistical learning theory (SLT) grounded in empirical process theory. Our end-to-end formal infrastructure implement the missing contents in latest Lean library, including a complete development of Gaussian Lipschitz concentration, Dudley’s entropy integral theorem for sub-Gaussian processes, and an application to least-squares (sparse) regression with a sharp rate. The project was carried out using a human-AI collaborative workflow, in which humans design proof strategies and AI agents execute tactical proof construction, leading to the human-verified Lean 4 toolbox for SLT. Beyond implementation, the formalization process exposes and resolves implicit assumptions and missing details in standard SLT textbooks, enforcing a granular, line-by-line understanding of the theory. This work establishes a reusable formal foundation and opens the door for future developments in machine learning theory. The code is provided in https://github.com/YuanheZ/lean-stat-learning-theory.

In Mixture-of-Experts (MoE) training, sparse routing, i.e., activating only the top-$K$ experts per token, is essential for balancing convergence speed and computational cost. However, existing works typically choose $K$ empirically, without theoretical guidance. To address this gap, we characterize the convergence behavior of MoE training using stochastic optimization theory. Specifically, we derive a convergence upper bound of $\mathcal{O}\left(\frac{1+M/K}{\sqrt{T}}\right)$, where $T$ is the number of training iterations and $M$ is the total number of experts per MoE layer. This result guarantees convergence and shows that increasing $K$ can accelerate training. By further fixing the total computational budget $R$ (in FLOPs), we obtain a refined bound of $\mathcal{O}\left(\sqrt{\frac{K}{R}} + \frac{M}{\sqrt{K R}}\right)$, which is convex in $K$ and implies the existence of an optimal $K^{*}\in[1,M]$ that achieves the best convergence performance. Extensive experiments validate our theoretical analysis under diverse settings.

The mean-field Langevin dynamics (MFLD) minimizes an entropy-regularized nonlinear convex functional on the Wasserstein space over $\mathbb{R}^d$, and has gained attention recently as a model for the gradient descent dynamics of interacting particle systems such as infinite-width two-layer neural networks. However, many problems of interest have constrained domains, which are not solved by existing mean-field algorithms due to the global diffusion term. We study the optimization of probability measures constrained to a convex subset of $\mathbb{R}^d$ by proposing the \emph{mirror mean-field Langevin dynamics} (MMFLD), an extension of MFLD to the mirror Langevin framework. We obtain linear convergence guarantees for the continuous MMFLD via a uniform log-Sobolev inequality, and uniform-in-time propagation of chaos results for its time- and particle-discretized counterpart.

We investigate decentralized online convex optimization with compressed communication, where $n$ learners connected by a network collaboratively minimize a sequence of global loss functions using only local information and compressed data from their neighbors. Prior work has established regret bounds of $O(\max\\{\omega^{-2}\rho^{-4}n^{1/2},\omega^{-4}\rho^{-8}\\}n\sqrt{T})$ and $O(\max\\{\omega^{-2}\rho^{-4}n^{1/2},\omega^{-4}\rho^{-8}\\}n\ln{T})$ for convex and strongly convex functions, respectively, where $\omega\in(0,1]$ is the compression quality factor and $\rho<1$ is the spectral gap of the communication matrix. However, these regret bounds suffer from a prohibitively high quadratic or even quartic dependence on $\omega^{-1}$. Moreover, the super-linear dependence on $n$ is also undesirable. To overcome these shortcomings, we propose a novel algorithm that achieves improved regret bounds of $\tilde{O}(\omega^{-1/2}\rho^{-1}n\sqrt{T})$ and $\tilde{O}(\omega^{-1}\rho^{-2}n\ln{T})$ for convex and strongly convex functions, respectively. The primary idea is to design a two-level blocking update framework incorporating two novel ingredients: an online gossip strategy and an error compensation scheme, which work together to promote better consensus among learners. Furthermore, we establish the first lower bounds for this problem, justifying the optimality of our results with respect to both $\omega$ and $T$. Additionally, we consider the bandit feedback scenario and extend our method with classical gradient estimators to enhance existing regret bounds.

Credal predictors are epistemic-uncertainty-aware models that produce a convex set of probabilistic predictions. They provide a principled framework for quantifying predictive epistemic uncertainty (EU) and have been shown to improve model robustness across a range of settings. However, most state-of-the-art (SOTA) methods primarily define EU as disagreement induced by random training initializations, which mainly reflects sensitivity to optimization randomness rather than uncertainty from more substantive sources. In response, we formulate EU as disagreement between models trained under different degrees of relaxation of the i.i.d. assumption between the training and test distributions. Building on this idea, we propose *CreDRO*, which learns an ensemble of plausible models via distributionally robust optimization. As a result, CreDRO captures EU arising not only from training randomness but also from informative disagreement due to potential train–test distribution shifts. Empirically, CreDRO consistently outperforms SOTA credal approaches on downstream tasks, including out-of-distribution detection on extensive benchmarks and selective classification in medical settings.

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast $ \widetilde{\mathcal{O}}(1/n) $ convergence rate under *unilateral concentrability*, improving over the standard $ \widetilde{\mathcal{O}}(1/\sqrt{n}) $ rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same $ \widetilde{\mathcal{O}}(1/n) $ statistical rate up to a vanishing optimization error of order $ \widetilde{\mathcal{O}}(1/\sqrt{T}) $ in the number of self-play iterations $T$.

Modern optimizers combine gradients from the current mini-batch with historical optimization state, such as momentum or adaptive moments. While highly effective, aggregating across the batch and incorporating this history can produce parameter updates that increase the loss of individual samples. We term this effect *harm* and formalize the parameter update as an optimization problem that explicitly minimizes the conflicting impact of both batch averaging and past optimization state on current data. Because the exact formulation is intractable, we introduce a highly efficient proxy. We first reduce the problem's dimensionality to the batch size, and then drastically cut memory and speed bottlenecks by successfully restricting the optimization to the last linear layer. This hinges on the unexpected finding that *this layer alone reliably captures the second-order statistics of the per-sample gradients*. The resulting surrogate problem integrates readily into standard optimizers like SGD and AdamW, and can be solved using a small number of GPU-friendly iterations. Crucially, the method exhibits *favorable scaling properties*, as the relative computational overhead shrinks as the model size or input grows. Experiments on image classification benchmarks confirm *reduced per-sample interference* and *improved generalization*.

We study optimization over non-convex constraint sets that are homeomorphic to a ball, encompassing important problem classes such as star-shaped sets that frequently arise in machine learning and engineering applications. We propose **Hom-PGD$^+$**, a learning-based and projection-efficient first-order method that efficiently solves such problems without requiring expensive projection or optimization oracles. Our approach leverages an invertible neural network (INN) to learn the homeomorphism between the non-convex constraint set and a unit ball, transforming the original problem into an equivalent ball-constrained optimization where projections admit efficient solutions. We establish that Hom-PGD$^+$ achieves an $\mathcal{O}(\epsilon^{-2})$ convergence rate to an ($\epsilon + \mathcal{O}(\sqrt{\epsilon_{\text{inn}}})$)-approximate stationary solution, where $\epsilon_{\text{inn}}$ denotes the homeomorphism learning error. This rate significantly improves upon existing methods for optimization over non-convex sets, while maintaining a per-iteration complexity of only $\mathcal{O}(W)$ for $W$ INN parameters. Extensive experiments, including QCQP, chance-constrained power-system optimization, and non-uniform adversarial attacks, demonstrate that Hom-PGD$^+$ achieves competitive solution quality while delivering speedups of up to one order of magnitude.

This article initiates the study of a basic question about model pruning. Given a vector s of importance scores assigned to model components, how many of the scored components could be discarded without sacrificing performance? We propose Effective Model Pruning (EMP), which derives the desired sparsity directly from the score distribution using the notion of effective sample size from particle filtering, also known as the inverse Simpson index. Rather than prescribe a pruning criterion, EMP supplies a universal adaptive threshold derived from the distribution of the score $s$ over the model components: EMP maps $s$ to a number $N_{eff} = N_{eff} (s)$, called the effective sample size. The $N − N_{eff}$ lowest scoring components are discarded. A tight lower bound on the preserved mass fraction seff (the sum of retained normalized scores) in terms of $N_{eff}$ is derived. This process yields models with a provable upper bound on the loss change relative to the original dense model. Numerical experiments are performed demonstrating this phenomenon across a variety of network architectures including MLPs, CNNs, Transformers, LLMs, and KAN. It is also shown that EMP addresses a rich set of pruning criteria such as weight magnitude, attention score, KAN importance score, and even feature-level signals such as image pixels.

Quantized neural network training optimizes a discrete, non-differentiable objective. The straight-through estimator (STE) enables backpropagation through surrogate gradients and is widely used. While previous studies have primarily focused on the properties of surrogate gradients and their convergence, the influence of quantization hyperparameters, such as bit width and quantization range, on learning dynamics remains largely unexplored. We theoretically show that in the high-dimensional limit, STE dynamics converge to a deterministic ordinary differential equation. This reveals that STE training exhibits a plateau followed by a sharp drop in generalization error, with plateau length depending on the quantization range. A fixed-point analysis quantifies the asymptotic deviation from the unquantized linear model. We also extend analytical techniques for stochastic gradient descent to nonlinear transformations of weights and inputs.

We introduce a constrained optimization framework for training transformers that behave like optimization descent algorithms. Specifically, we enforce layerwise descent constraints on the objective function and replace standard empirical risk minimization (ERM) with a primal-dual training scheme. This approach yields models whose intermediate representations decrease the loss monotonically in expectation across layers. We apply our method to both unrolled transformer architectures and conventional pretrained transformers on tasks of video denoising and text classification. Across these settings, we observe that constrained transformers achieve stronger robustness to perturbations and maintain higher out-of-distribution generalization, while preserving competitive in-distribution performance.

Resilience against Byzantine attackers and faster convergence on sparse networks are critical for decentralized optimization, yet existing methods fail to achieve both simultaneously. Existing DSGD-based Byzantine-resilient methods suffer from high transient complexity of $\mathcal{O}\left((1-\lambda)^{-6}\right)$, where $1-\lambda$ denotes the spectral gap of the network. While bias-correction methods such as Exact Diffusion can improve topology dependence, directly combining them with robust aggregators can lead to error accumulation. To address this issue, we introduce the scaled dual ascent (SDA) within the augmented Lagrangian framework for decentralized optimization, which mitigates error accumulation by scaling the dual update steps. Based on this, we propose BRED, which integrates Byzantine-robust Exact Diffusion with the SDA framework. We prove that BRED attains linear speedup, and achieves transient complexity of $\mathcal{O}\left((1-\lambda)^{-2}\right)$ when the Byzantine fraction $\delta$ is small. We further propose the momentum variant BRED-M, which reduces the Byzantine-affected transient complexity from $\mathcal{O}\left(\delta^2(1-\lambda)^{-6}\right)$ to $\mathcal{O}\left(\delta^2(1-\lambda)^{-4}\right)$. Empirical results on benchmark datasets demonstrate the efficacy of the proposed methods across diverse network topologies.

Learning rate warm-up -- increasing the learning rate at the beginning of training -- has become a ubiquitous heuristic in modern deep learning, yet its theoretical foundations remain poorly understood. In this work, we provide a principled explanation for why warm-up improves training. We rely on a generalization of the $(L_0, L_1)$-smoothness condition, which bounds local curvature as a linear function of the loss suboptimality and exhibits desirable closure properties. We show -- both theoretically and empirically -- that this condition is satisfied by common neural architectures and accurately captures the curvature of the optimization landscape early in training. Adapting the learning rate in response to this curvature condition naturally induces a warm-up–like schedule, and we show that this choice yields provably faster convergence guarantees than using a fixed learning rate. Experiments on language and vision models show that the resulting one-parameter warm-up schedule can match tuned linear warm-up and improve over no warm-up.

Shampoo is attracting considerable attention for its superior performance on large-scale optimization benchmarks; yet it faces a significant practical bottleneck: the prohibitive computational overhead of matrix inversion. To mitigate this, practitioners typically rely on stale preconditioner updates, creating a fundamental trade-off between computational efficiency and optimization fidelity. In this work, we provide a theoretical study of staleness through the complementary lenses of convergence and stability. While staleness improves computational efficiency, it inherently degrades performance and introduces numerical instability. Crucially, we identify that damping, acting as a numerical stabilizer, can effectively suppress these negative effects. Guided by this analysis, we propose FOAM, an adaptive algorithm that stabilizes training by dynamically controlling both the damping factor and the eigendecomposition frequency based on an approximation of the staleness-oriented error. Experimental results demonstrate that FOAM reduces wall-clock time compared to standard Shampoo while maintaining robust convergence.