ICML 2026 Papers

Zeroth-Order (ZO) optimization is pivotal for scenarios where backpropagation is unavailable, such as memory-constrained on-device learning and black-box optimization. However, existing methods face a stark trade-off: they are either sample-inefficient (e.g., standard finite differences) or suffer from high variance due to randomized estimation (e.g., random subspace methods). In this work, we propose Coherent Coordinate Descent (CoCD), a deterministic, sample-efficient, and budget-aware ZO optimizer. Theoretically, we formalize the notion of gradient coherence and demonstrate that CoCD is equivalent to Block Cyclic Coordinate Descent (BCCD) with ``warm starts,'' effectively converting historical (stale) gradients from a liability into a computational asset. This mechanism enables $O(1)$ query complexity per step while maintaining global descent directions. Furthermore, we derive error bounds revealing a counter-intuitive insight: larger finite-difference step sizes can induce an implicit smoothing effect on the optimization landscape by reducing the effective smoothness constant, thereby improving convergence stability. Experiments on MLP, CNN, and ResNet architectures (up to 270k parameters) demonstrate that CoCD significantly outperforms BCCD in terms of sample efficiency and convergence loss/accuracy, and exhibits superior stability over randomized ZO methods. Our results suggest that deterministic, structure-aware updates offer a superior alternative to randomization for lightweight ZO optimization.

Orthogonality constraints are ubiquitous in robust and probabilistic machine learning. Unfortunately, current optimizers are computationally expensive and do not scale to problems with hundreds or thousands of constraints. One notable exception is the Landing algorithm (Ablin et al., 2024) which, however comes at the expense of temporarily relaxing orthogonality. In this work, we revisit and improve on the ideas behind Landing, enabling the inclusion of modern adaptive optimizers while ensuring that orthogonal constraints are effectively met. Remarkably, these improvements come at little to no cost, and reduce the number of required hyperparemeters. Our algorithm POGO is fast and GPU-friendly, _consisting of only 5 matrix products_, and in practice maintains orthogonality at all times. On several challenging benchmarks, POGO greatly outperforms recent optimizers and shows it can optimize problems with thousands of orthogonal matrices in minutes while alternatives would take hours. As such, POGO sets a milestone to finally exploit orthogonality constraints in ML at scale. A public PyTorch implementation of POGO is available at https://github.com/adrianjav/pogo.

Stochastic optimization (SO) plays a central role in decision-making under uncertainty. Among SO problems, time-varying stochastic optimization (TV-SO) is particularly important due to its applications in adaptive control and machine learning. Non-parametric approaches have been proposed for time-varying deterministic optimization; however, they have not been developed for their stochastic counterparts. This work addresses that gap by developing a stochastic variational framework based on Malliavin calculus. This framework yields non-parametric optimality conditions for SO problems with stochastic decisions and supports the design of a scalable deep-learning algorithm that is insensitive to the parameterization dimension. This algorithm, called the Stochastic Path Follower (SPF), is applied to two important problems under distribution drift, namely least-squares recovery and logistic regression. Experimental results show that the proposed approach outperforms state-of-the-art learning-based and gradient-based methods in both performance and scalability.

The stochastic minimax optimization problem has widespread applications in machine learning. Recently, numerous distributed minimax optimization algorithms have been developed to handle distributed training data. However, most of these algorithms suffer from high communication costs. To address this issue, we develop a novel communication-efficient decentralized stochastic gradient descent ascent with momentum algorithm based on the error feedback mechanism. Importantly, our algorithm demonstrates how to balance the full-precision update and the compression residual with novel designs for coefficients regarding variables and gradients to guarantee convergence. However, compressing the primal and dual variables (and their gradients) of stochastic minimax optimization problems with the error feedback mechanism presents significant challenges for convergence analysis. In particular, it incurs the circle dependence among consensus errors and compression errors. To overcome this challenge, we propose novel strategies that enable the establishment of the convergence rate for our algorithm. Our theoretical results demonstrate how the compression operator influences the convergence rate. Finally, extensive experimental results confirm the efficacy of our proposed algorithm.

Graph neural networks (GNNs) have been widely used in various graph machine learning scenarios. Existing literature primarily assumes well-annotated training graphs, while the reliability of labels is not guaranteed in real-world scenarios. Recently, efforts have been made to address the problem of graph learning with label noise. However, existing methods often (i) struggle to distinguish between reliable and unreliable nodes, and (ii) overlook the relational information embedded in the graph topology. To tackle this problem, this paper proposes a novel method, Dual-Standard Semantic Homogeneity with Dynamic Optimization (DREAM), for reliable, relation-informed optimization on graphs with label noise. Specifically, we design a relation-informed dynamic optimization framework that iteratively reevaluates the reliability of each labeled node in the graph during the optimization process according to the relation of the target node and other nodes. To measure this relation comprehensively, we propose a dual-standard selection strategy that selects a set of anchor nodes based on both node proximity and graph topology. Subsequently, we compute the semantic homogeneity between the target node and the anchor nodes, which serves as guidance for optimization. We also provide a rigorous theoretical analysis to justify the design of DREAM. Extensive experiments are performed on six graph datasets across various domains under three types of graph label noise against competing baselines, and the results demonstrate the effectiveness of the proposed DREAM.

Nonnegative matrix factorization (NMF) seeks a low-rank approximation $X \approx UV^T$ with nonnegative factors and is commonly solved using *interior* methods that enforce feasibility throughout optimization. We show that such constraint-driven approaches can impede progress in the nonconvex landscape, leading to slow convergence or convergence to suboptimal stationary points. We propose an *exterior* framework for NMF (eNMF) that separates low-rank approximation from nonnegativity enforcement. Our method initializes from the optimal unconstrained factorization and introduces a rotation procedure that maps unconstrained factors to an exterior point closest to the nonnegative orthant. This viewpoint yields an algorithmic framework in which simple iterative updates converge to KKT-satisfying stationary points on the boundary of the positive orthant. The exterior formulation also enables a geometric interpretation of NMF solutions, clarifying equivalence classes of factorizations under permutation and orthogonal transformations. An intriguing numerical result, involving 400 NMF experiments across both real and synthetic datasets, show that in 99\% of the cases, different algorithms tend to converge towards equivalent factor matrices. We benchmark eNMF against 9 state-of-the-art NMF algorithms with 9 initialization schemes across 3 real-world and 2 synthetic datasets. eNMF consistently outperforms all 81 competitors, achieving up to 30\% lower reconstruction error under equal-time settings and up to 150\% speedup under equal-error settings. The downstream experiments further demonstrate substantial performance gains in audio processing and recommendation tasks, corroborating the practical benefits of the proposed exterior optimization framework. Code is available at https://github.com/roychowdhuryresearch/eNMF

Operator splitting algorithms are a cornerstone of modern first-order optimization, decomposing complex problems into simpler subproblems solved via proximal operators. However, most functions lack closed-form proximal operators, which has long restricted these methods to a narrow set of problems. Hamilton-Jacobi-based proximal operator (HJ-Prox) is a recent derivative-free Monte Carlo technique based on Hamilton-Jacobi PDE theory, that approximates proximal operators numerically. In this work, we introduce a unified framework for operator splitting via HJ-Prox, which allows for deployment of operator splitting even when functions are not proximable. We prove that replacing exact proximal steps with HJ-Prox in algorithms such as proximal point, proximal gradient descent, Douglas–Rachford splitting, Davis–Yin splitting, and primal–dual hybrid gradient preserves convergence guarantees under mild assumptions. Numerical experiments demonstrate HJ-Prox is competitive and effective on a wide variety of statistical learning tasks.

Backpropagation (BP) remains the dominant training paradigm for deep neural networks, yet its reliance on global gradient propagation fundamentally induces update locking problem, enforcing strong inter-layer dependencies in parameter updates. To address this limitation, we propose Depth-progressive Monotonic Learning (DMoL), a training scheme that assigns layer-wise local belief objectives and incrementally refines them across network depth, enabling unlocked parameter updates. As a result, DMoL supports dynamic modification of network depth during training, adapting to available compute and device resources while maintaining stable optimization. We provide theoretical guarantees that layer-wise local belief objectives improve monotonically with increasing depth and converge exponentially. Empirically, DMoL consistently matches or outperforms BP across diverse tasks, yielding a 4.3\% accuracy gain on CIFAR-100, mitigating over-smoothing in deep graph neural networks (+37.5\% on Cora), and reducing the final loss by over 35\% in diffusion model training, highlighting its robustness and flexibility as an alternative to BP. The code is publicly available at: https://github.com/ychAlbert/DMoL.

Optimization-based adversarial suffixes can jailbreak aligned large language models (LLMs) while remaining fluent, weakening static and windowed perplexity-based detectors. We cast adversarial suffix detection as an *online change-point detection* problem over the token-level next-token entropy stream. Using the LLM system prompt to estimate a robust baseline, we standardize user-token entropies and apply a one-sided CUSUM statistic. The resulting detector, *CPD Online* (CPD), is model-agnostic, training-free, runs online, and localizes the adversarial suffix onset. On a benchmark of 1,012 optimization-based suffix attacks (GCG, AutoDAN, AdvPrompter, BEAST, AutoDAN-HGA) and 1,012 perplexity-controlled benign prompts, CPD improves F1 over the strongest windowed-perplexity baseline on all six open-weight chat models (LLaMA-2-7B/13B, Vicuna-7B/13B, Qwen2.5-7B/14B). On LLaMA-2-7B at the canonical CUSUM setting ($k=0$), CPD reaches AUROC $0.88$ and F1 $0.82$. Beyond prompt-level detection, CPD concentrates 79.6% of its triggers inside the adversarial suffix, versus 17–46% for windowed perplexity. Finally, when used as a lightweight gate for LLaMA Guard, CPD reduces guard calls by 17–22% on a high-volume, benign-dominated deployment while preserving guard-level detection quality.

This paper approaches the fundamental challenge of accelerating the inherently autoregressive nature of gradient descent (GD) like SGD and Adam through a dynamic system perspective. Specifically, we introduce a unified framework that recasts the autoregressive GD process as solving a system of triangular nonlinear equations (TNEs), thereby enabling \textit{step-parallel} training, where gradients for different GD steps are computed concurrently without sequential dependencies. Within this generic framework, we establish that: (1) the TNE system admits a unique solution corresponding precisely to the autoregressive GD iterative trajectory; (2) solving the TNEs system guarantees convergence to the GD iterative trajectory in at most the equal iterations. Building on these insights, we present \textit{PASO}, the first step-parallel optimizer for accelerating a broad class of GD-based optimizers like SGD and Adam. Extensive experiments (\textit{e.g.}, Llama-3.2-1B and diffusion model) validate that PASO achieves up to \textbf{21}$\times$ reduction in GD steps and \textbf{4.5}$\times$ speedup in wall-clock time, with no model quality loss. Source code is available at: \url{https://github.com/Jianrong-Lu/PASO.git}.

Temporal difference (TD) learning is a fundamental algorithm for estimating value functions in reinforcement learning. Recent finite-time analyses of TD with linear function approximation quantify its theoretical convergence rate. However, they often require setting the algorithm parameters using problem-dependent quantities that are difficult to estimate in practice --- such as the minimum eigenvalue of the feature covariance ($\omega$) or the mixing time of the underlying Markov chain ($\tau_\text{mix}$). In addition, some analyses rely on nonstandard and impractical modifications, exacerbating the gap between theory and practice. To address these limitations, we use an exponential step-size schedule with the standard TD(0) algorithm. We analyze the resulting method under two sampling regimes: independent and identically distributed (i.i.d.) sampling from the stationary distribution, and the more practical Markovian sampling along a single trajectory. In the i.i.d. setting, the proposed algorithm does not require the knowledge of problem-dependent quantities such as $\omega$, and attains the optimal bias-variance trade-off for the last iterate. In the Markovian setting, we propose a regularized TD(0) algorithm with an exponential step-size schedule. The resulting algorithm achieves a comparable convergence rate to prior works, without requiring projections, iterate averaging, or knowledge of $\tau_\text{mix}$ or $\omega$.

Pipeline parallelism is essential for large-scale model training, but existing asynchronous approaches often degrade convergence due to parameter mismatch between forward and backward passes. We propose Asynchronous Multi-Directional Pipeline parallelism (AMDP) to mitigate this issue while sustaining high utilization. AMDP limits the first stage of each pipeline to process at most two minibatches before backpropagation, bounding the number of parameter updates between forward and backward passes. To alleviate the resulting pipeline bubbles, AMDP launches multiple concurrent pipelines and adapts their number according to pipeline depth. In addition, AMDP accumulates gradients across minibatches and applies them in a single update, ensuring that only a bounded number of minibatches experience parameter mismatch, limited to within one optimization step. Experiments on GPT- and BERT-style models demonstrate that AMDP significantly accelerates training while preserving convergence.

Decentralized SGD is a fundamental algorithm in decentralized learning, although the influence of an underlying network topology on its convergence behavior is not yet fully understood. Existing convergence analyses have shown that topologies with a small spectral gap significantly deteriorate the convergence rate of Decentralized SGD in both homogeneous and heterogeneous cases. However, many prior papers have reported that indeed the choice of the topology has a significant experimental impact in the heterogeneous case, but has little experimental impact on training behavior in the homogeneous case. In this paper, we present a tighter convergence analysis of Decentralized SGD, offering a more precise understanding of how topologies affect the convergence rate than the prior analysis. Specifically, unlike existing convergence analyses that used only the spectral gap as a property of the topology, our novel analysis shows that all eigenvalues of the mixing matrix affect the convergence rate. Throughout the experiments, we carefully evaluated the convergence behavior of Decentralized SGD and demonstrated that our novel convergence analysis can more accurately describe the effect of topology on the convergence rate.

We investigate the role of network architecture in shaping the inductive biases of modern score-based generative models. To this end, we introduce the Score Anisotropy Directions (SADs), architecture-dependent directions that reveal how different networks preferentially capture data structure. Our analysis suggests that SADs form adaptive bases aligned with the architecture's output geometry, providing a principled way to predict generalization ability in score models prior to training. Through both synthetic data and standard image benchmarks, we demonstrate that SADs reliably capture fine-grained model behavior and correlate with downstream performance, as measured by Wasserstein metrics. Our work offers a new lens for explaining and predicting directional biases of generative models.

Distributional learning problems optimize discrepancies between probability measures, including optimal transport or Sinkhorn divergence, yet are typically optimized using Euclidean first-order methods in parameter space. We show this mismatch is structural rather than algorithmic. We introduce geometry-misalignment, a local condition number that measures distortion between Euclidean geometry and the intrinsic geometry induced by a distributional objective. For a broad class of problems, we establish lower bounds demonstrating that Euclidean first-order methods incur an unavoidable convergence slowdown proportional to misalignment, even under intrinsic strong convexity and smoothness. We further prove geometry-aware preconditioned methods attain matching upper bounds independent of misalignment, yielding a sharp separation between Euclidean optimization and geometry-aware optimization. Beyond convergence rates, we show geometry-misalignment induces an optimization-dependent excess risk term under finite budgets, directly linking optimization geometry with statistical efficiency. We develop a geometry-calibrated optimization framework that estimates misalignment and selectively activates geometry-aware updates when necessary. Experiments on distribution matching for domain adaptation validate the theory, with improvements concentrated in high-misalignment regimes and negligible overhead.

We introduce a recursive AlphaZero style Monte--Carlo tree search algorithm, "RMCTS". It first generates the search tree using prior policies, and then recursively re-estimates action values by using the regularized optimal posterior policies from ``Monte--Carlo tree search as regularized policy optimization'' (Grill et al., 2020) at each node of the search tree, starting from the leaves and working back up to the root. We find that RMCTS matches or exceeds the quality of AlphaZero's MCTS-UCB in a tiny fraction of the time.

In this paper, we present the convergence analysis of the proximal Alternating Direction Method of Multipliers (ADMM) for problems with block anti-upper triangular constraints. While the linear constraints can be treated separately, most analyses of ADMM and its variants predominantly regard the linear constraints as one. Hence, it relies on assumptions related to the entire constraint matrix, such as the full column rank. However, some problems with block anti-upper triangular constraints that can be solved by ADMM do not satisfy these assumptions. To fill this gap, a new assumption is proposed and used to guarantee the global convergence of the proximal ADMM for nonconvex problems. In the strongly convex setting, we also prove the global convergence of the proximal ADMM and establish the linear convergence under four different scenarios. This work extends the theoretical understanding of the multi-block ADMM to more general cases with block anti-upper triangular constraints.

Task-vector–based model merging enables low-cost, training-free multi-task learning for large language models, but suffers from severe performance degradation due to task conflict. Prior mitigation strategies largely rely on validation data for costly hyperparameter tuning, limiting both interpretability and practicality. We therefore propose OPIC, an evolutionary optimization–based model merging framework. Our preliminary experiments reveal that the degradation of In-Context Learning (ICL) capabilities is a primary driver of task conflict. Motivated by this insight, we formulate model merging as an optimization problem with ICL preservation as the objective. OPIC introduces a hierarchical refinement operators and optimizes it using self-generated data, effectively eliminating the reliance on external validation sets. Experimental results demonstrate that OPIC achieves an average performance retention of 80.73%, outperforming SOTA methods and improving by up to 11.1% over recent validation-free approaches. In addition, OPIC is compatible with existing merging pipelines, offering a new alternative solution for deploying without validation dependencies. Code is available at: https://anonymous.4open.science/r/OPIC-CFFE.

In automated machine learning, scientific discovery, and other applications of Bayesian optimization, deciding when to stop evaluating expensive black-box functions in a cost-aware manner is an important but underexplored practical consideration. A natural performance metric for this purpose is the *cost-adjusted simple regret*, which explicitly captures the trade-off between solution quality and cumulative evaluation cost. Existing stopping rules for Bayesian optimization are either heuristic, or are theoretically grounded but designed to optimize simple regret without accounting for evaluation costs; as a result, they provide no guarantees against unnecessary evaluations when costs are high. We propose a *principled cost-aware stopping rule* for Bayesian optimization that adapts to varying evaluation costs without heuristic tuning. Our rule is grounded in a theoretical connection to state-of-the-art cost-aware acquisition functions, namely the Pandora's Box Gittins Index (PBGI) and log expected improvement per cost (LogEIPC). When paired with either acquisition function, we prove that the resulting policy satisfies a theoretical guarantee bounding the expected cost-adjusted simple regret. Across synthetic tasks and empirical benchmarks including hyperparameter optimization and neural architecture size search, pairing our stopping rule with PBGI or LogEIPC usually matches or outperforms other acquisition-function--stopping-rule pairs in terms of cost-adjusted simple regret.