ICML 2026 Papers

Synchronization over the special Euclidean group $\mathrm{SE}(3)$ aims to recover absolute poses from noisy pairwise relative transformations and is a core primitive in robotics and 3D vision. Standard approaches often require multi-step heuristic procedures to recover valid poses, which are difficult to analyze and typically lack theoretical guarantees. This paper adopts a dual quaternion representation and formulates $\mathrm{SE}(3)$ synchronization directly over the unit dual quaternion. A two-stage algorithm is developed: A spectral initializer computed via the power method on a Hermitian dual quaternion measurement matrix, followed by a dual quaternion generalized power method (DQGPM) that enforces feasibility through per-iteration projection. The estimation error bounds are established for spectral estimators, and DQGPM is shown to admits a finite-iteration error bound and achieves linear error contraction up to an explicit noise-dependent threshold. Experiments on synthetic benchmarks and real-world multi-scan point-set registration demonstrate that the proposed pipeline improves both accuracy and efficiency over representative matrix-based methods.

Mixed-integer optimization solvers often find optimal solutions early in the search, yet spend the majority of computation time proving optimality. We exploit this by learning when to terminate solvers early on distributions of similar problem instances. Our method trains a neural network to estimate the true optimality gap from the solver state, then uses conformal prediction to calibrate a stopping threshold with rigorous probabilistic guarantees on solution quality. On six problem families from the Distributional MIPLIB library, our method reduces solve time by over 60% while guaranteeing 0.1%-optimal solutions with 95% probability for new instances drawn from the same distribution.

Process reward models (PRMs) allow for fine-grained credit assignment in reinforcement learning (RL), and seemingly contrast with outcome reward models (ORMs), which assign a single reward to an entire trajectory. However, we provide theoretical proof in this work that the Group Relative Policy Optimization (GRPO) RL algorithm equipped with an ORM is in fact equivalent to a PRM-aware RL objective equipped with a non-trivial, Monte-Carlo-based PRM (given mild assumptions). Leveraging the framework of GRPO-as-a-PRM, we identify a flaw in the GRPO objective that interacts with imbalanced process steps and rewards to hinder both exploration and exploitation (under different conditions). We propose a simple modification to the algorithm to mitigate this defect ($\lambda$-GRPO), and show that LLMs tuned with $\lambda$-GRPO outperform LLMs tuned with standard GRPO on downstream reasoning tasks$\textemdash$and reach peak performance more rapidly. These results show that we can leverage the hidden, built-in PRM structure within the vanilla GRPO algorithm to boost model performance without employing an explicit PRM, and with a negligible impact on training time and cost.

Accurate Zeroth-Order (ZO) Hessian estimation is a cornerstone of derivative-free methods, essential for tasks such as bilevel optimization, Bayesian inference, and uncertainty quantification. However, obtaining a complete suite of low-variance estimators for the Hessian and its inverse in high-dimensional settings remains a significant challenge. To address this, we propose a unified framework that reinterprets ZO Hessian approximation through the lens of single-step Policy Optimization (PO). This perspective establishes a theoretical equivalence between general ZO Hessian estimators and the Hessian of a smoothed PO objective, unifying distinct classical randomized estimators as specific instances of baseline selection. Building on this foundation, we introduce ZoVH, a comprehensive suite of variance-reduced estimators for the full Hessian matrix, its regularized inverse, and the bias-corrected inverse Hessian-gradient product. ZoVH leverages two key techniques: (1) a unique optimal baseline derived to provably minimize variance, and (2) a query reuse strategy that incorporates historical function queries to enhance sample efficiency without inflating costs. Our rigorous theoretical analysis confirms the unbiasedness of the Hessian estimator, validates the variance optimality of our baseline, provides error bounds for the entire ZoVH suite, and establishes convergence guarantees for the resulting curvature-aware ZO algorithm. Extensive empirical results validate our theoretical findings, demonstrating that ZoVH achieves superior estimation accuracy and convergence performance in real-world applications.

Dueling optimization considers optimizing an objective with access to only a comparison oracle of the objective function. It finds important applications in emerging fields such as recommendation systems and robotics. Existing works on dueling optimization mainly focused on unconstrained problems in the Euclidean space. In this work, we study dueling optimization over Riemannian manifolds, which covers important applications that cannot be solved by existing dueling optimization algorithms. In particular, we propose a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establish its iteration complexity when the objective function is geodesically $L$-smooth or geodesically (strongly) convex. We also propose a projection-free algorithm, named Riemannian Dueling Frank–Wolfe (RDFW) method, to deal with the situation where projection is prohibited. We establish the iteration and oracle complexities for RDFW. We illustrate the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real applications.

Sign-based and LMO-inspired optimizers have recently attracted substantial attention in deep learning due to their strong performance and low memory footprint. However, their fixed-magnitude updates can hurt terminal convergence: they decouple update mechanisms from gradient magnitudes and fail to account for parameter heterogeneity, often leading to oscillation rather than convergence. We propose SoftSignum, a smooth relaxation of sign-based optimization that replaces the hard sign map with a temperature-controlled soft-sign transformation, enabling a parameter-wise transition from sign-like updates to magnitude-sensitive SGD-like steps. We complement it with an adaptive quantile-based temperature schedule and extend the same principle to matrix-valued optimizers, obtaining SoftMuon. We also develop a generalized geometry-relaxation framework based on strongly convex regularizers and Fenchel conjugates, proving convergence in stochastic non-convex setting. Experiments on diverse deep learning tasks, including LLM pretraining, show that SoftSignum and SoftMuon consistently improve over their hard sign-based counterparts and standard AdamW.

We study smooth nonconvex optimization using only noisy pairwise comparisons, without access to gradients or function values. We propose Noisy-Comparison Random Search (NCRS), a simple direct-search method that samples random directions and performs accept/reject updates from comparison feedback. Under a low-dimensional active-subspace structure, NCRS adapts to the intrinsic dimension $k\le d$ rather than the ambient dimension $d$. For a uniform-margin comparison oracle with advantage $p$, NCRS achieves $\epsilon$-first-order stationarity with comparison complexity $\mathcal{O}(k/(p^2\epsilon^2))$. We also introduce a gap-dependent confidence model, where comparison reliability decreases as the objective-value gap between the two candidates becomes small, and analyze a confidence-weighted voting variant of NCRS. For this oracle, the method achieves $\epsilon$-first-order stationarity with total comparison complexity $\mathcal{O}(k^2/\epsilon^4)$. These results provide intrinsic-dimension convergence guarantees for noisy comparison-based random search in smooth nonconvex optimization.

Large models recently are widely applied in machine learning, so efficient training of large models has received widespread attention. More recently, the useful Muon optimizer is specifically designed for matrix-structured parameters of large models. Although some works have begun to study the Muon optimizer, the existing Muon and its variants still suffer from high sample complexity or high memory for large models. To fill this gap, we propose a light and fast Muon (LiMuon) optimizer for training large models, which builds on the momentum-based variance reduced technique and randomized Singular Value Decomposition (SVD). In particular, our LiMuon simultaneously has a lower memory and lower sample complexity than the Muon and its variants. Moreover, we prove that our LiMuon with lower memory has a lower sample complexity of $O(\epsilon^{-3})$ for finding an $\epsilon$-stationary solution of non-convex stochastic optimization under the generalized smoothness condition. To further narrow practice and theory gap, we also prove that our LiMuon with Newton-Schulz steps has a lower sample complexity than the Muon with Newton-Schulz steps. Numerical experimental results on training Mamba-130M, Qwen2.5-0.5B and ViT models demonstrate effectiveness of our LiMuon.

Speculative decoding (SD) accelerates large language model inference by using a smaller draft model to propose draft tokens that are subsequently verified by a larger target model. However, the performance of standard SD is often limited by the strictly sequential execution of these drafting and verification stages. To address this, this paper proposes MineDraft, a batch parallel speculative decoding (PSD) framework designed to effectively hide drafting latency by overlapping it with verification. Our theoretical analysis shows that PSD is substantially more efficient than standard SD. MineDraft realizes the PSD through a novel batch-parallel design that maintains two batches of requests, overlapping drafting for one batch with verification for the other. Our experimental results show significant improvements of MineDraft in both throughput (up to 75%) and end-to-end latency (up to 39%) over standard SD. Furthermore, we have implemented MineDraft as a plugin for vLLM, demonstrating its practicality for production-ready inference systems. The code is publicly available in the [MineDraft GitHub repository](https://github.com/electron-shaders/MineDraft).

We analyze two classical algorithms for solving additively composite convex optimization problems where the objective is the sum of a smooth term and a nonsmooth regularizer. The first algorithm is the proximal stochastic gradient method for a single regularizer; the second is the randomized incremental proximal method, which uses the proximal operator of a randomly selected function when the regularizer is given as the sum of many nonsmooth functions. We focus on relaxing the bounded variance assumption that is common, yet stringent, for getting last iterate convergence rates. We prove the $\widetilde{O}(1/\sqrt{T})$ rate of convergence for the last iterate of both algorithms under componentwise convexity and smoothness, which is optimal up to log terms. Our results apply directly to graph-guided regularizers that arise in multi-task and federated learning, where the regularizer decomposes as a sum over edges of a collaboration graph.

Standard Set Representation Learning methods typically excel on curated data but often overlook the challenge of Inference-time Element Corruption. This refers to scenarios where deployed models encounter element-level degradations, such as outliers or missing components, that may distort set representation and degrade performance. We propose SW-DRSO, a distributionally robust optimization framework tailored for sets. Rather than minimizing loss solely on observed training data, SW-DRSO optimizes a tractable surrogate of the worst-case expected loss over a family of plausible inference-time variations. We introduce a barycentric adversary that approximates the intractable search over corrupted sets by a differentiable training-time optimization over simplex weights. Extensive experiments across four tasks demonstrate that SW-DRSO effectively enhances robustness against corruption while maintaining high overall performance.

Modern deep networks optimize features via cross-entropy without explicit margin maximization, while classical Support Vector Machines maximize the margin on a fixed feature space. Hybrid Deep-SVM approaches typically treat the deep feature extractor as a static input to an SVM head, so SVM geometry never flows back to shape representation learning. We propose the **Differentiable Deep Support Vector Machine (DDSVM)**, an iterative framework that alternates between re-fitting a linear SVM on $L_2$-normalized features and updating the backbone to pull each feature along the SVM-defined margin-widening direction, treating the decision boundary as a dynamic teacher for representation learning. Under spherical normalization the resulting feature gradient lies in the tangent space and optimization focuses exclusively on angular separability. DDSVM reduces top-1 error by **5.6--8.3\%** on image classification benchmarks under data scarcity.

High-fidelity vehicle aerodynamics analysis is bottlenecked by costly CFD simulations. Neural surrogates accelerate prediction but lack inverse design capabilities, while existing generative optimization methods suffer from unstable convergence and frequent engineering constraint violations. We present UniDrag, a unified framework bridging multi-field aerodynamic prediction with robust differentiable shape optimization. Given a vehicle geometry, UniDrag predicts surface pressure, volume flow fields, drag coefficient $C_d$, and a streamwise build-up profile localizing drag contributions. Our architecture combines enhanced Physics-Sliced Attention (ePSA), Gated Expert Routing, and Modality-Protected Learning to prevent negative transfer across output modalities. At deployment, the frozen surrogate enables gradient-based optimization via Free-Form Deformation with engineering constraints. We introduce Expectation-over-Transformation to prevent adversarial exploitation of surrogate fragility. We curate a large-scale dataset of 15,000 vehicle geometries spanning four body types with GPU-accelerated LBM simulations. On this benchmark, UniDrag achieves $C_d$ prediction $R^2$ of 0.937 (+7.6\% over baselines) and 13.7\% mean CFD-verified drag reduction with 100\% success rate and only 21.3~mm average displacement (0.46\% vehicle length).

The quantum approximate optimization algorithm (QAOA) holds promise for combinatorial optimization but is constrained by limited qubits. While divide-and-conquer frameworks like QAOA$^2$ address scalability by partitioning graphs into subgraphs, existing methods suffer from two fundamental limitations: i) misalignment between heuristic partitioning metrics and quantum optimization goals, and ii) topology-blind parameter initialization that leads to optimization cold starts. To bridge these gaps, we propose **Neural QAOA$^2$**, an end-to-end differentiable framework that jointly generates graph partitions and initial parameters. By integrating a generative evaluative network (GEN), our method utilizes a differentiable quantum evaluator as a high-fidelity performance surrogate to provide direct gradient guidance, enabling the joint generator to learn the intrinsic mapping from graph topology to high-quality partition and parameter configurations. Extensive experiments on 183 QUBO, Ising, and MaxCut instances (21 to 1000 variables) demonstrate that our gradient-driven approach broadly outperforms heuristic baselines, ranking first on 101 instances. It exhibits zero-shot generalization across out-of-distribution graph topologies and scales.

Low-Rank Adaptation (LoRA) is the most widely adopted method for fine-tuning large language models. Notably, LoRA is inherently overparameterized: multiple pairs of low-rank factors can yield the same adapted weight matrix. We show—both theoretically and empirically—that these pairs exhibit significantly different condition numbers. As a result, converging to different loss minimizers directly impacts the convergence rate of LoRA. Building on this observation, we introduce Balanced Low-Rank Adaptation (BaLoRA), a variant of LoRA that projects iterates onto a balanced manifold. This manifold improves the conditioning of the loss landscape while preserving the adapted matrix. The projection step is computationally lightweight and integrates seamlessly into existing fine-tuning pipelines. Empirically, BaLoRA converges faster than standard LoRA and achieves superior performance across a range of fine-tuning tasks.

Federated learning enables decentralized clients to collaboratively train models without sharing local data. However, heterogeneous client distributions often induce client drift and hinder convergence. This paper proposes FedCDWA, a decoupled hierarchical federated distillation framework. FedCDWA decouples client-side personalized distillation from server-side mutual distillation to mitigate distillation-induced optimization conflicts. It further adopts Hierarchical Wasserstein Aggregation to aggregate prototypes without restrictive parametric assumptions while preserving intra-class structure and inter-class geometry. To achieve finer-grained feature alignment, Prototype–Variance Dual Alignment matches feature means and variances in the feature space. We prove convergence guarantees for FedCDWA. Experiments on three datasets demonstrate that FedCDWA consistently improves both global and personalized accuracy across heterogeneity levels, with smaller performance degradation under more severe heterogeneity.

The stochastic Polyak step size (SPS) has proven to be a promising choice for stochastic gradient descent (SGD), delivering competitive performance relative to state-of-the-art methods on smooth convex and non-convex optimization problems, including deep neural network training. However, extensions of this approach to non-smooth settings remain in their early stages, often relying on interpolation assumptions or requiring knowledge of the optimal solution. In this work, we propose a novel SPS variant, Safeguarded SPS (SPS$_{safe}$), for the stochastic subgradient method, and provide rigorous convergence guarantees for non-smooth convex optimization with no need for strong assumptions. We further incorporate momentum into the update rule, yielding equally tight theoretical results. Comprehensive experiments on convex benchmarks and deep neural networks corroborate our theory: the proposed step size achieves competitive performance to existing adaptive baselines and exhibits stable behavior across a wide range of problem settings. Finally, in the context of deep neural network training, the gradient norms under our step size do not collapse to (near) zero, indicating robustness to vanishing gradients.

Deep learning optimization relies heavily on the assumption of smooth loss landscapes, a condition systematically violated by modern architectures due to non-smooth components like ReLU activations and quantization operators. In such non-smooth regimes, adaptive optimizers such as Adam suffer from **gradient chattering**—violent oscillations caused by conflicting signals within the Clarke subdifferential—leading to poor convergence and suboptimal generalization. To address this, we introduce **Singularity-aware Adam (S-Adam)**, a novel optimizer that stabilizes training by dynamically modulating step sizes based on local geometric instability. Our key contribution is the **Local Geometric Instability (LGI)** metric, a computationally efficient estimator of the Clarke subdifferential diameter derived from the variance of randomized directional derivatives. S-Adam incorporates an adaptive damping mechanism $\exp(-\lambda \rho_t)$ that decelerates updates in high-instability regions while preserving fast convergence in smooth basins. We provide a rigorous convergence analysis using differential inclusions, proving that S-Adam converges almost surely to $(\delta, \epsilon)$-Clarke stationary points at the optimal $\mathcal{O}(1/\sqrt{T})$ rate. Empirical evaluations on Quantization-Aware Training (QAT) and high-noise small-batch learning demonstrate that S-Adam consistently outperforms AdamW and Prox-SGD, achieving accuracy gains of up to +6% on CIFAR-100 and +3% on TinyImageNet while effectively mitigating gradient oscillations.

Meta Black-Box Optimization (MetaBBO) has emerged as a promising paradigm by employing meta learning to automatically optimize the configurations of low-level black-box optimizers. Despite its potential, the generalization of MetaBBO remains significantly constrained when facing unseen, complex objective landscapes. We identify that this bottleneck stems from a restricted performance upper bound inherent in current training mechanisms: by learning from scratch in a self-supervised or unsupervised manner, low-level learnable optimizers are never exposed to advanced, high-quality optimization behaviors, forcing them to converge on suboptimal strategies. In this paper, we propose \texttt{MetaDistill}, a general MetaBBO training framework designed to lift the strategy ceiling through pretraining and test-time fine-tuning. In the pretraining stage, we represent high-quality strategies from classical algorithms as expert optimization trajectories and utilize them for diversity-preserving distillation, enabling the learnable optimizer to internalize advanced optimization behaviors. In the optional fine-tuning stage, we perform self-supervised fine-tuning as a warm-start procedure to further refine the distilled knowledge on unseen tasks. We evaluate our \texttt{MetaDistill} framework on the BBOB test suite and three control tasks. The results demonstrate that \texttt{MetaDistill} significantly improves the generalization ability of various learnable optimizers compared to their original training paradigms. Our code is available at \href{https://github.com/Roki-Xing/MetaDistill}{here}.