NeurIPS 2026 论文列表

Quantizing Mixture-of-Experts (MoE) models is difficult for two coupled reasons: expert quantization error perturbs expert outputs, and sparse routing amplifies small perturbations into discrete expert selection changes. This coupling produces progressive output drift that is not captured by dense model quantization analyses. To address this gap, we propose a unified quantization framework that jointly handles both sources of error. First, we formulate compression-induced routing inconsistency as a form of training-inference router mismatch and use this connection to motivate limited routing replay, which replays BF16 routes during quantization-aware fine-tuning (QAT), freezes router parameters, and updates only expert-side adapter parameters. We then provide a standard convergence guarantee for the resulting fixed-route QAT objective, together with a routing-consistency bound under explicit router-margin conditions. Second, for expert restoration under a fixed rank budget, we propose load-aware low-rank compensation that assigns larger compensation rank to heavily loaded experts. Under this allocation rule, we prove monotone strict decrease of weighted compression error and characterize optimal rank assignment by marginal load-weighted spectral gains. Finally, experiments on the MXFP4 quantization of Mixtral, DeepSeek, and Qwen3 MoE models show that our method improves perplexity and downstream accuracy over prior methods. We also report system efficiency on Qwen3-30B, where uniform W4A8 serving accelerates decoding and our adapter path preserves most of that acceleration.

Latent space Bayesian optimization circumvents the curse of dimensionality over structured inputs by performing Gaussian process surrogate-guided search in a continuous latent space learned by deep generative models. However, existing latent space Bayesian optimization methods overlook both the inherent stochasticity of latent space representations and the geometric properties of the manifold induced by VAE, leading to suboptimal optimization performance. In this work, we propose a marginalized kernel that integrates out the latent variables through the VAE inference posterior, yielding a closed-form kernel on the observation space that naturally accounts for encoding uncertainty. We adopt a quadratic polynomial kernel as the base kernel in the latent space, which captures discrepancies in the first two moments of the latent posteriors while avoiding the bandwidth sensitivity inherent to the RBF kernel. Furthermore, inspired by the Riemannian interpretation of VAEs, we incorporate a geometry-aware sampling scheme that leverages the metric structure revealed by the learned posterior covariances to guide candidate acquisition toward high-information regions of the latent space. Empirical evaluations on molecular design and robot design tasks demonstrate that our method outperforms existing state-of-the-art baselines across the majority of tasks.

We introduce a unified framework that reinterprets first order optimization on a smooth parameter space as a closed loop controlled dynamical system on a Riemannian manifold. Within this framework, optimizers are specified by a quadruple $(M, g, \Phi, \eta)$ consisting of a Riemannian metric, an internal state transition map, a direction field generator, and a lifting parameter. Classical and modern methods including SGD, Adam, AdamW, Lion and Shampoo are recovered as specific instantiations of this quadruple. The central structural object of the framework is a time indexed family of target submanifolds in the extended state space of parameters, velocities, and internal memory, which we call the Normally Attracting Invariant Manifold (NAIM) family. These submanifolds organize training dynamics into two timescales: a fast contraction of the velocity state onto the current target submanifold, followed by a slow descent along it. Using Riemannian backstepping, we construct a strict Lyapunov function on the extended state space and prove Uniform Ultimate Boundedness (UUB) of the optimization trajectory under a descent alignment assumption and bounded stochastic disturbances. Under the additional Polyak-Lojasiewicz (PL) condition, the Lyapunov function converges linearly to a bounded neighborhood of the optimum, defined as the drift of the target direction field between consecutive iterates after parallel transport. To validate the framework's generativity, we derive three concrete instantiations of the quadruple, each exercising a distinct design axis, and show on geometric diagnostics, image classification benchmarks, and language modeling that their empirical behavior matches the theoretical predictions. The framework casts optimizer design as controller synthesis, complementing heuristic perspectives by providing a constructive template with provable guarantees.

This paper studies bilevel optimization on Riemannian manifolds where the upper-level objective is nonconvex and the lower-level problem satisfies a Riemannian Polyak - Łojasiewicz condition rather than geodesic strong convexity. The classical hypergradient formula then breaks down, since the lower-level Hessian may be singular or indefinite away from the minimizer. We address this with an intrinsic regularized tangent-space formulation based on spectral clipping, and develop a Riemannian bilevel algorithm that avoids Hessian-inverse solves and inner-loop differentiation. We establish an $O(1/T)$ rate on the averaged squared Riemannian gradient mapping and $O(\varepsilon^{-1})$ iteration complexity, and extend the method to the stochastic finite-sum setting. Experiments on Stiefel, Grassmannian, and Poincar\'e-ball problems with rank-deficient Hessians (BCI~IV-2a, UCI Superconductivity) and Stiefel-constrained meta-learning on MiniImageNet show that the proposed method is the only one whose linear-system error stays at machine precision and whose Riemannian gradient norm decreases monotonically, while Hessian-inversion and Neumann-series baselines fail to descend; the stochastic complexity exhibits the predicted $\Theta(1/B)$ decay.

Decentralized online convex optimization (D-OCO) is a popular framework for distributed applications with streaming data. To tackle the communication bottleneck, previous studies have investigated D-OCO with compressed communication and proposed several algorithms that are variants of online gradient descent (OGD). However, for D-OCO with exact communication, the best existing algorithms are variants of follow-the-regularized-leader (FTRL). In this paper, for the first time, we propose two FTRL-type algorithms for D-OCO with compressed communication. Compared with OGD-type algorithms, our algorithms are more elegant in both algorithmic design and theoretical analysis. The key insight is that the dual update mechanism of FTRL allows us to make a simple application of the technique for average consensus with communication compression. More specifically, our first algorithm considers the full-information setting, and can match the existing regret bounds. Our second algorithm is designed for the bandit setting, and can significantly improve both the regret bounds and communication costs of existing algorithms.

Scaling laws for large language models depend critically on the optimizer and parameterization. Existing hyperparameter transfer laws are mainly developed for first-order optimizers, and they do not structurally prevent training instability at scale. Recent hypersphere optimization methods constrain weight matrices to a fixed-norm hypersphere, offering a promising alternative for more stable scaling. We introduce HyperP (Hypersphere Parameterization), the first framework for transferring optimal learning rates across model width, depth, training tokens, and Mixture-of-Experts (MoE) granularity under the Frobenius-sphere constraint with the Muon optimizer. We prove that weight decay is a first-order no-op on the Frobenius sphere, show that Depth-$\mu$P remains necessary, and find that the optimal learning rate follows the same data-scaling power law with the ``magic exponent'' 0.32 previously observed for AdamW. A single base learning rate tuned at the smallest scale transfers across all compute budgets under HyperP, yielding $1.58\times$ compute efficiency over a strong Muon baseline at $6\times10^{21}$ FLOPs. Moreover, HyperP delivers transferable stability: all monitored instability indicators, including $Z$-values, output RMS, and activation outliers, remain bounded and non-increasing under training FLOPs scaling. We also propose SqrtGate, an MoE gating mechanism derived from the hypersphere constraint that preserves output RMS across MoE granularities for improved granularity scaling, and show that hypersphere optimization enables substantially larger auxiliary load-balancing weights, yielding both strong performance and good expert balance.

The recent empirical success of non-Euclidean optimizers, such as Muon (Jordan et al., 2024) and Shampoo (Gupta et al., 2018), has transformed the training of Large Language Models (LLMs) by exploiting the geometry of matrix spaces. Despite this progress, the theoretical understanding of stochastic optimization in general normed spaces remains limited, particularly in distributed settings. In this work, we present a novel analysis of Minibatch Stochastic Gradient Descent (SGD) in non-Euclidean geometries. Leveraging the framework of $\\kappa$-regular normed spaces (Juditsky & Nemirovski, 2008) and a light-tail noise model defined directly in the dual norm, we derive high-probability oracle complexity bounds for smooth non-convex optimization. Crucially, our analysis reveals that measuring stochastic variance within the intrinsic non-Euclidean geometry avoids the suboptimal dimension-dependent factors inherent in Euclidean norm equivalence. Building on this framework, we introduce Rennala-NSGD, a semi-asynchronous distributed algorithm, and establish, to the best of our knowledge, the first wall-clock time complexity guarantee for asynchronous non-Euclidean stochastic optimization.

Operations research practitioners typically tackle NP-hard combinatorial problems using large neighborhood search (LNS), a scalable heuristic that iteratively refines a current solution by locally re-optimizing subsets of its variables. In contrast, most existing approaches for integrating combinatorial optimization layers into neural networks still assume access to an exact global solution, which is computationally intractable. We bridge this gap by introducing regularized LNS (RLNS). By regularizing or perturbing local subproblems, we turn the LNS heuristic into an efficient MCMC sampler over the combinatorial set of feasible solutions, with associated Fenchel-Young losses. Under entropic regularization, we prove that RLNS performs exact block Gibbs sampling. Furthermore, adjusting the number of RLNS iterations allows us to interpolate between pseudolikelihood and exact maximum likelihood estimation, for end-to-end learning without global solvers. We demonstrate our approach on $k$-subset selection, generalized assignment, and stochastic vehicle scheduling problems.

Continuous DAG learning formulates combinatorial structure learning as differentiable optimization over weighted adjacency matrices. In these methods, the estimated adjacency matrix depends critically on a sparsity-controlling regularization parameter, whose magnitude governs the recovered graph skeleton. While extensive empirical evidence underscores the centrality of this parameter, existing practice relies on solving a sequence of independent optimization problems over a grid of tuning parameters, which is computationally wasteful, statistically unstable, and provides only a fragmented view of how the adjacency matrix evolves. In this work, we study the regularization path of stationary solutions in continuous DAG learning and propose DAG-flow, an exact path-following framework that compactly encodes the entire family of estimators. We show that for prominent formulations including NOTEARS, GOLEM, and DAGMA, stationary solutions evolve according to a piecewise-smooth matrix-valued dynamical system once the active support is fixed. Our analysis derives the explicit systems governing these branches, identifies the events at which the path changes regime, and clarifies how nonsmooth sparsity and nonconvex acyclicity interact along the path. Unlike grid-based search, DAG-flow exposes the geometry of model variation across regularization levels, which enables the construction of structured candidate graph sets and provides direct insight into edge-level stability. Across synthetic and real benchmarks, our DAG-flow outperforms grid-search baselines at a fraction of the compute and consistently improves structural stability.

Bayesian optimization (BO) is a widely used iterative black-box optimization method that utilizes Gaussian process (GP) surrogate models. In practice, BO is typically terminated after a fixed evaluation budget is exhausted, which can incur unnecessary cost and provides no optimality guarantee on solution quality. Recent research in developing a practical stopping criterion has made empirical progress, yet a theoretically sound stopping criterion remains a work in progress. In this work, we present provably tighter instantaneous regret bounds for GP upper confidence bound (GP-UCB) at any given iteration. Then, we propose stopping criteria for GP-UCB based on this tighter bound that ensures an $\epsilon$-optimal solution with high probability $1-\delta$ upon termination. Numerical experiments are performed to validate and demonstrate the effectiveness and efficiency of our stopping criteria.

We consider the optimization problem of an expensive-to-evaluate black-box function, in which we can obtain noisy function values in parallel. For this problem, parallel Bayesian optimization (PBO) is a promising approach, which aims to optimize with fewer function evaluations by selecting a diverse input set for parallel evaluation. However, existing PBO methods suffer from poor practical performance or lack theoretical guarantees. In this study, we propose a PBO method, called randomized kriging believer (KB), based on a well-known KB heuristic and inheriting the advantages of the original KB: low computational complexity, a simple implementation, versatility across various BO methods, and applicability to asynchronous parallelization. Furthermore, we show that our randomized KB achieves Bayesian expected regret guarantees. We demonstrate the effectiveness of the proposed method through experiments, including those on real-data emulators.

We study stochastic optimization with heavy-tailed gradient noise. We first propose a novel quantum mean estimator for multivariate heavy-tailed random variables that achieves lower query complexity than optimal classical estimators in the low-dimensional regime. We further develop an unbiased quantum mean estimator by applying a generalized multi-level Monte Carlo technique. We prove quantum lower bounds showing that, when the dimension $d$ of the random vector is small and can be viewed as a constant, our quantum estimators are optimal up to logarithmic factors; while for the high-dimensional regime, no quantum speedup is available compared to optimal classical mean estimators. Based on these estimators, we propose a quantum normalized stochastic gradient descent method ($\texttt{QNSGD}$), which finds an $\epsilon$-stationary point of a nonconvex objective using $\tilde{\mathcal{O}}\big(d^{\frac{p}{4(p-1)}}\epsilon^{-\frac{5p-4}{2p-2}}\big)$ queries to the stochastic gradient, where $p\in(1,2]$ is the tail index. For a convex objective function, we propose a quantum projected stochastic gradient descent method ($\texttt{QPSGD}$), which computes an $\epsilon$-optimal solution with query complexity $\tilde{\mathcal{O}}\big(d^{\frac{p}{4(p-1)}}\epsilon^{-\frac{3p-2}{2p-2}}\big)$. Our results improve upon the classical lower bounds $\Omega\big(\epsilon^{-\frac{3p-2}{p-1}}\big)$ for nonconvex problems and $\Omega\big(\epsilon^{-\frac{p}{p-1}}\big)$ for convex problems, demonstrating a quantum speedup when $d$ is relatively small.

The inference efficiency of diffusion large language models (dLLMs) is constrained by two challenges: bidirectional attention precludes efficient KV-cache reuse, while increasing decoding parallelism with static confidence thresholds can compromise generation quality. We observe that both challenges arise from a shared phenomenon: as tokens are decoded, their contextual integration through bidirectional attention causes token representations to drift (evolve) across decoding steps. This insight motivates Polestar, a training-free inference framework that uses token representation drift as a unified signal to jointly address both challenges. Polestar comprises two components: Polestar-Cache, which identifies stale KV-cache positions via drift and performs sparse KV-cache refreshes to enable efficient reuse, and Polestar-Commit, which detects sharp drift events to reliably identify commit-ready tokens. Across mathematics and coding benchmarks on several dLLM families, Polestar sets a new state of the art on the accuracy-throughput Pareto frontier, achieving up to 10.73% accuracy improvement, up to 3.7x higher throughput, and high decoding parallelism of 3.67 tokens per forward pass over existing baselines.

Learning across multiple inherently conflicting objectives requires selecting desirable trade-offs from the Pareto set. In this work, we study preference-guided multi-objective optimization, where a predefined preference function is minimized over the weak Pareto set of multiple nonconvex objectives. Leveraging a merit function that vanishes on the weak Pareto set, we propose a penalty-based min-max-min formulation, termed the M$^3$ problem, which unifies preference optimization and the enforcement of weak Pareto optimality in a single objective. We further develop ParetoM$^3$, a single-loop first-order algorithm for computing stationary solutions of the resulting M$^3$ problem. Under a local Kurdyka-{\L}ojasiewicz assumption, we prove that ParetoM$^3$ finds an $\epsilon$-stationary point of the M$^3$ problem that is also approximately weakly Pareto optimal for the original multi-objective problem when the penalty parameter is sufficiently large, with explicit non-asymptotic convergence rates. We also show that this stationarity notion recovers existing optimality criteria, including approximate preference stationarity and approximate Karush-Kuhn-Tucker conditions, under the same assumptions used in prior works. Experiments on a synthetic benchmark, image classification, regression, and math reasoning demonstrate that ParetoM$^3$ reliably attains preferred weak Pareto optimal solutions while achieving strong practical performance.

Direct pairwise AUC-surrogate training is attractive because it trains toward the ranking metric used at test time, especially under class imbalance. This creates a direct alignment between the training objective and the evaluation metric, but such metric alignment alone does not guarantee that stochastic training remains corrective. We study this gap through Corrective Power: whether pairwise mini-batch updates produce stronger and more reliable improvements for more severe positive-negative ranking errors. Our main observation is that direct pairwise training can enter regions where severe errors remain common but no longer produce useful corrective motion. In such regions, even the mini-batch noise that would normally help SGD move along the steepest severe-error correction direction can become much weaker. In this paper, we seek a theoretical explanation for this by exploring the degenerate property of the U-statistics formed from the pairwise gradients therein. Our results provide a mechanistic explanation for several empirical patterns: (a) squared-hinge loss is often an effective AUC surrogate; (b) cross-entropy warm-up can help models that enter the AUC phase from a poor starting region; and (c) parameter-efficient fine-tuning from a pretrained model can reduce that need by starting closer to a favorable ranking geometry. Experiments on 8 CV tasks and 8 NLP tasks, using ResNet50, DenseNet121, DistilBERT-base, and Qwen2.5-1.5B-Instruct, support these predictions.

We introduce Post-Optimization Model Edit (POME), a new algorithm that enhances the performance of fine-tuned large language models using only their pretrained and fine-tuned checkpoints, without requiring extra data or further optimization. The core idea is to apply a muon-style projection to $\Delta W$, the difference between the fine-tuned and pretrained weights. This projection uses truncated singular value decomposition (SVD) to equalize the influence of dominant update directions and prune small singular values, which often represent noise. As a simple post-processing step, POME is completely decoupled from the training pipeline. It requires zero modifications and imposes no overhead, making it universally compatible with any optimizer or distributed framework. POME delivers consistent gains, boosting average performance by +2.5\% on GSM8K and +1.0\% on code generation. Its broad applicability—from 7B foundation models to 72B RLHF-instructed models—establishes it as a practical, zero-cost enhancement for any fine-tuning pipeline.

We propose PACE, Partial-state Amortized Constraint Editing, which treats task-native feasible partial states as the common semantic object for neural combinatorial optimization, unifying edit learning, constraint-preserving editing, anytime closure, and test-time scaling across routing and graph combinatorial optimization. Existing neural combinatorial optimization methods expose complementary strengths: constructive and adaptive expansion solvers keep meaningful partial solutions but couple them to serial or method-specific growth, while global prediction, diffusion, masked reconstruction, and complete-solution refinement provide scalable guidance yet often refine heatmaps, noisy solutions, reconstruction targets, or perturbed full outputs. These semantics make learning, search, and test-time scaling act on the same state object. PACE learns amortized constraint editing by ranking candidate edits conditioned on the current feasible partial state; a constraint-preserving transition $\Gamma_I$ commits only edits that preserve legal continuation and extendability; and task-native closure $\mathrm{Comp}_I$ maps any extendable intermediate feasible partial state to a task-terminal feasible output. Budgeted refinement then spends extra inference-time compute through deeper edit steps, additional refinement rounds, or broader candidate sets along the same trajectory. We instantiate these semantics across TSP, ATSP, CVRP, MIS, MVC, MCL, and MCut, covering edge-oriented routing and node-oriented graph combinatorial optimization. Theoretical guarantees are structural rather than optimality claims, covering task-wise soundness, constructive completion, state-space closure in the extendable subset, terminal feasible completion at any editing depth, and elite-pool nondegradation. Empirically, PACE is competitive and budget-controllable, including ATSP-500 at 0.247\% Drop, MIS RB-[800-1200] at 2.00\%, and MVC at 0.07\%.

Single-loop minimax methods are appealing for modern machine learning, but standard stochastic gradient updates can be unstable and oscillatory under tightly coupled primal--dual dynamics, making performance notoriously sensitive to stepsizes. To fundamentally improve stability, we develop an orthogonal update framework tailored to minimax optimization, which applies updates along orthogonalized directions to reduce update anisotropy. However, orthogonalization reshapes the optimization geometry and makes coordinating primal--dual stepsizes even more delicate. To overcome this challenge, we develop \textbf{AdaSGDA}, which equips orthogonal updates with an adaptive stepsize rule driven by accumulated gradient norms. This mechanism automatically balances coupled primal--dual progress under orthogonalization, yielding an effectively parameter-free method for nonconvex-strongly-concave minimax optimization. We prove that AdaSGDA avoids problem-dependent tuning while attaining the state-of-the-art $\mathcal{O}\left(T^{-1/4}\right)$ convergence rate. We further propose \textbf{AdaSGDA-VR}, which incorporates a carefully designed variance reduction scheme and achieves the faster $\mathcal{O}\left(T^{-1/3}\right)$ rate. Extensive experiments demonstrate that our methods deliver competitive performance compared to existing minimax optimization methods.

Contrastive embedding models trained with scale-invariant losses are typically paired with distance metrics like cosine similarity, effectively ignoring embedding magnitudes. However, surprisingly, empirical studies reveal that despite this, these "discarded" norms seem to correlate with semantic properties such as concept specificity, token frequency, and human uncertainty. In this work, we provide a formal theoretical framework explaining this phenomenon. By analyzing the optimization dynamics, we derive an analytic formula demonstrating that embedding length naturally encodes this information as a byproduct of the training process. We also show how this gives rise to signals that can serve as "free" calibration tools in specific models and retrieval tasks, providing a grounded explanation for a previously heuristic observation.