Nonconvex Optimization

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10 papers in the last four weeks, up 100% on the four weeks before. 0.1% of all new papers.

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Latest papers 96

Jul 10, 2026cs.LG

Understanding Schedule-Free Methods in Nonconvex Optimization: Rate Guarantees and Escaping Saddles

Schedule-Free methods have attracted growing interest for alleviating the burden of designing and tuning a learning rate scheduler, while matching and sometimes even outperforming optimizers with tuned schedulers. Despite their strong empirical results, their convergence theory in nonconvex optimization, where modern machine learning objectives typically arise, has remained largely unexplored. In this paper, we provide worst-case analyses of Schedule-Free gradient descent and Schedule-Free stochastic gradient descent, in their standard form and without auxiliary modifications or restrictive conditions, for smooth but possibly nonconvex objectives. Based on a Lyapunov analysis derived from the continuous-time limiting ordinary differential equation associated with these methods, we show that Schedule-Free gradient descent and Schedule-Free stochastic gradient descent achieve the optimal worst-case convergence rates attainable among first-order methods. We further formulate Schedule-Free gradient descent as a nonautonomous dynamical system and prove strict-saddle avoidance under an arbitrarily small one-time perturbation. These theoretical results provide a better understanding of the strong performance that Schedule-Free methods demonstrate.

Jiseok Chae, Donghwan Kim

Natural Science Research Institute KAIST Daejeon, Republic of Korea · Department of Mathematical Sciences KAIST Daejeon, Republic of Korea

Learning Rate Scheduling Optimization Convergence Analysis Nonconvex Optimization Gradient Descent Dynamics Stochastic Gradient Descent

Jul 9, 2026math.OC

Nonconvex Composite Functional Constraints via First-Order Augmented Lagrangian Methods under Local Regularity

We study nonasymptotic convergence of primal-dual methods for a class of nonconvex constrained optimization problems with a convex-composite structure. In this class, both the objective and the functional inequality constraints are given by convex Lipschitz outer functions composed with smooth nonlinear inner mappings. The analysis is complicated by constraint violation in a nonconvex functional inequality system and by the lack of an a priori bound on the multipliers. To address these issues, we restrict the dual variable to an auxiliary compact set and analyze a smoothed prox-linear augmented Lagrangian method through a nonsmooth nonconvex-concave minimax reformulation. The main contribution is a finite-time mechanism for converting stationarity of the truncated minimax problem into a KKT certificate for the original constrained problem. We show that, for a sufficiently large penalty parameter, all but a controlled number of iterates enter a near-feasible region. On this region, a local conic regularity condition uniformly bounds the associated prox-linear multipliers and thereby makes the artificial dual truncation inactive at the selected iterates. Building on this mechanism, we establish explicit convergence rates for the proposed method in terms of the KKT residual. With dual regularization, a global dual error bound together with a bias-balancing argument gives an

O(K^{-1/3})

rate. In the unregularized case, under additional local structural assumptions including piecewise linearity of the outer functions, a local dual error bound yields the sharper

O(K^{-1/2})

rate.

Linglingzhi Zhu, Jiajin Li

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA. · Sauder School of Business, University of British Columbia, Vancouver, BC, Canada.

Primal-Dual Optimization Nonconvex Optimization Augmented Lagrangian Method

Jul 8, 2026cs.LG

Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates

Nonlinear least-squares objectives form the foundation of scientific machine-learning tasks, yet even curvature-aware optimizers remain geometrically inconsistent at finite step sizes: Levenberg-Marquardt (LM) derives its direction from local Riemannian metrics but realizes it as a straight parameter update. Here we introduce RNC-LM, which carries the LM direction along a locally constructed curved trajectory in Riemann normal coordinates. A recursive reformulation of the geodesic equation generates arbitrary finite-order corrections while reusing the same damped Gauss-Newton matrix factorization, and curve length is controlled separately from damping. On a reaction-diffusion physics-informed neural network benchmark, RNC-LM reduces relative

L^2

errors below

8\times10^{-3}

, whereas L-BFGS, LM and LM with geodesic acceleration remain near one. On a large-scale machine-learning potential fitting task with 985,160 configurations, fourth-order RNC-LM reaches a fixed training-error target with a

34\times

wall-clock speedup over LM. These results establish finite-step geometric realization as a distinct optimizer design principle.

Jianing Liu, Dong H. Zhang

State Key Laboratory of Chemical Reaction Dynamics and Department of Chemical Physics, University of Science and Technology of China, Hefei, 230026, China · State Key Laboratory of Chemical Reaction Dynamics, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, 116023, China. · University of Chinese Academy of Sciences, Beijing, 100049, China +1

Gauss-Newton Optimization Riemannian Optimization Nonconvex Optimization

Jul 7, 2026math.OC

On the Condition Number Upper Bound of the L-BFGS Inverse Hessian Approximation Matrix with a Two-Sided Geometric Envelope Safeguarding Mechanism

The limited-memory BFGS (L-BFGS) algorithm is a cornerstone of large-scale optimization due to its linear memory and computational costs. However, in ill-conditioned or non-convex landscapes, the implicit inverse Hessian approximation can suffer from an exploding condition number, leading to numerical instability and degraded convergence. To address this, we propose Two-Sided L-BFGS, a safeguarded variant that dynamically constrains the condition number of the inverse Hessian operator via a two-sided geometric envelope. Moreover, we show that Two-Sided L-BFGS preserves accumulated curvature information and maintains standard

O(mn)

memory and per-iteration time complexities. We prove that this geometric envelope yields a uniform bound on the condition number of every inverse Hessian approximation generated by the algorithm. By tracking the algebraic evolution of the extreme eigenvalues through

m

consecutive quasi-Newton updates starting from a scaled identity matrix, the resulting bound is expressed explicitly as a function of the memory depth, problem dimension, and envelope hyperparameters. Moreover, we show that Two-Sided L-BFGS preserves asymptotic global convergence in non-convex regimes under standard smoothness and strong Wolfe line-search assumptions, matching the theoretical guarantees of L-BFGS variants utilizing the Li-Fukushima cautious update rule. Numerical experiments on high-dimensional optimization problems demonstrate that the proposed method maintains well-conditioned inverse Hessian approximations and improves robustness and convergence behavior on ill-conditioned benchmarks.

Don Li

Department of Mathematics & Statistics Portland State University Portland, OR 97201, USA

Second-Order Optimization Nonconvex Optimization Quasi-Newton Methods

Jul 6, 2026cs.LG

Non-Convex Sparse Reinforcement Learning via Non-Monotone Inclusions

This work delivers two key contributions: one to efficient feature selection in reinforcement learning (RL), the other to the theory of non-monotone inclusions. On the RL side, the estimation bias inherent in conventional regularization schemes is addressed by augmenting classical least-squares temporal-difference (LSTD) policy evaluation with the sparsity-inducing, non-convex projected minimax concave (PMC) penalty. Because the PMC penalty is weakly convex, the resulting fixed-point problem is no longer monotone; instead, it falls under a broader class of non-monotone inclusions involving the sum of a monotone Lipschitz operator and a hypomonotone operator. On the theory side, novel convergence conditions are developed for the forward-reflected-backward splitting (FRBS) method applied to this broader class of non-monotone inclusion problems. Under mild conditions, Lyapunov stability and the existence of a limit point of the sequence of FRBS iterates are established; alternatively, under the weak Minty variational inequality assumption, exact convergence is guaranteed. Numerical tests on benchmark datasets show that the proposed FRBS iterates, applied to the non-convexly regularized LSTD problem, substantially outperform state-of-the-art feature-selection methods, especially when many noisy features are present.

Kyohei Suzuki, Konstantinos Slavakis

Department of Information and Communications Engineering, Institute of Science Tokyo, Yokohama, 226-8501, Japan

Temporal-Difference Learning Nonconvex Optimization

Jul 2, 2026math.OC

Decentralized Stochastic Subgradient-type Methods with Communication Compression for Nonsmooth Nonconvex Optimization

In this paper, we consider the nonsmooth nonconvex decentralized optimization problem, where inter-agent communication is compressed. We propose a general framework that unifies various decentralized stochastic subgradient-type methods with unbiased compression and contractive compression with error compensation. By relating the consensus-error iterates and the averaged iterates to the trajectories of continuous-time differential inclusions, we establish global convergence for all methods encompassed by our framework when the objective functions are nonsmooth and lack Clarke regularity. Based on our framework, we further develop several compression-based methods, including decentralized stochastic subgradient methods utilizing sign-based regularization and gradient-tracking momentum. Preliminary numerical experiments empirically support our theoretical results and highlight the communication-accuracy trade-off of the newly developed methods.

Siyuan Zhang, Nachuan Xiao, Xin Liu

Stochastic Optimization Nonconvex Optimization Stochastic Optimization Convergence Nonsmooth Convex Optimization Decentralized Optimization

Jun 30, 2026math.OC

Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics

Low-rank matrix optimization is often carried out via the Burer-Monteiro (BM) formulation, but choosing the factorization rank

r

is delicate and can substantially slow optimization. We propose a unified framework, termed direction-magnitude decomposition (DMD), that decomposes the optimization variable to improve optimization efficiency even when the target rank is unknown. We develop two DMD-based approaches and establish their theoretical advantages on the canonical problem of matrix factorization. The first, overparameterized DMD, uses a rank

r

larger than necessary and enjoys faster convergence as

r

increases. The second, recursive DMD, is motivated by the incremental eigenpair learning, or saddle-to-saddle, behavior of overparameterized DMD. It achieves lower memory and computational costs, complementing overparameterized DMD. Both approaches are exponentially faster than gradient descent applied to the BM formulation. Numerical experiments on matrix factorization, sensing, and completion corroborate our theoretical findings and demonstrate the practical effectiveness of DMD.

Yudong Wei, Liang Zhang, Bingcong Li +1

Department of Mathematics, ETH Zurich, Zurich, Switzerland · Department of Computer Science, ETH Zurich, Zurich, Switzerland

Low-Rank Matrix Decomposition Nonconvex Optimization Matrix Optimization Matrix Factorization Gradient Descent Dynamics

Jun 26, 2026cs.MA

A Fast Convergent Algorithm for Solving Non-convex Partially-Decoupled Generalized Nash Equilibrium Problems

Solving multi-agent optimal control problems in aerospace such as pursuit-evasion and contested space operations can be modeled as non-convex differential games for which, there are limited algorithms. In this work, a relaxation of generalized Nash Equilibrium problems (GNEPs) to exclude inter-agent control coupling in dynamics, which is representative of many multi-agent systems is introduced. The main contribution is an algorithm for solving a broad class of differential games named FALCON: Fast Augmented Lagrangian Convexification for Open-loop Nash equilibria is presented. Methodologically, sequential convex programming (SCP) is utilized to create tractable convex sub-games which can then be solved via standard convex programming methods involving a potential game reformulation. FALCON is demonstrated to have global convergence guarantees to an open-loop Nash equilibrium for non-convex differential games under mild assumptions. This is numerically shown through both cooperative and competitive differential games.

Bennet Outland, Vishala Arya

Smead Aerospace Engineering Sciences Department, 3775 Discovery Drive, Boulder, CO, USA

Game Theory Nonconvex Optimization Optimal Control Nash Equilibrium Computation

Jun 26, 2026math.OC

Second-Order KKT Guarantees for Bregman ADMM in Nonconvex and Non-Lipschitz Optimization

We analyze Bregman ADMM for nonconvex linearly constrained problems under two-sided relative smoothness, a condition that replaces the standard Lipschitz gradient assumption with a Hessian comparison relative to a Bregman kernel. This setting covers polynomial objectives arising in matrix and tensor models for which a global Lipschitz-gradient constant need not exist. We show that on an invariant open state-space domain, one iteration of Bregman ADMM defines a smooth primal--dual fixed-point map whose strict-saddle KKT points are unstable fixed points; consequently, from random initialization the iterates converge to a strict saddle with probability zero. Combined with existing first-order convergence results, this yields almost-sure second-order stationarity of limiting KKT points. We extend the analysis to a multi-block star consensus formulation for distributed optimization. The technical novelty lies in a determinant reduction with a Bregman-specific symmetrization and scaling step in the two block spectral argument, together with a null space cancellation exploiting the star graph structure in the consensus case. Numerical experiments on distributed matrix factorization illustrate the theory, and a symmetric tensor factorization example demonstrates the broader Bregman proximal splitting idea beyond the separable consensus setting.

Shuang Li, Zhihui Zhu, Qiuwei Li

Department of Electrical and Computer Engineering, Iowa State University, Ames, Iowa 50014, USA · Department of Computer Science and Engineering, Ohio State University, Columbus, Ohio 43210, USA · DAMO Academy, Alibaba Group US, Bellevue, Washington 98004, USA

Optimization Convergence Analysis Distributed Optimization Alternating Direction Method of Multipliers Nonconvex Optimization

Jun 26, 2026cs.LG

Difference of Convex Programming in the Wasserstein Space with Applications to MMD Optimization

Optimizing functionals over the space of probability measures is now ubiquitous in machine learning. A widely used approach is to perform the optimization directly over the Wasserstein space, but many objective functionals of practical interest are non-convex along Wasserstein geodesics, making the analysis of standard first-order methods challenging. In this work, we study a class of objectives over the Wasserstein space that admit a difference-of-convex (DC) decomposition and we lift the classical convex-concave procedure (CCCP) to this setting. Under smoothness and strong convexity assumptions on the convex components of the decomposition, we prove almost stationarity along the iterates of the resulting algorithm. Our main focus is on the Maximum Mean Discrepancy (MMD) and the Energy Distance (ED) functionals, for which we develop explicit Wasserstein DC decompositions, and establish local convergence of the scheme under mild assumptions. Empirically, we show that well-chosen DC decompositions yield faster and more stable convergence than Wasserstein gradient descent on these MMD objectives.

Clément Bonet, Pierre-Cyril Aubin-Frankowski, Youssef Mroueh

CMAP, CNRS, Ecole Polytechnique, IP Paris · CERMICS, CNRS, ENPC, IP Paris · IBM Research

Maximum Mean Discrepancy Nonconvex Optimization Optimal Transport

Jun 25, 2026cs.LG

Finding Stationary Points by Comparisons

We study the problem of finding stationary points of non-convex functions when access to the objective is provided only through a comparison oracle that, given two points, outputs which has the larger function value. For a twice differentiable

f\colon\mathbb R^n\to\mathbb R

with Lipschitz gradient and Hessian, we develop an algorithm that visits an

ε

-stationary point using

\widetilde O(n^2/ε^{1.5})

queries. Our approach uses a subroutine that estimates the normalized Hessian to accuracy

δ

using

\widetilde O(n^2\log(1/δ))

queries. We further study this problem with a quantum comparison oracle model where queries can be made in superpositions, and develop the first quantum algorithm that finds an

ε

-stationary point, which takes

\widetilde O(n/ε^{1.5})

queries.

Helin Wang, Chenyi Zhang, Xiwen Tao +2

School of Electronics Engineering and Computer Science, Peking University · Center on Frontiers of Computing Studies, Peking University · Computer Science Department, Stanford University +1

Black-Box Optimization Zeroth-Order Optimization Nonconvex Optimization

Jun 21, 2026math.OC

Adam Converges in Nonsmooth Nonconvex Optimization

Adam is one of the most widely implemented and influential modern optimizers. Why is it effective across different optimization problems in practice? This question arguably lies at the center of the optimization community over the last decade and has motivated a substantial body of work aimed at understanding its convergence behavior. However, existing studies have mainly focused on the convergence rate of Adam in smooth nonconvex optimization, which unfortunately does not adequately capture practical settings, since many real-world problems are nonsmooth, such as those arising in training neural networks. Thus, these studies cannot fully explain the popularity and empirical success of Adam. Recently, an insightful and powerful framework called Online-to-Nonconvex Conversion has opened a new way to analyze Adam for nonsmooth nonconvex optimization. Unfortunately, prior works along this line share two common limitations. First, all of them ignore the important bias-correction term in the original Adam algorithm. Second and more importantly, many of them require extra operations that are not used in Adam, such as a clipping step. Therefore, the convergence guarantee for the original Adam method still remains unclear. In this work, we present the first finite-time analysis for the classical form of Adam, i.e., with the bias-correction step and without further algorithmic modifications, and prove that a randomly scaled learning rate ensures a convergence rate of

1/T^{\frac{2}{13}}

for nonsmooth nonconvex optimization. Moreover, our result provably applies to the modern heavy-tailed noise regime, which is closer to practice. Interestingly, our theory is established under the parameter choice

β_1=β_2

, aligning with the recent empirical studies.

Zijian Liu

Department of Technology, Operations, and Statistics Leonard N. Stern School of Business New York University New York, NY 10012, USA

Heavy-Tailed Noise Nonconvex Optimization Stochastic Optimization Convergence Adaptive Gradient Methods Nonsmooth Convex Optimization

Jun 18, 2026cs.LG

On the Oracle Complexity of Interpolation-Based Gradient Descent

Recent work on first-order optimizers for empirical risk minimization (ERM) has suggested that smoothness of ERM loss functions in the training data, rather than in the optimization parameters, can be leveraged to improve the oracle complexity of gradient descent (GD) methods. In this paper, we propose an inexact gradient method, piecewise polynomial interpolation-based gradient descent (PPI-GD), which approximates the full gradient in each iteration by querying the first-order oracle at equidistant points in the data domain to construct polynomial interpolants of the resulting gradient samples over appropriately sized patches of the data domain. We analyze the oracle complexity of PPI-GD for strongly convex and non-convex loss functions when the data space dimension is bounded by a polylogarithmic function of the number of training samples, and find it to outperform several GD variants in key regimes when the loss function is sufficiently smooth. Furthermore, our analysis extends several techniques from the error analysis of bicubic spline interpolants to the setting of

d

-variate tensor product polynomial interpolants which may be of independent interest in interpolation analysis.

Dongmin Lee, William Lu, Anuran Makur

Department of Computer Science, Purdue University, West Lafayette, IN 47907, USA · Department of Computer Science and the Elmore Family School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA

Oracle Complexity Gradient Descent Nonconvex Optimization Statistical Learning Theory Empirical Risk Minimization

Jun 18, 2026math.OC

Semiglobal Input-Delay Tolerance Algorithm for Distributed Nonconvex Optimization of Networked Nonlinear Systems

This paper studies a class of distributed optimization problems in networked nonlinear systems (NNSs) subject to input delays and consensus constraints. It introduces input-delay tolerant semiglobal convergence (IDTSC), meaning that for any prescribed compact initial set there exists an admissible delay bound under which the optimal solution is computed within consensus constraints and all node states converge to the solution. Building on a hierarchical design and input-to-state stability analysis, a new semiglobal input-delay tolerant (SIDT) algorithm is developed that practically achieves IDTSC for distributed optimization under the coupling between input delays and nonlinear dynamics. Further, by relaxing strict convexity requirements through the Polyak-Łojasiewicz condition, the SIDT algorithm broadens its applicability to nonconvex optimization. Finally, numerical experiments corroborate the theory on NNSs with input delays.

Jing-Zhe Xu, Zhi-Wei Liu, Ming-Feng Ge +2

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan 430074, China · School of Mechanical Engineering and Electronic Information, China University of Geosciences, Wuhan 430074, China

Dynamical Systems Optimization Convergence Analysis Distributed Optimization Nonconvex Optimization

Jun 17, 2026cs.CV

Low-Rank Tensor Completion Based on Fractional Regularization with Ky Fan p-k Norm

This paper addresses low-rank tensor completion (LRTC) by proposing a novel nonconvex surrogate, namely the ratio of the tensor nuclear norm to the tensor Ky Fan p-k norm (TNPK), to accurately approximate the tensor tubal rank. The TNPK possesses appealing properties, including scale invariance, parameter flexibility, and the existence of closed-form solutions under specific choices of p and k. With specific parameter settings of p and k, it reduces to the ratio of the tensor nuclear norm to the tensor Ky Fan k norm (TNK) or the ratio of the tensor nuclear norm to the tensor Frobenius norm (TNF). We construct a LRTC model and, under the tensor null space property (NSP), prove that low-rank tensors are local minimizers of the proposed model. Moreover, we derive the proximal operator of the Ky Fan p-k inverse-norm and further develop an efficient alternating direction method of multipliers (ADMM) algorithm with guaranteed subsequential convergence under mild conditions. Extensive experiments on synthetic and real-world datasets validate the superior performance of our method against state-of-the-art competitors.

Shan Fan, Feng Zhang, Jianjun Wang +2

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China · School of Mathematical Sciences/Research Center for Image and Vision Computing, University of Electronic Science and Technology of China, Chengdu 611731, China · Faculty of Computer Science and Control Engineering, Shenzhen University of Advanced Technology, Shenzhen 518055, China

Tensor Completion Nonconvex Optimization Low-Rank Approximation

Jun 15, 2026cs.CC

The Complexity of Min-Max Optimization for Quadratic Polynomials

We prove that computing approximate stationary points of min-max optimization over the hypercube is PPAD-hard for quadratic polynomials. This holds even when the polynomials are multilinear, each variable appears in at most three monomials, and the approximation factor is inverse polynomial. As a direct consequence, we obtain the first PPAD-hardness results for two-team zero-sum polymatrix games.

Martino Bernasconi, Matteo Castiglioni, Andrea Celli +1

Bocconi University · Politecnico di Milano · University of Oxford

Min-Max Optimization Nonconvex Optimization

Jun 13, 2026cs.LG

When to use what Schatten- $p$ norm in deep learning?

Schatten-

\infty

based optimizers such as Muon have shown promising empirical performance, but there remains seemingly conflicting observations regarding whether they are beneficial. We resolve this conflict by showing that the conclusion is regime dependent. Even when the objective is smooth in the Schatten-

\infty

geometry, smaller Schatten-

p

geometries can be optimal, specifically in the low-dimensional regime, which we show includes Chinchilla scaling. This conclusion follows from a new noise-robust acceleration result for the SODA framework for

p>2

. The same analysis explains why Muon-like methods do not require warmup, why they naturally favor large batches, and yields a batch size scaling rule for arbitrary

p

.

Thomas Pethick

Deep Learning Optimization Muon Optimizer Nonconvex Optimization Stochastic Optimization Convergence Accelerated Gradient Methods

Jun 12, 2026math.OC

Operator Calculus for Population-Based Optimization: Modular Convergence and Finite-Population Guarantees

Population-based optimizers combine update rules such as mutation, selection, and recombination. When one rule changes, it is often unclear which convergence guarantees survive or how the new combination should be assessed. We develop an operator calculus: an operator is a population-update rule, and the calculus specifies how separately checked effects can be combined. Under explicit regularity and small-step conditions, the leading changes caused by the updates add, yielding reusable building blocks for convergence analysis. The framework distinguishes finding and retaining a good solution, reducing the population's mean objective, and concentrating candidates near an optimizer, and identifies the extra approximation conditions needed for finite evaluation-budget guarantees. Applications include distribution adaptation, recombinative evolution, and consensus dynamics, with verified nonconvex cases. Controlled experiments on a common nonconvex problem collection show how component effects change with population geometry and the performance measure: a rule can worsen the mean objective yet produce better candidates.

Pekka Malo, Lauri Viitasaari, Patrik Nummi +3

Aalto University, Department of Information and Service Management, Finland. · Aalto University, Department of Information and Communications Engineering, Finland. · Indian Institute of Management Ahmedabad, Operations and Decision Sciences, India. +1

Evolutionary Optimization Optimization Convergence Analysis Metaheuristic Optimization Nonconvex Optimization Mean-Field Theory

Jun 3, 2026cs.LG

Sharp First-Order Lower Bounds for Higher-Order Smooth Nonconvex Optimization

We study the deterministic first-order oracle complexity of finding

ε

-stationary points in smooth nonconvex optimization when the objective satisfies higher-order smoothness assumptions. While the classical

ε^{-2}

rate is optimal under only Lipschitz gradients, higher-order smoothness leads to accelerated first-order upper bounds, most notably the

ε^{-7/4}

rate under Lipschitz Hessians and the

ε^{-5/3}

rate under Lipschitz third derivatives. The matching lower bounds, however, have remained open. We resolve this gap by proving a new dimension-free first-order lower bound for higher-order smooth nonconvex functions, valid for every finite smoothness order. In particular, our construction gives a matching

Ω(ε^{-7/4})

lower bound in the Hessian-Lipschitz case and a matching

Ω(ε^{-5/3})

lower bound in the third-order-smooth regime. The hard instance is based on a \emph{block-chain} mechanism that enforces blockwise oracle revelation while preserving the smoothness structure needed for the scalar hard instance. The lower-bound construction was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors.

Dongruo Zhou

Department of Computer Science, Indiana University Bloomington, IN 47408, USA

Optimization Convergence Analysis Nonconvex Optimization

Jun 1, 2026cs.LG

From Non-Convex to Strongly Convex: Curvature-Adaptive FTPL for Online Optimization

Curvature adaptivity is a classical theme in online optimization: for convex Lipschitz losses, adaptive methods interpolate between the optimal

O(\sqrt{T})

regret for general convex losses and

O(\log T)

regret under strong convexity. Recent work has shown that Follow-the-Perturbed-Leader (FTPL) achieves optimal

O(\sqrt{T})

regret even for online non-convex Lipschitz losses, assuming access to an approximate offline-optimization oracle, but these guarantees do not exploit curvature. We show that FTPL can be made curvature-adaptive in the non-convex setting, without knowing in advance how curvature will accumulate over time. Our algorithm replaces the fixed perturbation scale of standard FTPL with a time-varying scale chosen using only past information. We give a simple follow-the-leader tuning rule for this scale and show that it competes, up to constants, with the best choice in hindsight. The resulting method achieves

O(\sqrt{T})

regret for arbitrary non-convex Lipschitz losses and improves as cumulative curvature grows; with sufficiently accurate oracle calls, it achieves

O(\log T)

regret when cumulative curvature grows linearly, which includes the classical strongly convex regime. We complement these upper bounds with matching lower bounds for prescribed cumulative-curvature sequences, already for one-dimensional convex losses, showing that the tradeoff between worst-case non-convex regret and curvature-driven fast rates is intrinsic.

Moses Charikar, Chirag Pabbaraju, Ambuj Tewari

Stanford University. · University of Michigan, Ann Arbor.

Online Convex Optimization Nonconvex Optimization

May 31, 2026cs.LG

Learning-based Directed Graph Abstraction of Combinatorial Spaces for Order-Preserving Search in Mixed-Combinatorial Nonlinear Optimization

Mixed-combinatorial nonlinear programming (MCNLP) problems arise in many engineering design and planning applications, e.g., due to categorical, component, and geometric design choices, as well as joint task and motion planning. Traditional representations of combinatorial spaces, such as integer or binary encoding, often introduce spurious relations, increase dimensionality, and require additional compatibility constraints. Instead, this paper draws on recent developments in robot planning and vehicle/network routing domains that aim to learn search heuristics over combinatorial spaces using graph neural networks (GNNs). More specifically, this paper presents a first-of-its-kind structured abstraction of the combinatorial space by learning a mapping from an undirected fully connected graph of combinations to a directed graph indicating improvement directions using an Edge Field Graph Network (EFGN). To demonstrate the utility of this new way of abstracting the combinatorial space in solving MCNLPs, we adopt a recent optimization framework that purely searches over the non-combinatorial (e.g., continuous) variables and retrieves the best-suited combination for each candidate design by using the abstraction model, akin to a recommender system. The presented direction-aware abstraction model provides a potentially more scalable and interpretable retrieval of combinations compared to the original recommendation system in that framework. For evaluation, the proposed method is integrated with a well-known particle swarm optimization and genetic algorithm solvers on three benchmark nonlinear problems with varying numbers of combinations and variables. Compared to baseline solvers using indexified combinations, the GNN-based recommender consistently achieves better mean optimum values and robustness across multiple runs.

Gishnu Madhu, Feng Liu, Souma Chowdhury

University at Buffalo Buffalo, New York, 14260

Graph Structure Learning Graph Neural Networks Nonconvex Optimization Neural Combinatorial Optimization

May 29, 2026cs.LG

GLENS: Global Search via Learning from Solver Iterates with Diffusion Models

We consider the problem of generating a large collection of initial guesses for local minima of multimodal non-convex continuous optimization problems. The goal is for these initial guesses to be high-quality (i.e., a numerical solver converges quickly) and diverse (i.e., represent many different local minima). Identifying multiple locally optimal solutions enables flexible downstream decision-making, but typically requires expensive global search. Existing data-driven methods predict initial guesses using only the final converged optima from offline solver runs, which discards information about the local neighborhoods of solutions and limits the available training data. We propose GLENS (Global Search via Learning from Solver Iterates), a data-efficient global search method that leverages intermediate solver iterates as free data augmentation. GLENS consists of two components: a neighborhood structure model that uses diffusion models to learn the local geometry around optima conditioned on problem parameters, and a solver behavior model that learns refinement directions to further guide samples towards nearby optima during diffusion sampling. Experiments on modified non-convex benchmark problems and a two-robot obstacle-avoidance navigation problem show that GLENS generates high-quality initial guesses while preserving the multimodal distribution of diverse local optima. The resulting initial guesses lead to faster solver convergence across different problem settings and solvers. We also analyze how key hyperparameter choices affect the performance.

Anjian Li, Bartolomeo Stellato, Ryne Beeson

Department of Electrical and Computer Engineering, Princeton University, Princeton, 08544, NJ, USA. · Department of Operations Research and Financial Engineering, Princeton University, Princeton, 08544, NJ, USA. · Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, 08544, NJ, USA.

Learning to Optimize Diffusion Sampling Global Optimization Multimodal Optimization Nonconvex Optimization

May 28, 2026cs.LG

Convergence of Steepest Descent and Adam under Non-Uniform Smoothness

Recent work has analyzed the convergence of first-order methods under non-uniform smoothness assumptions that better model the loss landscape in machine learning tasks. We generalize this assumption to objectives whose curvature is an affine function of the objective value. This property is satisfied by a broad class of problems, including logistic regression, generalized linear models with a logistic link function, softmax policy gradient in reinforcement learning, and a class of neural networks. Under this assumption and gradient domination conditions, we establish a general convergence rate for the steepest descent method, and deterministic, diagonal variants of RMSProp and Adam. Our results imply that for logistic regression on separable data and the softmax policy gradient objective, sign GD converges linearly and is provably faster than GD. Furthermore, we show that for a class of two-layer neural networks on separable data, RMSProp and Adam can converge at a linear rate with a constant step-size and momentum parameter. Finally, we present a lower bound demonstrating that, under our assumption, RMSProp and Adam are provably faster than AdaGrad, AMSGrad, gradient descent, and heavy-ball momentum.

Sharan Vaswani, Yifan Sun, Reza Babanezhad

1Simon Fraser University · 2Stony Brook University · 3Samsung AI, Montreal.

Optimization Convergence Analysis Gradient Descent Nonconvex Optimization Adaptive Gradient Methods

May 28, 2026cs.RO

Caspar: CUDA Accelerator for Symbolic Programming with Adaptive Reordering

We present Caspar, a library that makes the power of modern GPUs more accessible in robotics and provides a state-of-the-art nonlinear GPU solver that can be applied to a wide range of different optimization problems. Caspar bridges the gap between expressive symbolic programming in Python and high-performance GPU runtimes in C++ by automatically generating optimized CUDA kernels from symbolic expressions. Building on the SymForce library, users can easily define and combine symbolic expressions, including Lie group operations, to generate custom CUDA kernels. To use Caspar as a solver, users need only define the symbolic residual functions; Caspar then uses symbolic differentiation to generate the necessary GPU kernels and interfaces to perform nonlinear optimization. In this paper, we present the core components of Caspar and showcase its performance by performing bundle adjustment on the Bundle Adjustment in the Large (BAL) dataset. We benchmark Caspar against other state-of-the-art bundle adjusters and show that it is 5 to 20 times faster than the best alternative, requires less memory, and achieves similar accuracy. This illustrates the benefit of our symbolic GPU programming approach. Caspar is released as part of SymForce and is freely available at https://github.com/symforce-org/symforce

Emil Martens, Aaron Miller, Matias Varnum +1

Norwegian University of Science and Technology · Skydio

Bundle Adjustment GPU Kernel Optimization GPU Acceleration High-Performance Computing Nonconvex Optimization GPU Kernel Generation

May 28, 2026math.OC

MoSSP: A Momentum-Based Single-Loop Stochastic Penalty Method for Nonconvex Constrained DC-Regularized Optimization

In this paper, we study a structured class of nonconvex constrained stochastic problems with difference-of-convex (DC) regularization, where the feasible set is possibly nonconvex and the concave part of the DC regularizer is allowed to be nonsmooth. The fundamental challenge lies in maintaining feasibility for nonconvex constraints while achieving favorable oracle complexity. Although single-loop algorithms efficiently solve unconstrained DC optimization problems, their potential for constrained optimization with DC structure remains largely unexplored. To address this gap, we develop MoSSP, a Momentum-based Single-loop Stochastic Penalty method for such problems with provable complexity guarantees. The key idea is to apply a single stochastic proximal-gradient step to the Moreau envelope of the penalty plus the convex DC part, with the concave part's proximal mapping computed in parallel. We derive two algorithm variants: a Polyak-momentum version with

O(\varepsilon^{-4})

oracle complexity for finding stochastic

\varepsilon

-KKT points, and an improved

O(\varepsilon^{-3})

version incorporating recursive momentum. Experimental results demonstrate the effectiveness of the proposed algorithms.

Luxuan Li, Chunfeng Cui, Xiao Wang

School of Mathematical Sciences, Beihang University, Beijing 100191, China · School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China.

Stochastic Optimization Nonconvex Stochastic Optimization Nonconvex Optimization

May 27, 2026cs.LG

Designing Active Tether-Net Systems for Space Debris Capture with Graph-Learning-Aided Mixed-Combinatorial Optimization

Active tether-net systems are a promising solution for capturing large non-cooperative targets, such as space debris, by deploying a flexible net manipulated by maneuverable units (MUs). However, concurrent systematic explorations of design and control choices of the tether-net system to understand its full potential remain limited, partly due to the complex, constrained, nonlinear optimization problem that it presents -- one that involves a mixture of continuous, integer and categorical variables, with the latter two arising from net connectivity and component choices, respectively. Classical binary encoding methods are often ineffective for solving highly nonlinear and multimodal Mixed Combinatorial Nonlinear Programmings (MCNLPs) in engineering design, while integer coding approaches can introduce spurious relations among combinations. Given the graph-structured characteristics of the combinatorial space, this paper adopts and extends a new graph-learning-aided optimization approach to solve this MCNLP problem. Here, a Graph Neural Network (GNN) is trained to score (as output) and thereof recommend candidate combinations represented as nodes in a graph, with the continuous variable vector portion of a candidate design given as input. As a result, the MCNLP optimization reduces to an NLP, which can be solved using standard solvers. While this reduction approach is agnostic to the choice of the NLP solver, here a state-of-the-art Particle Swarm Optimization (PSO) algorithm with gradient-based fine-tuning is used as the solver. Demonstrated on the problem of concurrently designing the morphology of the net, choice of mass and thrusters in the MUs and aiming points used by the controller of the tether-net system, the GNN-based recommender is shown to provide significantly faster convergence to similar optimal solutions, compared to direct solution of the MCNLP problem.

Feng Liu, Achira Boonrath, Gishnu Madhu +2

University at Buffalo Buffalo, New York, 14260

Metaheuristic Optimization Nonconvex Optimization Neural Combinatorial Optimization

May 24, 2026cs.LG

Theoretical Analysis of Sparse Optimization with Reparameterization, Weight Decay, and Adaptive Learning Rate

Sparse optimization is a fundamental challenge in various practical applications. A popular approach to sparse optimization is

\ell_p

regularization. However, it may encounter optimization instability due to the unbounded gradients when

0<p<1

. In this paper, we introduce a novel approach to sparse optimization termed ReWA, based on Reparameterization, Weight decay, and Adaptive learning rate. ReWA is closely connected to

\ell_p

-regularization, yet it unveils a distinct optimization landscape that helps mitigate instability issues. Experiments on CIFAR-10 and ImageNet with ResNets demonstrate that ReWA leads to significant sparsity improvements over the

\ell_1

-regularization approach while preserving test accuracy.

Huangyu Xu, Jingqin Yang, Qianqian Xu +1

State Key Laboratory of AI Safety, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China · School of Computer Science and Technology, University of Chinese Academy of Sciences, Beijing, China · IIIS, Tsinghua University, Beijing, China +3

Nonconvex Optimization

May 23, 2026cs.LG

Zeroth-Order Nonconvex Nonsmooth Optimization with Heavy-Tailed Noise

This paper considers the nonconvex nonsmooth problem in which the objective function is Lipschitz continuous. We focus on the stochastic setting where the algorithm can access stochastic function value evaluations with heavy-tailed noise, which is prevalent in many popular machine learning applications. We propose a stochastic zeroth-order algorithm that refines the framework of online-to-nonconvex conversion by clipping the two-point gradient estimator. The theoretical analysis shows that our algorithm can find a

(δ, ε)

-Goldstein stationary point with zeroth-order oracle complexity of

{\mathcal O}(d^{\frac{p}{2(p-1)}}δ^{-1}ε^{-\frac{2p-1}{p-1}})

, where

d

is the problem dimension and

p\in(1,2]

is the order of bounded moments. Note that our dependence on dimension

d

matches the best-known results of stochastic zeroth-order optimization for finding the sub-optimal solution of a stochastic convex nonsmooth problem. In addition, our dependence on accuracy parameters

δ

and

ε

is consistent with that of the best-known stochastic first-order algorithms for stochastic nonconvex nonsmooth problems. Finally, we conduct numerical experiments to demonstrate the effectiveness of the proposed method.

Zhuanghua Liu, Luo Luo

Stochastic Optimization Gradient Clipping Zeroth-Order Optimization Heavy-Tailed Noise Nonconvex Optimization Nonsmooth Convex Optimization+1

May 22, 2026math.OC

RA-DCA: A Randomized Active-Set DCA for Directional Stationarity in Max-Structured DC Programs

We study nonsmooth difference-of-convex programs whose subtracted convex term is a finite maximum of smooth convex functions. In this setting, standard DCA iterations may converge to critical points that are not directionally stationary, whereas exact active-vertex screening can be expensive when active sets are large or combinatorial. We propose RA-DCA, a vertex-first randomized active-set DCA that projects active gradients onto sampled directions, checks a sampled vertex residual, and uses a small linear program only as a low-residual convex-combination fallback. The method preserves the descent structure of DCA and reduces the randomized screening layer to matrix multiplications. Under the stated regularity, numerical active-set consistency, and random-embedding assumptions, every accumulation point generated by the safeguarded method is directionally stationary with probability one. MATLAB experiments first test the theorem on degenerate max-affine, max-quadratic, and sparse support-function models, where the safeguard avoids nonstationary critical points and closely tracks a full active-vertex scan. Block top-k tests then show that the same screening idea remains useful when exact aggregate enumeration is combinatorial. Trimmed-regression, complementarity, and QUBO diagnostics separate cases where active-set selection helps from cases dominated by multistart search, the DC split, or other problem-specific features.

Yi-Shuai Niu

Beijing Institute of Mathematical Sciences and Applications (BIMSA), Beijing, China

Nonconvex Optimization

May 22, 2026cs.LG

Convex Compositional Reasoning Models

Compositional energy-based models can generalize to larger combinatorial reasoning problems by reusing a learned factor energy across many local constraints. In our paper, we show that a key bottleneck in compositional reasoning is not composition itself, but the non-convex geometry of the learned energy landscape. To solve this problem, we introduce Convex Compositional Energy Minimization (CCEM), a framework that parameterizes each factor with an input-convex neural network and optimizes the composed energy over a tight convex relaxation of the feasible set. Because convexity is preserved under summation, the global relaxed objective remains convex, enabling deterministic projected first-order optimization. CCEM is trained in two stages: factor-level contrastive learning to shape local energy basins, followed by end-to-end refinement through an unrolled projected solver. Our experiments show that our models trained on small subproblems or a single problem size transfer to larger instances without retraining.

Meir Roketlishvili, Semyon Semenov, Maksim Bobrin +7

Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, UAE · Applied AI Institute, Computational Imaging Lab · AXXX

Compositional Reasoning Energy-Based Models Nonconvex Optimization Input Convex Neural Networks Neural Combinatorial Optimization