arXiv subjects

Matthew Hough

Publications and source records attributed to Matthew Hough.

4 recordsLinked to original sources

arXiv2026

Convergence of the Frank-Wolfe Algorithm for Monotone Variational Inequalities

We consider the Frank-Wolfe algorithm for solving variational inequalities over compact, convex sets under a monotone $C^1$ operator and vanishing, nonsummable step sizes. We introduce a continuous-time interpolation of the discrete iteration and use tools from dynamical systems theory to analyze its asymptotic behavior. This allows us to derive convergence results for the original discrete algorithm. Consequently, every cluster point of the iterates is a solution of the underlying variational inequality, the distance from the iterates to the solution set converges to zero, and the Frank-Wolfe gap vanishes asymptotically. In the strongly monotone case, the solution is unique and the iterates converge to it. In particular, this proves Hammond's conjecture on the convergence of generalized fictitious play. We also discuss rates of convergence and under what assumptions rates can be shown.

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arXiv2025

Data-Dependent Complexity of First-Order Methods for Binary Classification

Large-scale problems in data science are often modeled with optimization, and the optimization model is usually solved with first-order methods that may converge at a sublinear rate. Therefore, it is of interest to terminate the optimization algorithm as soon as the underlying data science task is accomplished. We consider FISTA for solving two binary classification problems: the ellipsoid separation problem (ESP), and the soft-margin support-vector machine (SVM). For the ESP, we cast the dual second-order cone program into a form amenable to FISTA and show that the FISTA residual converges to the infimal displacement vector of the primal-dual hybrid gradient (PDHG) algorithm, that directly encodes a separating hyperplane. We further derive a data-dependent iteration upper bound scaling as $\mathcal{O}(1/\delta_{\mathcal{A}}^2)$, where $\delta_{\mathcal{A}}$ is the minimal perturbation that destroys separability. For the SVM, we propose a strongly-concave perturbed dual that admits efficient FISTA updates under a linear time projection scheme, and with our parameter choices, the objective has small condition number, enabling rapid convergence. We prove that, under a reasonable data model, early-stopped iterates identify well-classified points and yield a hyperplane that exactly separates them, where the accuracy required of the dual iterate is governed by geometric properties of the data. In particular, the proposed early-stopping criteria diminish the need for hard-to-select tolerance-based stopping conditions. Our numerical experiments on ESP instances derived from MNIST data and on soft-margin SVM benchmarks indicate competitive runtimes and substantial speedups from stopping early.

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arXiv2024

A Primal-Dual Frank-Wolfe Algorithm for Linear Programming

We present two first-order primal-dual algorithms for solving saddle point formulations of linear programs, namely FWLP (Frank-Wolfe Linear Programming) and FWLP-P. The former iteratively applies the Frank-Wolfe algorithm to both the primal and dual of the saddle point formulation of a standard-form LP. The latter is a modification of FWLP in which regularizing perturbations are used in computing the iterates. We show that FWLP-P converges to a primal-dual solution with error $O(1/\sqrt{k})$ after $k$ iterations, while no convergence guarantees are provided for FWLP. We also discuss the advantages of using FWLP and FWLP-P for solving very large LPs. In particular, we argue that only part of the matrix $A$ is needed at each iteration, in contrast to other first-order methods.

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arXiv2022

Model-Based Derivative-Free Methods for Convex-Constrained Optimization

We present a model-based derivative-free method for optimization subject to general convex constraints, which we assume are unrelaxable and accessed only through a projection operator that is cheap to evaluate. We prove global convergence and a worst-case complexity of $O(ε^{-2})$ iterations and objective evaluations for nonconvex functions, matching results for the unconstrained case. We introduce new, weaker requirements on model accuracy compared to existing methods. As a result, sufficiently accurate interpolation models can be constructed only using feasible points. We develop a comprehensive theory of interpolation set management in this regime for linear and composite linear models. We implement our approach for nonlinear least-squares problems and demonstrate strong practical performance compared to general-purpose solvers.

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