A Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron

• URL: http://arxiv.org/abs/2503.01441v1
• Date: Mon, 03 Mar 2025 11:46:46 GMT
• Title: A Linearly Convergent Frank-Wolfe-type Method for Smooth Convex Minimization over the Spectrahedron
• Abstract summary: We consider the problem of minimizing a smooth and convex function over the $n$-dimensional spectrahedron.<n>Standard first-order methods often require high-rank matrix computations which are prohibitive when the dimension $n$ is large.<n>We present the first Frank-Wolfe-based algorithm that only applies efficient rank-one matrix computations.
• Score: 18.035791732015802
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the problem of minimizing a smooth and convex function over the $n$-dimensional spectrahedron -- the set of real symmetric $n\times n$ positive semidefinite matrices with unit trace, which underlies numerous applications in statistics, machine learning and additional domains. Standard first-order methods often require high-rank matrix computations which are prohibitive when the dimension $n$ is large. The well-known Frank-Wolfe method on the other hand, only requires efficient rank-one matrix computations, however suffers from worst-case slow convergence, even under conditions that enable linear convergence rates for standard methods. In this work we present the first Frank-Wolfe-based algorithm that only applies efficient rank-one matrix computations and, assuming quadratic growth and strict complementarity conditions, is guaranteed, after a finite number of iterations, to converges linearly, in expectation, and independently of the ambient dimension.

Related papers

• Hardness of High-Dimensional Linear Classification [58.29089693778071]
  We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem.<n>Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms.
  arXiv  Detail & Related papers  (2026-03-19T15:53:41Z)
• Accelerated Frank-Wolfe Algorithms: Complementarity Conditions and Sparsity [10.491266065801982]
  We develop new accelerated first-order algorithms for smooth convex functions over compact convex sets.<n>A key technical ingredient is a complementarity condition that captures solution sparsity.
  arXiv  Detail & Related papers  (2025-11-04T18:47:07Z)
• Inertial Quadratic Majorization Minimization with Application to Kernel Regularized Learning [1.0282274843007797]
  We introduce the Quadratic Majorization Minimization with Extrapolation (QMME) framework and establish its sequential convergence properties.<n>To demonstrate practical advantages, we apply QMME to large-scale kernel regularized learning problems.
  arXiv  Detail & Related papers  (2025-07-06T05:17:28Z)
• Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization [19.32160757444549]
  In practical instances of nonspecified matrix factorization, the rank of the true solutionrstar$ is often unknown, so the rankr$ of the model can be singular as $r>rstar$. We propose an inexpensive suber for matrix sensing variant non matrix factorization that restores the convergence factor back to linear, even in agnosticized case. Our numerical experiments find that PrecGD works equally well in restoring the convergence of other variants non matrix factorization.
  arXiv  Detail & Related papers  (2025-04-13T20:06:49Z)
• Online Learning Guided Quasi-Newton Methods with Global Non-Asymptotic Convergence [20.766358513158206]
  We prove a global convergence rate of $O(min1/k,sqrtd/k1.25)$ in terms of the duality gap. These results are the first global convergence results to demonstrate a provable advantage of a quasi-Newton method over the extragradient method.
  arXiv  Detail & Related papers  (2024-10-03T16:08:16Z)
• Stochastic Optimization for Non-convex Problem with Inexact Hessian Matrix, Gradient, and Function [99.31457740916815]
  Trust-region (TR) and adaptive regularization using cubics have proven to have some very appealing theoretical properties. We show that TR and ARC methods can simultaneously provide inexact computations of the Hessian, gradient, and function values.
  arXiv  Detail & Related papers  (2023-10-18T10:29:58Z)
• Pseudonorm Approachability and Applications to Regret Minimization [73.54127663296906]
  We convert high-dimensional $ell_infty$-approachability problems to low-dimensional pseudonorm approachability problems. We develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $ell$ and other norms.
  arXiv  Detail & Related papers  (2023-02-03T03:19:14Z)
• Explicit Second-Order Min-Max Optimization Methods with Optimal Convergence Guarantee [86.05440220344755]
  We propose and analyze inexact regularized Newton-type methods for finding a global saddle point of emphcon unconstrained min-max optimization problems. We show that the proposed methods generate iterates that remain within a bounded set and that the iterations converge to an $epsilon$-saddle point within $O(epsilon-2/3)$ in terms of a restricted function.
  arXiv  Detail & Related papers  (2022-10-23T21:24:37Z)
• Near-Optimal Algorithms for Linear Algebra in the Current Matrix Multiplication Time [46.31710224483631]
  We show how to bypass the main open question of Nelson and Nguyen (FOCS, 2013) regarding the logarithmic factors in the sketching dimension for existing constant factor approximation oblivious subspace embeddings. A key technique we use is an explicit mapping of Indyk based on uncertainty principles and extractors. For the fundamental problems of rank computation and finding a linearly independent subset of columns, our algorithms improve Cheung, Kwok, and Lau (JACM, 2013) and are optimal to within a constant factor and a $loglog(n)$-factor, respectively.
  arXiv  Detail & Related papers  (2021-07-16T19:34:10Z)
• On the Efficient Implementation of the Matrix Exponentiated Gradient Algorithm for Low-Rank Matrix Optimization [26.858608065417663]
  Convex optimization over the spectrahedron has important applications in machine learning, signal processing and statistics. We propose efficient implementations of MEG, which are tailored for optimization with low-rank matrices, and only use a single low-rank SVD on each iteration. We also provide efficiently-computable certificates for the correct convergence of our methods.
  arXiv  Detail & Related papers  (2020-12-18T19:14:51Z)
• Quantum algorithms for spectral sums [50.045011844765185]
  We propose new quantum algorithms for estimating spectral sums of positive semi-definite (PSD) matrices. We show how the algorithms and techniques used in this work can be applied to three problems in spectral graph theory.
  arXiv  Detail & Related papers  (2020-11-12T16:29:45Z)
• Adaptive extra-gradient methods for min-max optimization and games [35.02879452114223]
  We present a new family of minmax optimization algorithms that automatically exploit the geometry of the gradient data observed at earlier iterations. Thanks to this adaptation mechanism, the proposed method automatically detects whether the problem is smooth or not. It converges to an $varepsilon$-optimal solution within $mathcalO (1/varepsilon)$ iterations in smooth problems, and within $mathcalO (1/varepsilon)$ iterations in non-smooth ones.
  arXiv  Detail & Related papers  (2020-10-22T22:54:54Z)
• Linear-Sample Learning of Low-Rank Distributions [56.59844655107251]
  We show that learning $ktimes k$, rank-$r$, matrices to normalized $L_1$ distance requires $Omega(frackrepsilon2)$ samples. We propose an algorithm that uses $cal O(frackrepsilon2log2fracepsilon)$ samples, a number linear in the high dimension, and nearly linear in the matrices, typically low, rank proofs.
  arXiv  Detail & Related papers  (2020-09-30T19:10:32Z)
• Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization [56.05635751529922]
  We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
  arXiv  Detail & Related papers  (2020-06-10T15:00:09Z)
• Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent [34.0533596121548]
  Low-rank matrix estimation converges convex problem that finds numerous applications in signal processing, machine learning and imaging science. We show that ScaledGD achieves the best of the best in terms of the number of the low-rank matrix. Our analysis is also applicable to general loss that are similar to low-rank gradient descent.
  arXiv  Detail & Related papers  (2020-05-18T17:17:16Z)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.