Frank--Wolfe Methods with an Unbounded Feasible Region and Applications to Structured Learning
Name
20m1387869.pdf
Description
Published version
Size
582.06 KB
Format
Adobe PDF
Checksum (MD5)
0f9f079db2821d29e5c86d854d3ab0c5
Author(s)
Wang, Haoyue
Lu, Haihao
Mazumder, Rahul
Date Issued
December 2022
Journal
SIAM Journal on Optimization
Publisher
Society for Industrial & Applied Mathematics (SIAM)
Version
Final published version
Abstract
The Frank--Wolfe method is a popular algorithm for solving large-scale convex optimization problems appearing in structured statistical learning. However, the traditional Frank--Wolfe method can only be applied when the feasible region is bounded, which limits its applicability in practice. Motivated by two applications in statistical learning, the ℓ1 trend filtering problem and matrix optimization problems with generalized nuclear norm constraints, we study a family of convex optimization problems where the unbounded feasible region is the direct sum of an unbounded linear subspace and a bounded constraint set. We propose two new Frank--Wolfe methods, the unbounded Frank--Wolfe method and the unbounded away-step Frank--Wolfe method, for solving a family of convex optimization problems with this class of unbounded feasible regions. We show that under proper regularity conditions, the unbounded Frank--Wolfe method has an 𝑂(1/𝑘) sublinear convergence rate, and the unbounded away-step Frank--Wolfe method has a linear convergence rate, matching the best-known results for the Frank--Wolfe method when the feasible region is bounded. Furthermore, computational experiments indicate that our proposed methods appear to outperform alternative solvers.
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DOI of Published Version
https://doi.org/10.1137/20M1387869
