A Deterministic Nonsmooth Frank Wolfe Algorithm with Coreset Guarantees

TL;DR

This paper introduces a deterministic nonsmooth Frank-Wolfe algorithm with coreset guarantees, offering efficient approximate solutions for large-scale machine learning problems without smoothing or proximal steps.

Contribution

The paper presents a novel deterministic Frank-Wolfe variant applicable to nonsmooth problems, with theoretical coreset guarantees and practical efficiency demonstrated through experiments.

Findings

Provides convergence bounds for the algorithm.

Achieves input size-independent time complexity.

Demonstrates practical efficiency on large datasets.

Abstract

We present a new Frank-Wolfe (FW) type algorithm that is applicable to minimization problems with a nonsmooth convex objective. We provide convergence bounds and show that the scheme yields so-called coreset results for various Machine Learning problems including 1-median, Balanced Development, Sparse PCA, Graph Cuts, and the $\ell_1$-norm-regularized Support Vector Machine (SVM) among others. This means that the algorithm provides approximate solutions to these problems in time complexity bounds that are not dependent on the size of the input problem. Our framework, motivated by a growing body of work on sublinear algorithms for various data analysis problems, is entirely deterministic and makes no use of smoothing or proximal operators. Apart from these theoretical results, we show experimentally that the algorithm is very practical and in some cases also offers significant computational advantages on large problem instances. We provide an open source implementation that can be adapted for other problems that fit the overall structure.