Sebastian Pokutta, Andrea Walther, Zev Woodstock
01.01.2023 − 31.12.2025
Motivated by nonsmooth problems in machine learning, we solve the problem of minimizing an abs-smooth function subject to closed convex constraints. New theory and algorithms are developed using linear minimization oracles to enforce constraints and abs-linearization methods to handle nonsmoothness.
Timo Kreimeier, Sebastian Pokutta, Andrea Walther, and Zev Woodstock
On a Frank-Wolfe Approach for Abs-smooth Functions [arXiv preprint]
Abstract: We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our nonsmooth nonconvex problem setting is motivated by machine learning, since the broad class of abs-smooth functions includes, for instance, the squared ℓ2-error of a neural network with ReLU or hinge loss activation. To overcome the nonsmoothness in our problem, we propose a generalization to the traditional Frank-Wolfe gap and prove that first-order minimality is achieved when it vanishes. We derive a convergence rate for our algorithm which is identical to the smooth case. Although our algorithm necessitates the solution of a subproblem which is more challenging than the smooth case, we provide an efficient numerical method for its partial solution, and we identify several applications where our approach fully solves the subproblem. Numerical and theoretical convergence is demonstrated, yielding several conjectures.