First-order optimization on stratified sets

Guillaume Olikier; Kyle A. Gallivan; P. -A. Absil

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arXiv · 2303.16040

First-order optimization on stratified sets

Guillaume Olikier·Kyle A. Gallivan·P. -A. Absil

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Abstract

We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on a stratified set and present a first-order algorithm designed to find a stationary point of that problem. Our assumptions on the stratified set are satisfied notably by the determinantal variety (i.e., matrices of bounded rank), its intersection with the cone of positive-semidefinite matrices, and the set of nonnegative sparse vectors. The iteration map of the proposed algorithm applies a step of projected-projected gradient descent with backtracking line search, as proposed by Schneider and Uschmajew (2015), to its input but also to a projection of the input onto each of the lower strata to which it is considered close, and outputs a point among those thereby produced that maximally reduces the cost function. Under our assumptions on the stratified set, we prove that this algorithm produces a sequence whose accumulation points are stationary, and therefore does not follow the so-called apocalypses described by Levin, Kileel, and Boumal (2022). We illustrate the apocalypse-free property of our method through a numerical experiment on the determinantal variety.

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Guillaume Olikier, Kyle A. Gallivan, P. -A. Absil. 2023-03-28. First-order optimization on stratified sets. https://arxiv.org/abs/2303.16040

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