Stochastic Zeroth order Descent with Structured Directions

• URL: http://arxiv.org/abs/2206.05124v3
• Date: Tue, 08 Oct 2024 11:37:51 GMT
• Title: Stochastic Zeroth order Descent with Structured Directions
• Authors: Marco Rando, Cesare Molinari, Silvia Villa, Lorenzo Rosasco,
• Abstract summary: We introduce and analyze Structured Zeroth order Descent (SSZD), a finite difference approach that approximates a gradient on a set $lleq d directions, where $d is the dimension of the ambient space. For convex convex we prove almost sure convergence of functions on $O( (d/l) k-c1/2$)$ for every $c1/2$, which is arbitrarily close to the one of the Gradient Descent (SGD) in terms of one number of iterations.
• Score: 10.604744518360464
• License:
• Abstract: We introduce and analyze Structured Stochastic Zeroth order Descent (S-SZD), a finite difference approach that approximates a stochastic gradient on a set of $l\leq d$ orthogonal directions, where $d$ is the dimension of the ambient space. These directions are randomly chosen and may change at each step. For smooth convex functions we prove almost sure convergence of the iterates and a convergence rate on the function values of the form $O( (d/l) k^{-c})$ for every $c<1/2$, which is arbitrarily close to the one of Stochastic Gradient Descent (SGD) in terms of number of iterations. Our bound shows the benefits of using $l$ multiple directions instead of one. For non-convex functions satisfying the Polyak-{\L}ojasiewicz condition, we establish the first convergence rates for stochastic structured zeroth order algorithms under such an assumption. We corroborate our theoretical findings with numerical simulations where the assumptions are satisfied and on the real-world problem of hyper-parameter optimization in machine learning, achieving competitive practical performance.

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