A gradient estimator via L1-randomization for online zero-order optimization with two point feedbackDownload PDF

Published: 31 Oct 2022, Last Modified: 12 Jan 2023NeurIPS 2022 AcceptReaders: Everyone
Keywords: zero-order optimization, online learning
TL;DR: We propose a new gradient estimator for zero-order optimisation and study its theoretical and practical aspects
Abstract: This work studies online zero-order optimization of convex and Lipschitz functions. We present a novel gradient estimator based on two function evaluations and randomization on the $\ell_1$-sphere. Considering different geometries of feasible sets and Lipschitz assumptions we analyse online dual averaging algorithm with our estimator in place of the usual gradient. We consider two types of assumptions on the noise of the zero-order oracle: canceling noise and adversarial noise. We provide an anytime and completely data-driven algorithm, which is adaptive to all parameters of the problem. In the case of canceling noise that was previously studied in the literature, our guarantees are either comparable or better than state-of-the-art bounds obtained by~\citet{duchi2015} and \citet{Shamir17} for non-adaptive algorithms. Our analysis is based on deriving a new weighted Poincaré type inequality for the uniform measure on the $\ell_1$-sphere with explicit constants, which may be of independent interest.
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