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Zeroth-order Nonconvex Stochastic Optimization: Handling Constraints, High-Dimensionality, and Saddle-Points

Krishnakumar Balasubramanian (kbala***at***ucdavis.edu)
Saeed Ghadimi (sghadimi***at***princeton.edu)

Abstract: In this paper, we propose and analyze zeroth-order stochastic approximation algorithms for nonconvex and convex optimization, with a focus on addressing constrained optimization, high-dimensional setting, and saddle-point avoiding. To handle constrained optimization, we first propose generalizations of the conditional gradient algorithm achieving rates similar to the standard stochastic gradient algorithm using only zeroth-order information. To facilitate zeroth-order optimization in high-dimensions, we explore the advantages of structural sparsity assumptions. Specifically, (i) we highlight an implicit regularization phenomenon where the standard stochastic gradient algorithm with zeroth-order information adapts to the sparsity of the problem at hand by just varying the step-size and (ii) propose a truncated stochastic gradient algorithm with zeroth-order information, whose rate of convergence depends only polylogarithmically on the dimensionality. We next focus on avoiding saddle-points in non-convex setting. Towards that, we interpret the Gaussian smoothing technique for estimating gradient based on zeroth-order information as an instantiation of first-order Steinís identity. Based on this, we provide a novel linear-(in dimension) time estimator of the Hessian matrix of a function using only zeroth-order information, which is based on second-order Steinís identity. We then provide an algorithm for avoiding saddle-points, which is based on a zeroth-order cubic regularization Newtonís method and discuss its convergence rates.

Keywords: Nonconvex optimization, Stochastic optimization, Zeroth-order algorithms, Complexity, Newton method, Conditional gradient

Category 1: Nonlinear Optimization

Category 2: Stochastic Programming

Citation:

Download: [PDF]

Entry Submitted: 01/14/2019
Entry Accepted: 01/14/2019
Entry Last Modified: 01/15/2019

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