Abstract
This paper deals with a natural stochastic optimization procedure derived from the so-called Heavy-ball method differential equation, which was introduced by Polyak in the 1960s with his seminal contribution [Pol64]. The Heavy-ball method is a second-order dynamics that was investigated to minimize convex functions f. The family of second-order methods recently received a large amount of attention, until the famous contribution of Nesterov [Nes83], leading to the explosion of large-scale optimization problems. This work provides an in-depth description of the stochastic heavy-ball method, which is an adaptation of the deterministic one when only unbiased evalutions of the gradient are available and used throughout the iterations of the algorithm. We first describe some almost sure convergence results in the case of general non-convex coercive functions f. We then examine the situation of convex and strongly convex potentials and derive some non-asymptotic results about the stochastic heavy-ball method. We end our study with limit theorems on several rescaled algorithms.
Highlights
Finding the minimum of a function f over a set Ω with an iterative procedure is very popular among numerous scientific communities and has many applications in optimization, image processing, economics and statistics, to name a few
The most widespread approaches rely on some first-order strategies, with a sequence pXkqkě0 that evolves over Ω with a first-order recursive formula Xk1 “ ΨrXk, f pXkq, ∇f pXkqs that uses a local approximation of f at point Xk, where this approximation is built with the knowledge of f pXkq and ∇f pXkq alone
The complexity of each update involved in first-order methods is relatively limited and useful when dealing with a large-scale optimization problem, which is generally expensive in the case of Interior Point and Newton-like methods
Summary
Finding the minimum of a function f over a set Ω with an iterative procedure is very popular among numerous scientific communities and has many applications in optimization, image processing, economics and statistics, to name a few. Among the available interpretations of NAGD, some recent advances have been proposed concerning the second-order dynamical system by [WSC16], being a particular case of the generalized Heavy Ball with Friction method (referred to as HBF in the text), as previously pointed out in [CEG09a, CEG09?b]. Even though the Robbins-Monro algorithm is able to achieve an optimal Op1{nq rate of convergence for strongly convex functions, its ability is highly sensitive to the step sizes used This remark led [PJ92] to develop an averaging method that makes it possible to use longer step sizes of the Robbins-Monro algorithm, and to average these iterates with a Cesaro procedure so that this method produces optimal results in the minimax sense (see [NY83]) for convex and strongly convex minimization problems, as pointed out in [BM11]. Other authors [GL13, GL16] obtained convergence results for the stochastic version of a variant of NAGD for non-convex optimization for gradient Lipschitz functions but these methods cannot be used for the analysis of the Heavy-ball algorithm. Appendix A consists of some important results on the supremum of certain random variables needed for the non-convex case
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