Stochastic heavy ball

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.

Non asymptotic controls on a recursive superquantile approximation

“…[8]) and some details are skipped for the sake of convenience. The reader may found similar arguments in Section 5 of [24].…”

“…Proof of . The proof is briefly sketched since it exactly follows the same lines as those of Theorem 2.4 of [24] (Section 5.3).…”

Convergence and Dynamical Behavior of the ADAM Algorithm for Nonconvex Stochastic Optimization

“…The convergence towards the critical points of the objective function is shown under some hypotheses. The paper [22] studies a stochastic version of the celebrated heavy ball algorithm.…”

“…for all t > 0. In the sense of (5.4), x(t) can be interpreted as the solution to a generalized HBF problem, where both the mass of the particle and the viscosity depend on time [1,4,14,22,21].…”

Stochastic approximation algorithms for superquantiles estimation

“…Bardou et al [2] have previously studied the averaged version [22,27] of a onetime-scale stochastic algorithm in order to estimate θ α and ϑ α . Here, we have chosen to investigate a two-time-scale stochastic algorithm [5,11,17,20] which performs pretty well and offers more flexibility than the one-time-scale algorithm. Let (X n ) be a sequence of independent and identically distributed random variables sharing the same distribution as X.…”