Stochastic Heavy Ball

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.

Regret bounds for Narendra-Shapiro bandit algorithms

Narendra-Shapiro (NS) algorithms are bandit-type algorithms that have been introduced in the sixties (with a view to applications in Psychology or learning automata), whose convergence has been intensively studied in the stochastic algorithm literature. In this paper, we adress the following question: are the Narendra-Shapiro (NS) bandit algorithms competitive from a \textit{regret} point of view? In our main result, we show that some competitive bounds can be obtained for such algorithms in their penalized version (introduced in \cite{Lamberton_Pages}). More precisely, up to an over-penalization modification, the pseudo-regret $\bar{R}_n$ related to the penalized two-armed bandit algorithm is uniformly bounded by $C \sqrt{n}$ (where $C$ is made explicit in the paper). \noindent We also generalize existing convergence and rates of convergence results to the multi-armed case of the over-penalized bandit algorithm, including the convergence toward the invariant measure of a Piecewise Deterministic Markov Process (PDMP) after a suitable renormalization. Finally, ergodic properties of this PDMP are given in the multi-armed case.