MINIBATCH STOCHASTIC THREE POINTS METHOD FOR UNCONSTRAINED SMOOTH MINIMIZATION

Abstract

In this paper, we propose a new zero order optimization method called minibatch stochastic three points (MiSTP) method to solve an unconstrained minimization problem in a setting where only an approximation of the objective function evaluation is possible. It is based on the recently proposed stochastic three points (STP) method (Bergou et al., 2020). At each iteration, MiSTP generates a random search direction in a similar manner to STP, but chooses the next iterate based solely on the approximation of the objective function rather than its exact evaluations. We also analyze our method's complexity in the nonconvex and convex cases and evaluate its performance on multiple machine learning tasks.

1. INTRODUCTION

In this paper we consider the following unconstrained finite-sum optimization problem: min x∈R d f (x) def = 1 n n i=1 f i (x) where each f i : R d → R is a smooth objective function. Such kind of problems arise in a large body of machine learning (ML) applications including logistic regression (Conroy & Sajda, 2012), ridge regression (Shen et al., 2013) , least squares problems (Suykens & Vandewalle, 1999) , and deep neural networks training. The formulation (1) can express the distributed optimization problem across n agents, where each function f i represents the objective function of agent i, or the optimization problem where each f i is the objective function associated with the data point i. We assume that we work in the Zero Order (ZO) optimization settings, i.e., we do not have access to the derivatives of any function f i and only functions evaluations are available. Such situation arises in many fields and may occur due to multiple reasons, for example: (i) In many optimization problems, there is only availability of the objective function as the output of a black-box or simulation oracle and hence the absence of derivative information (Conn et al., 2009) . (ii) There are situations where the objective function evaluation is done through an old software. Modification of this software to provide firstorder derivatives may be too costly or impossible (Conn et al., 2009; Nesterov & Spokoiny, 2017) . (iii) In some situations, derivatives of the objective function are not available but can be extracted. This necessitates access and a good understanding of the simulation code. This process is considered invasive to the simulation code and also very costly in terms of coding efforts (Kramer et al., 2011) . (IV) In the case of using a commercial software that evaluates only the functions, it is impossible to compute the derivatives because the simulation code is inaccessible (Kramer et al., 2011; Conn et al., 2009) . (V) In the case of having access only to noisy function evaluations, computing derivatives is useless because they are unreliable (Conn et al., 2009) . ZO optimization has been used in many ML applications, for instance: hyperparameters tuning of ML models (Turner et al., 2021; P.Koch et al., 2018) , multi-agent target tracking (Al-Abri et al., 2021) , policy optimization in reinforcement learning algorithms (Malik et al., 2020; Li et al., 2020) 



, maximization of the area under the curve

