Parallel stochastic line search methods with feedback for minimizing finite sums

We consider unconstrained minimization of a finite sum of N continuously differentiable, not necessarily convex, cost functions. Several gradientlike (and more generally, line search) methods, where the full gradient (the sum of N component costs’ gradients) at each iteration k is replaced with an inexpensive approximation based on a sub-sample Nk of the component costs’ gradients, are available in the literature. However, a vast majority of the methods considers pre-determined (either deterministic or random) rules for selecting subsets Nk; these rules are unrelated with the actual progress of the algorithm along iterations. In this paper, we propose a very general framework for nonmonotone line search algorithms with an adaptive choice of sub-samplesNk. Specifically, we consider master-worker architectures with one master and N workers, where each worker holds one component function fi. The master maintains the solution estimate xk and controls the states of the workers (active or inactive) through a single scalar control parameter pk. Each active worker sends to the master the value and the gradient of its compoDragana Bajović Biosense Institute, University of Novi Sad Zorana Djindjića 3, 21000 Novi Sad, Serbia E-mail: [email protected] Dušan Jakovetić Biosense Institute, University of Novi Sad Zorana Djindjića 3, 21000 Novi Sad, Serbia E-mail: [email protected] Nataša Krejić Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad Trg Dositeja Obradovića 3, 21000 Novi Sad, Serbia E-mail: [email protected] Nataša Krklec Jerinkić Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad Trg Dositeja Obradovića 3, 21000 Novi Sad, Serbia E-mail: [email protected] 2 Dragana Bajović et al. nent cost, while inactive workers stay idle. Parameter pk is proportional to the expected (average) number of active workers (which equals the average sample size), and it can increase or decrease along iterations based on a computationally inexpensive estimate of the algorithm progress. Hence, through parameter pk, the master sends feedback to the workers about the desired sample size at the next iteration. For the proposed algorithmic framework, we show that each accumulation point of sequence {xk} is a stationary point for the desired cost function, almost surely. Simulations on both synthetic and real world data sets illustrate the benefits of the proposed framework with respect to the existing non-adaptive rules.