Convergence of ease-controlled Random Reshuffling gradient Algorithms under Lipschitz smoothness

TL;DR

This paper introduces ease-controlled modifications of Incremental Gradient and Random Reshuffling algorithms that adaptively manage stepsizes and convergence criteria, improving efficiency in large-scale non-convex optimization.

Contribution

It proposes novel ease-controlled IG/RR schemes with watchdog and linesearch mechanisms that ensure convergence without pre-set stepsize rules, validated through theoretical proofs and extensive experiments.

Findings

Algorithms achieve convergence under Lipschitz smoothness.

Comparable computational effort to existing online methods.

Potential for faster objective function decrease.

Abstract

In this work, we consider minimizing the average of a very large number of smooth and possibly non-convex functions, and we focus on two widely used minibatch frameworks to tackle this optimization problem: Incremental Gradient (IG) and Random Reshuffling (RR). We define ease-controlled modifications of the IG/RR schemes, which require a light additional computational effort {but} can be proved to converge under {weak} and standard assumptions. In particular, we define two algorithmic schemes in which the IG/RR iteration is controlled by using a watchdog rule and a derivative-free linesearch that activates only sporadically to guarantee convergence. The two schemes differ in the watchdog and the linesearch, which are performed using either a monotonic or a non-monotonic rule. The two schemes also allow controlling the updating of the stepsize used in the main IG/RR iteration, avoiding the use of pre-set rules that may drive the stepsize to zero too fast, reducing the effort in designing effective updating rules of the stepsize. We prove convergence under the mild assumption of Lipschitz continuity of the gradients of the component functions and perform extensive computational analysis using different deep neural architectures and a benchmark of varying-size datasets. We compare our implementation with both a full batch gradient method (i.e. L-BFGS) and an implementation of IG/RR methods, proving that our algorithms require a similar computational effort compared to the other online algorithms and that the control on the learning rate may allow a faster decrease of the objective function.