Convergence Conditions for Stochastic Line Search Based Optimization of Over-parametrized Models

TL;DR

This paper analyzes convergence conditions for stochastic line search algorithms in over-parametrized models, providing theoretical guarantees and insights for algorithm design in machine learning optimization.

Contribution

It introduces conditions on search directions ensuring finite termination and linear convergence for stochastic line search methods on PL functions.

Findings

Conditions for finite termination of stochastic line search algorithms.

Bounds for backtracking procedures in over-parametrized models.

Insights into integrating line searches with momentum and preconditioning methods.

Abstract

In this paper, we deal with algorithms to solve the finite-sum problems related to fitting over-parametrized models, that typically satisfy the interpolation condition. In particular, we focus on approaches based on stochastic line searches and employing general search directions. We define conditions on the sequence of search directions that guarantee finite termination and bounds for the backtracking procedure. Moreover, we shed light on the additional property of directions needed to prove fast (linear) convergence of the general class of algorithms when applied to PL functions in the interpolation regime. From the point of view of algorithms design, the proposed analysis identifies safeguarding conditions that could be employed in relevant algorithmic frameworks. In particular, it could be of interest to integrate stochastic line searches within momentum, conjugate gradient or adaptive preconditioning methods.