A KL-based Analysis Framework with Applications to Non-Descent Optimization Methods

TL;DR

This paper introduces a new analysis framework based on the Kurdyka-Lojasiewicz property for non-descent optimization algorithms, enabling convergence analysis of stochastic and distributed methods without requiring descent or bounded iterates.

Contribution

It presents a novel framework for analyzing non-descent optimization algorithms, including stochastic and distributed methods, establishing convergence and rates under mild assumptions.

Findings

Proves convergence of decentralized gradient and federated averaging methods.

Shows convergence of stochastic gradient descent without bounded iterates.

Provides a unified analysis approach for non-descent algorithms.

Abstract

We propose a novel analysis framework for non-descent-type optimization methodologies in nonconvex scenarios based on the Kurdyka-Lojasiewicz property. Our framework allows covering a broad class of algorithms, including those commonly employed in stochastic and distributed optimization. Specifically, it enables the analysis of first-order methods that lack a sufficient descent property and do not require access to full (deterministic) gradient information. We leverage this framework to establish, for the first time, iterate convergence and the corresponding rates for the decentralized gradient method and federated averaging under mild assumptions. Furthermore, based on the new analysis techniques, we show the convergence of the random reshuffling and stochastic gradient descent method without necessitating typical a priori bounded iterates assumptions.