Continuized Nesterov Acceleration for Non-Convex Optimization

TL;DR

This paper extends the continuized framework for analyzing momentum algorithms to non-convex optimization, providing tighter convergence guarantees and broader applicability, especially for strongly quasiconvex functions.

Contribution

It introduces an extended continuized analysis that handles non-smooth Lyapunov functions and improves convergence bounds for non-convex optimization.

Findings

Achieves finite-time high-probability convergence bounds.

Strengthens trajectory-wise guarantees for linear convergence.

Improves convergence rate constants for strongly quasiconvex functions.

Abstract

In convex optimization, continuous-time counterparts have been a fruitful tool for analyzing momentum algorithms. Fewer such examples are available when the function to minimize is non-convex. In several cases, discrepancies arise between the existing discrete-time results, namely those obtained for momentum algorithms, and their continuous-time counterparts, with the latter typically yielding stronger guarantees. We argue that the continuized framework (Even et al., 2021), mixing continuous and discrete components, can tighten the gap between known continuous and discrete results. This framework relies on computations akin to standard Lyapunov analyses, from which are deduced convergence bounds for an algorithm that can be written as a Nesterov momentum algorithm with stochastic parameters. In this work, we extend the range of applicability of the continuized framework, e.g. by allowing it to handle non-smooth Lyapunov functions. We then strengthen its trajectory-wise guarantees for linear convergence rate, deriving finite time bounds with high probability and asymptotic almost sure bounds. We apply this framework to the non-convex class of strongly quasar convex functions. Adapting continuous-time results that have weaker discrete equivalents to the continuized method, we improve by a constant factor the known convergence rate, and relax the existing assumptions on the set of minimizers.