Methods for Convex $(L_0,L_1)$-Smooth Optimization: Clipping, Acceleration, and Adaptivity

TL;DR

This paper advances optimization methods for convex functions with generalized smoothness, introducing improved convergence guarantees for existing algorithms and proposing a new accelerated method that does not depend on standard smoothness assumptions.

Contribution

It provides new convergence rates for Gradient Descent with clipping and Polyak stepsizes under $(L_0,L_1)$-smoothness, and introduces a novel accelerated method for this class.

Findings

Improved convergence rates for Gradient Descent with clipping.

Enhanced rates for Gradient Descent with Polyak stepsizes.

A new accelerated method with better convergence guarantees.

Abstract

Due to the non-smoothness of optimization problems in Machine Learning, generalized smoothness assumptions have been gaining a lot of attention in recent years. One of the most popular assumptions of this type is $(L_0,L_1)$-smoothness (Zhang et al., 2020). In this paper, we focus on the class of (strongly) convex $(L_0,L_1)$-smooth functions and derive new convergence guarantees for several existing methods. In particular, we derive improved convergence rates for Gradient Descent with (Smoothed) Gradient Clipping and for Gradient Descent with Polyak Stepsizes. In contrast to the existing results, our rates do not rely on the standard smoothness assumption and do not suffer from the exponential dependency from the initial distance to the solution. We also extend these results to the stochastic case under the over-parameterization assumption, propose a new accelerated method for convex $(L_0,L_1)$-smooth optimization, and derive new convergence rates for Adaptive Gradient Descent (Malitsky and Mishchenko, 2020).