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

2024-09-23

AI Summary

We address convex optimization problems in machine learning that are nonsmooth yet satisfy $(L_0,L_1)$-smoothness—a structural generalization of classical $C^{1,1}$ smoothness. We propose a suite of novel algorithms that dispense with the standard $C^{1,1}$ assumption and achieve convergence rates independent of the initial point’s distance to the optimum. Methodologically, we (i) establish the first tight deterministic and stochastic convergence bounds for gradient clipping and the Polyak stepsize method, eliminating exponential dependence on initialization; (ii) design the first Nesterov-type accelerated algorithm for $(L_0,L_1)$-smooth convex functions, extended to stochastic overparameterized settings; and (iii) integrate Adaptive Gradient Descent (within the Malitsky–Mishchenko framework) for fully adaptive stepsize selection. All results hold for both strongly convex and general convex objectives, with rigorous theoretical guarantees. Our methods significantly improve upon state-of-the-art performance under nonsmooth yet structured smoothness assumptions.

Technology Category

Search and Optimization: Non-convex OptimizationMachine Learning: OptimizationNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Security and privacy of machine learning and AI applicationsResponsible Web: Human-perceived consequences of algorithmic deployment on the web

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).

Problem

• Machine Learning Optimization
• Non-smooth Functions
• Convergence Rate

Innovation

• (L_0,L_1)-smooth optimization
• accelerated methods for stochastic optimization
• convergence rate analysis for adaptive gradient descent