AI summaryⓘ
The authors studied Schedule-Free optimization methods, which keep learning rates constant rather than adjusting them over time, and are popular because they reduce tuning efforts. They focused on understanding how these methods perform theoretically when optimizing smooth but nonconvex functions, common in machine learning. Using mathematical tools from dynamical systems, the authors showed these methods achieve the best possible convergence speeds among similar algorithms without needing extra tricks or assumptions. They also demonstrated that Schedule-Free gradient descent can avoid getting stuck at saddle points when given a tiny initial nudge. Overall, the work helps explain why Schedule-Free methods work well in practice.
Schedule-Free methodsGradient descentStochastic gradient descentNonconvex optimizationConvergence rateLyapunov analysisDynamical systemsSaddle pointsLearning rate scheduler
Authors
Jiseok Chae, Donghwan Kim
Abstract
Schedule-Free methods have attracted growing interest for alleviating the burden of designing and tuning a learning rate scheduler, while matching and sometimes even outperforming optimizers with tuned schedulers. Despite their strong empirical results, their convergence theory in nonconvex optimization, where modern machine learning objectives typically arise, has remained largely unexplored. In this paper, we provide worst-case analyses of Schedule-Free gradient descent and Schedule-Free stochastic gradient descent, in their standard form and without auxiliary modifications or restrictive conditions, for smooth but possibly nonconvex objectives. Based on a Lyapunov analysis derived from the continuous-time limiting ordinary differential equation associated with these methods, we show that Schedule-Free gradient descent and Schedule-Free stochastic gradient descent achieve the optimal worst-case convergence rates attainable among first-order methods. We further formulate Schedule-Free gradient descent as a nonautonomous dynamical system and prove strict-saddle avoidance under an arbitrarily small one-time perturbation. These theoretical results provide a better understanding of the strong performance that Schedule-Free methods demonstrate.