解析无调度优化方法的收敛性，揭示其高效原因

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.