Learning Rate Annealing Improves Tuning Robustness in Stochastic Optimization

The learning rate in stochastic gradient methods is a critical hyperparameter that is notoriously costly to tune via standard grid search, especially for training modern large-scale models with billions of parameters. We identify a theoretical advantage of learning rate annealing schemes that decay the learning rate to zero at a polynomial rate, such as the widely-used cosine schedule, by demonstrating their increased robustness to initial parameter misspecification due to a coarse grid search. We present an analysis in a stochastic convex optimization setup demonstrating that the convergence rate of stochastic gradient descent with annealed schedules depends *sublinearly* on the multiplicative misspecification factor $\rho$ (i.e., the grid resolution), achieving a rate of $\smash{O(\rho^{1/(2p+1)}/\sqrt{T})}$ where $p$ is the degree of polynomial decay and $T$ is the number of steps. This is in contrast to the $\smash{O(\rho/\sqrt{T})}$ rate obtained under the inverse-square-root and fixed stepsize schedules, which depend linearly on $\rho$. Experiments confirm the increased robustness compared to tuning with a fixed stepsize, that has significant implications for the computational overhead of hyperparameter search in practical training scenarios.