High Probability Guarantees for Random Reshuffling

We consider the stochastic gradient method with random reshuffling ($\mathsf{RR}$) for tackling smooth nonconvex optimization problems. $\mathsf{RR}$ finds broad applications in practice, notably in training neural networks. In this work, we provide high probability complexity guarantees for this method. First, we establish a high probability ergodic sample complexity result (without taking expectation) for finding an $\varepsilon$-stationary point. Our derived complexity matches the best existing in-expectation one up to a logarithmic term while imposing no additional assumptions nor modifying $\mathsf{RR}$'s updating rule. Second, building on this analysis, we propose a simple stopping criterion embedded with a computable stopping test for $\mathsf{RR}$ (denoted as $\mathsf{RR}$-$\mathsf{sc}$). This criterion is guaranteed to be triggered after a finite number of iterations, enabling us to prove the same order high probability complexity for the returned last iterate. The fundamental ingredient in deriving the aforementioned results is a new concentration property for random reshuffling, which could be of independent interest. Finally, we conduct numerical experiments on small neural network training to support our theoretical findings.