Random Reshuffling with Momentum: Complexity Bounds and Last-iterate Convergence

Random reshuffling with momentum (RRM) corresponds to the SGD optimizer with the 'momentum' option enabled, as found in many machine learning libraries such as PyTorch and TensorFlow. Despite its widespread use, the convergence properties of RRM do not seem to be well understood. This work establishes new complexity bounds and asymptotic convergence guarantees for popular versions of RRM using stochastic heavy-ball momentum, Nesterov acceleration, and mini-batches in a general nonconvex setting. In particular, we prove that the base variant of RRM achieves the complexity $O(n^{-1/3}((1-\beta^n)T)^{-2/3})$, where $n$ denotes the number of samples, $\beta \in [0,1)$ is a momentum parameter, and $T$ is the total number of epochs. On the asymptotic side, we show that every accumulation point of the iterates $\{x^k\}_k$ generated by RRM is a stationary point of the problem. For definable objectives -- a broad and common class of functions including, e.g., semialgebraic, globally subanalytic, and log-exp functions -- we strengthen this subsequential result to last-iterate convergence to a single stationary point. Moreover, improved asymptotic complexity bounds are presented that are based on the additional geometric properties of definable functions.