Convergence of Sign-based Random Reshuffling Algorithms for Nonconvex Optimization

Abstract: signSGD is popular in nonconvex optimization due to its communication efficiency. Yet, existing analyses of signSGD rely on assuming that data are sampled with replacement in each iteration, contradicting the practical implementation where data are randomly reshuffled and sequentially fed into the algorithm. We bridge this gap by proving the first convergence result of signSGD with random reshuffling (SignRR) for nonconvex optimization. Given the dataset size $n$, the number of epochs of data passes $T$, and the variance bound of a stochastic gradient $\sigma^2$, we show that SignRR has the same convergence rate $O(\log(nT)/\sqrt{nT} + |\sigma|_1)$ as signSGD \citep{bernstein2018signsgd}. We then present SignRVR and SignRVM, which leverage variance-reduced gradients and momentum updates respectively, both converging at $O(\log (nT)/\sqrt{nT} + \log (nT)\sqrt{n}/\sqrt{T})$. In contrast with the analysis of signSGD, our results do not require an extremely large batch size in each iteration to be of the same order as the total number of iterations \citep{bernstein2018signsgd} or the signs of stochastic and true gradients match element-wise with a minimum probability of 1/2 \citep{safaryan2021stochastic}. We also extend our algorithms to cases where data are distributed across different machines, yielding dist-SignRVR and dist-SignRVM, both converging at $O(\log (n_0T)/\sqrt{n_0T} + \log (n_0T)\sqrt{n_0}/\sqrt{T})$, where $n_0$ is the dataset size of a single machine. We back up our theoretical findings through experiments on simulated and real-world problems, verifying that randomly reshuffled sign methods match or surpass existing baselines.