On the Comparison between Cyclic Sampling and Random Reshuffling

When applying a stochastic/incremental algorithm, one must choose the order to draw samples. Among the most popular approaches are cyclic sampling and random reshuffling, which are empirically faster and more cache-friendly than uniform-iid-sampling. Cyclic sampling draws the samples in a fixed, cyclic order, which is less robust than reshuffling the samples periodically. Indeed, existing works have established worst case convergence rates for cyclic sampling, which are generally worse than that of random reshuffling. In this paper, however, we found a certain cyclic order can be much faster than reshuffling and one can discover it at a low cost! Studying and comparing different sampling orders typically require new analytic techniques. In this paper, we introduce a norm, which is defined based on the sampling order, to measure the distance to solution. Applying this technique on proximal Finito/MISO algorithm allows us to identify the optimal fixed ordering, which can beat random reshuffling by a factor up to log(n)/n in terms of the best-known upper bounds. We also propose a strategy to discover the optimal fixed ordering numerically. The established rates are state-of-the-art compared to previous works.