Improved Analysis and Rates for Variance Reduction under Without-replacement Sampling Orders

TL;DR

This paper improves the theoretical understanding and convergence rates of variance reduction methods under without-replacement sampling orders, introducing Prox-DFinito and optimal cyclic ordering to achieve state-of-the-art results.

Contribution

It develops a damped variant of Finito called Prox-DFinito with improved convergence guarantees under various without-replacement sampling schemes.

Findings

Prox-DFinito matches full-batch gradient descent rates.

Optimal cyclic sampling can achieve data-heterogeneity independent convergence.

The analysis guides the design of optimal sampling orders.

Abstract

When applying a stochastic algorithm, one must choose an order to draw samples. The practical choices are without-replacement sampling orders, which are empirically faster and more cache-friendly than uniform-iid-sampling but often have inferior theoretical guarantees. Without-replacement sampling is well understood only for SGD without variance reduction. In this paper, we will improve the convergence analysis and rates of variance reduction under without-replacement sampling orders for composite finite-sum minimization. Our results are in two-folds. First, we develop a damped variant of Finito called Prox-DFinito and establish its convergence rates with random reshuffling, cyclic sampling, and shuffling-once, under both convex and strongly convex scenarios. These rates match full-batch gradient descent and are state-of-the-art compared to the existing results for without-replacement sampling with variance-reduction. Second, our analysis can gauge how the cyclic order will influence the rate of cyclic sampling and, thus, allows us to derive the optimal fixed ordering. In the highly data-heterogeneous scenario, Prox-DFinito with optimal cyclic sampling can attain a sample-size-independent convergence rate, which, to our knowledge, is the first result that can match with uniform-iid-sampling with variance reduction. We also propose a practical method to discover the optimal cyclic ordering numerically.