Stochastic Extragradient with Flip-Flop Shuffling & Anchoring: Provable Improvements

TL;DR

This work tackles convergence challenges of stochastic extragradient methods for unconstrained finite-sum minimax problems. It introduces SEG-FFA, a small modification that combines flip-flop sampling with an anchoring step to achieve second-order matching with EG/EG+, yielding provable improvements in convergence rates for convex-concave and strongly monotone settings. Specifically, SEG-FFA achieves a rate of $ ilde{O}(1/K^{1/3})$ in the convex-concave case and $ ilde{O}(1/(nK^{4}))$ in the strongly monotone case, with lower bounds showing clear advantages over SEG variants based on random reshuffling or without anchoring. The results are supported by theoretical analyses of within-epoch errors and experiments on monotone and strongly monotone quadratic problems, illustrating the practical impact of second-order matching for shuffling-based SEG methods.

Abstract

In minimax optimization, the extragradient (EG) method has been extensively studied because it outperforms the gradient descent-ascent method in convex-concave (C-C) problems. Yet, stochastic EG (SEG) has seen limited success in C-C problems, especially for unconstrained cases. Motivated by the recent progress of shuffling-based stochastic methods, we investigate the convergence of shuffling-based SEG in unconstrained finite-sum minimax problems, in search of convergent shuffling-based SEG. Our analysis reveals that both random reshuffling and the recently proposed flip-flop shuffling alone can suffer divergence in C-C problems. However, with an additional simple trick called anchoring, we develop the SEG with flip-flop anchoring (SEG-FFA) method which successfully converges in C-C problems. We also show upper and lower bounds in the strongly-convex-strongly-concave setting, demonstrating that SEG-FFA has a provably faster convergence rate compared to other shuffling-based methods.

Figures (4)

• Figure 1: Experimental results on the (left) monotone and (right) strongly monotone examples, comparing the variants of SEG. For a fair comparison, we take the number of passes over the full dataset as the abscissae. In other words, we plot ${\|{{\bm{F}} {\bm{z}}_0^{t/2}}\|^2}/\|{{\bm{F}} {\bm{z}}_0^0}\|^2$ for SEG-FFA and SEG-FF, as they pass through the whole dataset twice every epoch, and ${\|{{\bm{F}} {\bm{z}}_0^{t}}\|^2}/\|{{\bm{F}} {\bm{z}}_0^0}\|^2$ for the other methods, as they pass once every epoch.
• Figure 2: Experimental results in the monotone example, comparing the performance of SEG-RRA and SEG-USA with the results displayed in \ref{['fig:cc-shortlist']}. Because SEG-FFA and SEG-FF use two passes per epoch, for those two methods, we plot $\|{\bm{F}} {\bm{z}}_0^{t/2}\|^2/\|{\bm{F}} {\bm{z}}_0^0\|^2$.
• Figure 3: Experimental results in the monotone example, comparing SEG-FFA and the methods proposed by Hsie20. By the same reason as in \ref{['fig:monotone-result']}, we plot $\|{\bm{F}} {\bm{z}}_0^{t/2}\|^2/\|{\bm{F}} {\bm{z}}_0^0\|^2$ for SEG-FFA only.
• Figure 4: Experimental results on the strongly monotone problems with different stepsizes. Notice that \ref{['subfig:1e-3']} is exactly the plot that is included in \ref{['sec:experiments-shortlist']}. The only difference between the experiments conducted is the choice of the stepsize.

Theorems & Definitions (101)

• Theorem 4.1
• Proposition 5.1
• Proposition 5.2
• Proposition 5.3
• Theorem 5.4
• Theorem 5.5
• Theorem 5.6
• proof
• Lemma C.1: Polarization identity
• proof