RAMPAGE: RAndomized Mid-Point for debiAsed Gradient Extrapolation

TL;DR

This paper introduces RAMPAGE and RAMPAGE+ algorithms that address discretization bias in extragradient methods for variational inequalities, providing unbiased, variance-reduced solutions with provable convergence guarantees.

Contribution

The paper proposes RAMPAGE and RAMPAGE+, novel unbiased algorithms with variance reduction for VIs, improving convergence and extending to constrained and stochastic settings.

Findings

RAMPAGE and RAMPAGE+ are unbiased methods for VIs.

Both algorithms achieve $ ext{O}(1/k)$ convergence guarantees.

RAMPAGE+ reduces variance and removes internal first-order terms.

Abstract

A celebrated method for Variational Inequalities (VIs) is Extragradient (EG), which can be viewed as a standard discrete-time integration scheme. With this view in mind, in this paper we show that EG may suffer from discretization bias when applied to non-linear vector fields, conservative or otherwise. To resolve this discretization shortcoming, we introduce RAndomized Mid-Point for debiAsed Gradient Extrapolation (RAMPAGE) and its variance-reduced counterpart, RAMPAGE+, which leverages antithetic sampling. In contrast with EG, both methods are unbiased. Furthermore, leveraging negative correlation, RAMPAGE+ acts as an unbiased, geometric path-integrator that completely removes internal first-order terms from the variance, provably improving upon RAMPAGE. We further demonstrate that both methods enjoy provable $\mathcal{O}(1/k)$ convergence guarantees for a range of problems including root finding under co-coercive, co-hypomonotone, and generalized Lipschitzness regimes. Furthermore, we introduce symmetrically scaled variants to extend our results to constrained VIs. Finally, we provide convergence guarantees of both methods for stochastic and deterministic smooth convex-concave games. Somewhat interestingly, despite being a randomized method, RAMPAGE+ attains purely deterministic bounds for a number of the studied settings.