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 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.
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Figure 1: Comparison of ( EG ) and ( RAMPAGE+ ). See Appendix H for details. (a) denotes an unconstrained optimization task with a nonconvex 4th-order polynomial objective, (b) denotes a nonconvex-nonconcave min-max game involving high-frequency sinusoids (motivated by infinite-width GANs), (c) and (d) denote a 2-dimensional nonconvex-nonconcave min-max game involving high-frequency sinusoids (again motivated by infinite-width GANs), and (e) and (f) correspond to 30-dimensional logistic regression tasks on the Breast Cancer Wisconsin dataset [ 100 ] under KL-regularized distributionally robust optimization and adversarial training formulations, respectively. In all settings, we find two stepsizes for ( EG ) on the edge of stability. The chosen larger stepsize makes ( EG ) unstable ( blue curves ), consistent with the discretization-bias mechanism discussed above, while ( RAMPAGE+ ) enjoys convergence ( red curves ). Furthermore, ( RAMPAGE+ ) allows larger stepsizes due to its higher stability ( green curves ).