Efficient Interpolation between Extragradient and Proximal Methods for Weak MVIs

Thomas Pethick, Ioannis Mavrothalassitis, Volkan Cevher

Keywords: Weak Minty variational inequalities, cohypomonotone, nonmonotone, first-order methods, extragradient method, proximal point algorithm

TL;DR: We show the best known complexity for nonmontone games satisfying the weak Minty variational inequality using an adaptive scheme

Abstract: We study nonmonotone games satisfying the weak Minty variational inequality (MVI) with parameter $\rho \in (-\tfrac{1}{L}, \infty)$, where $L$ is the Lipschitz constant of the gradient operator. An error corrected version of the inexact proximal point algorithm is proposed, with which we establish the first $\mathcal O(1/\epsilon)$ rate for the entire range $\rho \in (-\tfrac{1}{L}, \infty)$, thus removing a logarithmic factor compared with the complexity of existing methods. The scheme automatically selects the needed accuracy for the proximal computation, and can recover the relaxed extragradient method when $\rho > -\tfrac{1}{2L}$ and the relaxed proximal point algorithm (rPPA) when $\rho > -\tfrac{1}{L}$. Due to the error correction, the scheme inherits the strong properties of the _exact_ rPPA. Specifically, we show that linear convergence is automatically achieved under appropriate conditions. Tightness for the range of $\rho$ is established through a lower bound for rPPA. Central to the algorithmic construction is a halfspace projection, where the key insight is that the allowed error tolerance can both be used to correct for the proximal approximation and to enlarge the problem class.

Primary Area: optimization

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Submission Number: 9992