Mirror Frameworks for Relatively Lipschitz and Monotone-Like Variational Inequalities

Abstract

Nonconvex–nonconcave saddle-point optimization in machine learning has triggered lots of research for studying non-monotone variational inequalities (VIs). In this work, we introduce two mirror frameworks, called mirror extragradient method and mirror extrapolation method, for approximating solutions to relatively Lipschitz and monotone-like VIs. The former covers the well-known Nemirovski’s mirror prox method and Nesterov’s dual extrapolation method, and the recently proposed Bregman extragradient method; all of them can be reformulated into a scheme that is very similar to the original form of extragradient method. The latter includes the operator extrapolation method and the Bregman extrapolation method as its special cases. The proposed mirror frameworks allow us to present a unified and improved convergence analysis for all these existing methods under relative Lipschitzness and monotone-like conditions that may be the currently weakest assumptions guaranteeing (sub)linear convergence.

Appendix

Proof of Lemma 3

Fix $u, v, z\in {\mathbb {R}}^d$. By Cauchy-Schwartz inequality, Lipschitzness of F, and strong convexity of $\omega$, we derive that for any $u^*\in \partial \omega (u), v^*\in \partial \omega (v)$,

$$\begin{aligned}{} & {} \qquad \langle F(v)-F(u),v-z\rangle \\{} & {} \quad \leqslant \Vert F(v)-F(u)\Vert _*\Vert z-v\Vert \leqslant L\Vert v-u\Vert \Vert z-v\Vert \\{} & {} \quad \leqslant L(\frac{1}{2}\Vert v-u\Vert ^2 +\frac{1}{2}\Vert z-v\Vert ^2) \leqslant \frac{L}{\mu }(D_\omega ^{u^*}(v,u)+ D_\omega ^{v^*}(z,v)) \end{aligned}$$

from which the result follows.

Proof of Lemma 4

Fix $u, v, z\in {\mathbb {R}}^d$. By the definition Bregman distance and relative smoothness of $\phi$, we derive that for any $u^*\in \partial \omega (u), v^*\in \partial \omega (v)$,

$$\begin{aligned}{} & {} \,\quad L(D_\omega ^{u^*}(v,u)+ D_\omega ^{v^*}(z,v))) \\{} & {} \geqslant \phi (v)-[\phi (u)+\langle \nabla \phi (u),v-u\rangle ]+ \phi (z)-[\phi (v)+\langle \nabla \phi (v), z-v\rangle ]\\{} & {} = \phi (z)-\phi (u)-\langle \nabla \phi (u),v-u\rangle - \langle \nabla \phi (v), z-v\rangle \\{} & {} = D_\phi (z,u)+\langle \nabla \phi (u),z-u\rangle -\langle \nabla \phi (u),v-u\rangle - \langle \nabla \phi (v),z-v\rangle \\{} & {} = D_\phi (z,u)+\langle \nabla \phi (v)-\nabla \phi (u),v-z\rangle . \end{aligned}$$

Note that $F=\nabla \phi$ and $D_\phi (z,u)\geqslant 0$ due to the convexity of $\phi$. The conclusion follows.