Optimistic Dual Extrapolation for Coherent Non-monotone Variational Inequalities

The optimization problems associated with training generative adversarial neural networks can be largely reduced to certain non-monotone variational inequality problems (VIPs), whereas existing convergence results are mostly based on monotone or strongly monotone assumptions. In this paper, we propose optimistic dual extrapolation (OptDE), a method that only performs one gradient evaluation per iteration. We show that OptDE is provably convergent to a strong solution under different coherent non-monotone assumptions. In particular, when a weak solution exists, the convergence rate of our method is $O(1/{\epsilon^{2}})$, which matches the best existing result of the methods with two gradient evaluations. Further, when a $\sigma$-weak solution exists, the convergence guarantee is improved to the linear rate $O(\log\frac{1}{\epsilon})$. Along the way--as a byproduct of our inquiries into non-monotone variational inequalities--we provide the near-optimal $O\big(\frac{1}{\epsilon}\log \frac{1}{\epsilon}\big)$ convergence guarantee in terms of restricted strong merit function for monotone variational inequalities. We also show how our results can be naturally generalized to the stochastic setting, and obtain corresponding new convergence results. Taken together, our results contribute to the broad landscape of variational inequality--both non-monotone and monotone alike--by providing a novel and more practical algorithm with the state-of-the-art convergence guarantees.