VI problems; logarithmic factors ignored throughout. A box indicates a contribution from this paper.

The Past Extragradient (PEG) [Popov, 1980] method, also known as the Optimistic Gradient method, has known a recent gain in interest in the optimization community with the emergence of variational inequality formulations for machine learning. Recently, in the unconstrained case, Golowich et al. [2020] proved that a $O(1/N)$ last-iterate convergence rate in terms of the squared norm of the operator can be achieved for Lipschitz and monotone operators with a Lipchitz Jacobian. In this work, by introducing a novel analysis through potential functions, we show that (i) this $O(1/N)$ last-iterate convergence can be achieved without any assumption on the Jacobian of the operator, and (ii) it can be extended to the constrained case, which was not derived before even under Lipschitzness of the Jacobian. The proof is significantly different from the one known from Golowich et al. [2020], and its discovery was computer-aided. Those results close the open question of the last iterate convergence of PEG for monotone variational inequalities.

VI problems; logarithmic factors ignored throughout. A box indicates a contribution from this paper.

Recently, in the unconstrained case, Golowich et al. [2020a] proved that a

We study the convergence behavior of a generalized Frank-Wolfe algorithm in constrained (stochastic) monotone variational inequality (MVI) problems. In recent years, there have been numerous efforts to design algorithms for solving constrained MVI problems due to their connections with optimization, machine learning, and equilibrium computation in games. Most work in this domain has focused on extensions of simultaneous gradient play, with particular emphasis on understanding the convergence properties of extragradient and optimistic gradient methods. In contrast, we examine the performance of an algorithm from another well-known class of optimization algorithms: Frank-Wolfe. We show that a generalized variant of this algorithm achieves a fast \mathcal{O}(T {-1/2}) last-iterate convergence rate in constrained MVI problems.

The Past Extragradient (PEG) [Popov, 1980] method, also known as the Optimistic Gradient method, has known a recent gain in interest in the optimization community with the emergence of variational inequality formulations for machine learning. Recently, in the unconstrained case, Golowich et al. [2020] proved that a O(1/N) last-iterate convergence rate in terms of the squared norm of the operator can be achieved for Lipschitz and monotone operators with a Lipchitz Jacobian. In this work, by introducing a novel analysis through potential functions, we show that (i) this O(1/N) last-iterate convergence can be achieved without any assumption on the Jacobian of the operator, and (ii) it can be extended to the constrained case, which was not derived before even under Lipschitzness of the Jacobian. The proof is significantly different from the one known from Golowich et al. [2020], and its discovery was computer-aided. Those results close the open question of the last iterate convergence of PEG for monotone variational inequalities.

Variational inequalities (VIs) are a broad class of optimization problems encompassing machine learning problems ranging from standard convex minimization to more complex scenarios like min-max optimization and computing the equilibria of multi-player games. In convex optimization, strong convexity allows for fast statistical learning rates requiring only $\Theta(1/\epsilon)$ stochastic first-order oracle calls to find an $\epsilon$-optimal solution, rather than the standard $\Theta(1/\epsilon^2)$ calls. In this paper, we explain how one can similarly obtain fast $\Theta(1/\epsilon)$ rates for learning VIs that satisfy strong monotonicity, a generalization of strong convexity. Specifically, we demonstrate that standard stability-based generalization arguments for convex minimization extend directly to VIs when the domain admits a small covering, or when the operator is integrable and suboptimality is measured by potential functions; such as when finding equilibria in multi-player games.

We provide improved convergence rates for constrained convex-concave min-max problems and monotone variational inequalities with higher-order smoothness. In min-max settings where the $p^{th}$-order derivatives are Lipschitz continuous, we give an algorithm HigherOrderMirrorProx that achieves an iteration complexity of $O(1/T^{\frac{p+1}{2}})$ when given access to an oracle for finding a fixed point of a $p^{th}$-order equation. We give analogous rates for the weak monotone variational inequality problem. For $p>2$, our results improve upon the iteration complexity of the first-order Mirror Prox method of Nemirovski [2004] and the second-order method of Monteiro and Svaiter [2012]. We further instantiate our entire algorithm in the unconstrained $p=2$ case.

In this paper, we consider first-order algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex (resp. weakly concave) in terms of the variable of minimization (resp. maximization). It has many important applications in machine learning, statistics, and operations research. One such example that attracts tremendous attention recently in machine learning is training Generative Adversarial Networks. We propose an algorithmic framework motivated by the inexact proximal point method, which solves the weakly monotone variational inequality corresponding to the original min-max problem by approximately solving a sequence of strongly monotone variational inequalities constructed by adding a strongly monotone mapping to the original gradient mapping. In this sequence, each strongly monotone variational inequality is defined with a proximal center that is updated using the approximate solution of the previous variational inequality. Our algorithm generates a sequence of solution that provably converges to a nearly stationary solution of the original min-max problem. The proposed framework is flexible because various subroutines can be employed for solving the strongly monotone variational inequalities. The overall computational complexities of our methods are established when the employed subroutines are subgradient method, stochastic subgradient method, gradient descent method and Nesterov's accelerated method and variance reduction methods for a Lipschitz continuous operator. To the best of our knowledge, this is the first work that establishes the non-asymptotic convergence to a nearly stationary point of a non-convex non-concave min-max problem.