Curvature-Independent Last-Iterate Convergence for Games on Riemannian Manifolds

Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before.

Accelerated Algorithms for Monotone Inclusions and Constrained Nonconvex-Nonconcave Min-Max Optimization

We study monotone inclusions and monotone variational inequalities, as well as their generalizations to non-monotone settings. We first show that the Extra Anchored Gradient (EAG) algorithm, originally proposed by Yoon and Ryu [2021] for unconstrained convex-concave min-max optimization, can be applied to solve the more general problem of Lipschitz monotone inclusion. More specifically, we prove that the EAG solves Lipschitz monotone inclusion problems with an \emph{accelerated convergence rate} of $O(\frac{1}{T})$, which is \emph{optimal among all first-order methods} [Diakonikolas, 2020, Yoon and Ryu, 2021]. Our second result is a new algorithm, called Extra Anchored Gradient Plus (EAG+), which not only achieves the accelerated $O(\frac{1}{T})$ convergence rate for all monotone inclusion problems, but also exhibits the same accelerated rate for a family of general (non-monotone) inclusion problems that concern negative comonotone operators. As a special case of our second result, EAG+ enjoys the $O(\frac{1}{T})$ convergence rate for solving a non-trivial class of nonconvex-nonconcave min-max optimization problems. Our analyses are based on simple potential function arguments, which might be useful for analysing other accelerated algorithms.

Tight Last-Iterate Convergence of the Extragradient Method for Constrained Monotone Variational Inequalities

The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient method by Korpelevich [1976] is one of the most popular methods for solving monotone variational inequalities. Despite its long history and intensive attention from the optimization and machine learning community, the following major problem remains open. What is the last-iterate convergence rate of the extragradient method for monotone and Lipschitz variational inequalities with constraints? We resolve this open problem by showing a tight $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence rate for arbitrary convex feasible sets, which matches the lower bound by Golowich et al. [2020]. Our rate is measured in terms of the standard gap function. The technical core of our result is the monotonicity of a new performance measure -- the tangent residual, which can be viewed as an adaptation of the norm of the operator that takes the local constraints into account. To establish the monotonicity, we develop a new approach that combines the power of the sum-of-squares programming with the low dimensionality of the update rule of the extragradient method. We believe our approach has many additional applications in the analysis of iterative methods.