Accuracy Certificates for Convex Optimization at Accelerated Rates via Primal-Dual Averaging

Abstract: Many works in convex optimization provide rates for achieving a small primal gap. However, this quantity is typically unavailable in practice. In this work, we show that solving a regularized surrogate with algorithms based on simple primal-dual averaging provides non-asymptotic convergence guarantees for a \textit{computable} optimality certificate. We first analyze primal and dual methods based on one average, namely modified dual averaging and generalized conditional gradient, and establish $\tilde{O}(\varepsilon^{-1})$ certificate complexities. Motivated by asymmetries in the one-average case, we analyze a self-dual, two-average method that preserves symmetry while losing certificate guarantees. To recover certificate convergence, we propose a three-average method that achieves an accelerated $\tilde{O}(\varepsilon^{-1/2})$ certificate complexity. Furthermore, we prove primal-dual algorithm correspondences for the one, two, and three-average cases. In particular, the primal three-average accelerated method mirrors the well-known gradient extrapolation method in the dual. By interpreting our results through the lens of zero-sum matrix games and Fisher markets, we further connect primal-dual averaging methods to game theory and market dynamics.