Accuracy Certificates for Convex Optimization at Accelerated Rates via Primal-Dual Averaging
Matthew X. Burns, Jiaming Liang

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 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 certificate complexity. Furthermore, we prove primal-dual…
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