Last Iterate is Slower than Averaged Iterate in Smooth Convex-Concave Saddle Point Problems

2020/01/31 by Golowich, Noah, Pattathil, Sarath, Daskalakis, Constantinos +1 · 2 citations
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2002.00057

Abstract

In this paper we study the smooth convex-concave saddle point problem. Specifically, we analyze the last iterate convergence properties of the Extragradient (EG) algorithm. It is well known that the ergodic (averaged) iterates of EG converge at a rate of O(1/T) (Nemirovski, 2004). In this paper, we show that the last iterate of EG converges at a rate of O(1/√(T)). To the best of our knowledge, this is the first paper to provide a convergence rate guarantee for the last iterate of EG for the smooth convex-concave saddle point problem. Moreover, we show that this rate is tight by proving a lower bound of Ω(1/√(T)) for the last iterate. This lower bound therefore shows a quadratic separation of the convergence rates of ergodic and last iterates in smooth convex-concave saddle point problems.

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