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

In this paper we study the smooth convex-concave saddle point problem.\nSpecifically, we analyze the last iterate convergence properties of the\nExtragradient (EG) algorithm. It is well known that the ergodic (averaged)\niterates of EG converge at a rate of $O(1/T)$ (Nemirovski, 2004). In this\npaper, we show that the last iterate of EG converges at a rate of\n$O(1/\\sqrt{T})$. To the best of our knowledge, this is the first paper to\nprovide a convergence rate guarantee for the last iterate of EG for the smooth\nconvex-concave saddle point problem. Moreover, we show that this rate is tight\nby proving a lower bound of $\\Omega(1/\\sqrt{T})$ for the last iterate. This\nlower bound therefore shows a quadratic separation of the convergence rates of\nergodic and last iterates in smooth convex-concave saddle point problems.\n