Optimal complexity and certification of Bregman first-order methods

We provide a lower bound showing that the O(1/k) convergence rate of NoLips method (a.k.a. Bregman Gradient or Mirror Descent) is optimal for class problems satisfying relative smoothness assumption. This assumption appeared in recent developments around method, where acceleration remained an open issue. The main inspiration behind this stems from extension performance estimation framework Drori and Teboulle (Mathematical Programming, 2014) to first-order methods. technique allows computing worst-case scenarios context relatively-smooth minimization. In particular, we used numerically generated examples as basis obtaining general bound.