A Few Accelerated Algorithms for Convex Optimization under $(H_0,H_1)$-Smoothness

Abstract

We develop accelerated algorithms for convex $(H_0,H_1)$-smooth optimization, where $\|\nabla^2 f(x)\|\le H_0+H_1(f(x)-f^*)$. This class generalizes standard smoothness and contains the $(L_0,L_1)$-smooth class. Combining a Nesterov-type accelerated gradient scheme with small-dimensional relaxation and phase restarts, we obtain a full-gradient method with iteration complexity $\widetilde O(\sqrt{H_0\widetilde R^2/\varepsilon}+\sqrt{H_1\widetilde R^2}\log(F_0/\varepsilon))$. We extend the same approach to randomized coordinate optimization, obtaining a coordinate method with uniform sampling whose iteration complexity carries the standard factor $d$, and a coordinate method with non-uniform sampling whose iteration complexity is governed by $S_{1/2}^{(j)}=\sum_i\sqrt{H_{j,i}}$. These results provide, to our knowledge, the first accelerated full-gradient and coordinate guarantees for this convex class. We also provide practical implementation recommendations. Experiments confirm the predicted acceleration, gains from non-uniform sampling, and the viability of inexact relaxation.