Accelerated Bregman Proximal Gradient Methods for Relatively Smooth Convex Optimization

Abstract

We consider the problem of minimizing the sum of two convex functions: one isdifferentiable and relatively smooth with respect to a reference convexfunction, and the other can be nondifferentiable but simple to optimize. Therelatively smooth condition is much weaker than the standard assumption ofuniform Lipschitz continuity of the gradients, thus significantly increases thescope of potential applications. We present accelerated Bregman proximalgradient (ABPG) methods that employ the Bregman distance of the referencefunction as the proximity measure. These methods attain an $O(k^{-\gamma})$convergence rate in the relatively smooth setting, where $\gamma\in [1, 2]$ isdetermined by a triangle scaling property of the Bregman distance. We developadaptive variants of the ABPG method that automatically ensure the bestpossible rate of convergence and argue that the $O(k^{-2})$ rate is attainablein most cases. We present numerical experiments with three applications:D-optimal experiment design, Poisson linear inverse problem, andrelative-entropy nonnegative regression. In all experiments, we obtainnumerical certificates showing that these methods do converge with the$O(k^{-2})$ rate.