Block-proximal methods with spatially adapted acceleration

We study and develop (stochastic) primal–dual block-coordinate descent methods for convex problems based on the method due to Chambolle and Pock. Our methods have known convergence rates for the iterates and the ergodic gap: $O(1\mathrm{/}{N}^2)$ if each block is strongly convex, $O(1\mathrm{/}N)$ if no convexity is present, and more generally a mixed rate $O(1\mathrm{/}{N}^2)+O(1\mathrm{/}N)$ for strongly convex blocks, if only some blocks are strongly convex. Additional novelties of our methods include blockwise-adapted step lengths and acceleration, as well as the ability to update both the primal and dual variables randomly in blocks under a very light compatibility condition. In other words, these variants of our methods are doubly-stochastic. We test the proposed methods on various image processing problems, where we employ pixelwise-adapted acceleration.