Accelerated Primal-Dual Proximal Block Coordinate Updating Methods for Constrained Convex Optimization
This work provides an incremental improvement for researchers and practitioners in optimization, enhancing efficiency for large-scale problems with block structures.
The paper tackles the problem of solving linearly constrained convex optimization with multi-block variables by proposing an accelerated primal-dual block coordinate updating method, achieving an O(1/t^2) convergence rate under strong convexity and linear convergence under specific independence conditions, with numerical experiments showing stable performance without parameter tuning.
Block Coordinate Update (BCU) methods enjoy low per-update computational complexity because every time only one or a few block variables would need to be updated among possibly a large number of blocks. They are also easily parallelized and thus have been particularly popular for solving problems involving large-scale dataset and/or variables. In this paper, we propose a primal-dual BCU method for solving linearly constrained convex program in multi-block variables. The method is an accelerated version of a primal-dual algorithm proposed by the authors, which applies randomization in selecting block variables to update and establishes an $O(1/t)$ convergence rate under weak convexity assumption. We show that the rate can be accelerated to $O(1/t^2)$ if the objective is strongly convex. In addition, if one block variable is independent of the others in the objective, we then show that the algorithm can be modified to achieve a linear rate of convergence. The numerical experiments show that the accelerated method performs stably with a single set of parameters while the original method needs to tune the parameters for different datasets in order to achieve a comparable level of performance.