A2BCD: Asynchronous Acceleration with Optimal Complexity

In this paper, we propose the Asynchronous Accelerated Nonuniform Randomized Block Coordinate Descent algorithm (A2BCD). We prove A2BCD converges linearly to a solution of the convex minimization problem at the same rate as NU_ACDM, so long as the maximum delay is not too large. This is the first asynchronous Nesterov-accelerated algorithm that attains any provable speedup. Moreover, we then prove that these algorithms both have optimal complexity. Asynchronous algorithms complete much faster iterations, and A2BCD has optimal complexity. Hence we observe in experiments that A2BCD is the top-performing coordinate descent algorithm, converging up to 4-5x faster than NU_ACDM on some data sets in terms of wall-clock time. To motivate our theory and proof techniques, we also derive and analyze a continuous-time analog of our algorithm and prove it converges at the same rate.