A2BCD: an Asynchronous Accelerated Block Coordinate Descent Algorithm With Optimal Complexity

Robert Hannah, Fei Feng, and Wotao Yin

Submitted

Overview

We propose the Asynchronous Accelerated Nonuniform Randomized Block Coordinate Descent algorithm (A2BCD), the first asynchronous Nesterov-accelerated algorithm that achieves optimal complexity. This parallel algorithm solves the unconstrained convex minimization problem, using p computing nodes which compute updates to shared solution vectors, in an asynchronous fashion with no central coordination. Nodes in asynchronous algorithms do not wait for updates from other nodes before starting a new iteration, but simply compute updates using the most recent solution information available. This allows them to complete iterations much faster than traditional ones, especially at scale, by eliminating the costly synchronization penalty of traditional algorithms.

We first prove that A2BCD converges linearly to a solution with a fast accelerated rate that matches the recently proposed NU_ACDM, so long as the maximum delay is not too large. Somewhat surprisingly, A2BCD pays no complexity penalty for using outdated information. We then prove lower complexity bounds for randomized coordinate descent methods, which show that A2BCD (and hence NU_ACDM) has optimal complexity to within a constant factor. We confirm with numerical experiments that A2BCD outperforms NU_ACDM, which is the current fastest coordinate descent algorithm, even at small scale.

We also derive and analyze a second-order ordinary differential equation,

 ddot Y + 2n^{-1}kappa^{-1}dot Y + 2n^{-2}kappa^{-1}nabla f(hat{Y}) = 0,

where the time dependent variable Y has n blocks, kappa is the condition number of the objective function f, and hat{Y} is a time-delayed copy of Y. This equation is the continuous-time limit of our accelerated algorithm. f(Y(t))-min{f} converges to zero at an accelerated linear rate of O(exp(-t/n/sqrt{kappa})).

Citation

R. Hannah, F. Feng, and W. Yin, A2BCD: an Asynchronous Accelerated Block Coordinate Descent Algorithm With Optimal Complexity, UCLA CAM Report 18-18, 2018.


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