Randomized Dual Coordinate Ascent with Arbitrary Sampling

TL;DR

This paper introduces Quartz, a primal-dual coordinate ascent method with arbitrary sampling, providing new theoretical bounds and practical speedups for large-scale convex optimization, including serial, parallel, and distributed settings.

Contribution

The paper presents a novel primal-dual method with arbitrary sampling, offering direct primal-dual error bounds and efficient variants for various computational architectures.

Findings

Bounds match the best known for SDCA with importance sampling

Predicts initial and data-driven speedups with mini-batching

First distributed SDCA-like method with analysis for non-separable data

Abstract

We study the problem of minimizing the average of a large number of smooth convex functions penalized with a strongly convex regularizer. We propose and analyze a novel primal-dual method (Quartz) which at every iteration samples and updates a random subset of the dual variables, chosen according to an arbitrary distribution. In contrast to typical analysis, we directly bound the decrease of the primal-dual error (in expectation), without the need to first analyze the dual error. Depending on the choice of the sampling, we obtain efficient serial, parallel and distributed variants of the method. In the serial case, our bounds match the best known bounds for SDCA (both with uniform and importance sampling). With standard mini-batching, our bounds predict initial data-independent speedup as well as additional data-driven speedup which depends on spectral and sparsity properties of the data. We calculate theoretical speedup factors and find that they are excellent predictors of actual speedup in practice. Moreover, we illustrate that it is possible to design an efficient mini-batch importance sampling. The distributed variant of Quartz is the first distributed SDCA-like method with an analysis for non-separable data.