Multi-Timescale Primal Dual Hybrid Gradient with Application to Distributed Optimization

TL;DR

This paper introduces multi-timescale variants of the PDHG algorithm that are robust to delays and heterogeneity, improving distributed optimization efficiency and achieving optimal gradient similarity dependency.

Contribution

The paper develops novel multi-timescale PDHG algorithms with convergence guarantees under arbitrary update rates and applies them to distributed optimization with improved communication efficiency.

Findings

Algorithms converge under arbitrary dual update rates.

Enhanced efficiency in distributed optimization with heterogeneous objectives.

Achieves linear, optimal dependency on gradient similarity for non-smooth objectives.

Abstract

We propose two variants of the Primal Dual Hybrid Gradient (PDHG) algorithm for saddle point problems with block decomposable duals, hereafter called Multi-Timescale PDHG (MT-PDHG) and its accelerated variant (AMT-PDHG). Through novel mixtures of Bregman divergence and multi-timescale extrapolations, our MT-PDHG and AMT-PDHG converge under arbitrary updating rates for different dual blocks while remaining fully deterministic and robust to extreme delays in dual updates. We further apply our (A)MT-PDHG, augmented with the gradient sliding techniques introduced in Lan et al. (2020), Lan (2016), to distributed optimization. The flexibility in choosing different updating rates for different blocks allows a more refined control over the communication rounds between different pairs of agents, thereby improving the efficiencies in settings with heterogeneity in local objectives and communication costs. Moreover, with careful choices of penalty levels, our algorithms show linear and thus optimal dependency on function similarities, a measure of how similar the gradients of local objectives are. This provides a positive answer to the open question whether such dependency is achievable for non-smooth objectives (Arjevani and Shamir 2015).