Quantization for decentralized learning under subspace constraints

TL;DR

This paper introduces a decentralized learning method with quantization to reduce communication costs, ensuring stability and accuracy under subspace constraints and general task relatedness models.

Contribution

It proposes an adaptive decentralized strategy using differential randomized quantizers for communication compression in subspace-constrained optimization.

Findings

Strategy is stable with small step-sizes and quantization noise.

Estimation errors are proportional to step-size $.

Decentralized learning is effective with minimal bit usage.

Abstract

In this paper, we consider decentralized optimization problems where agents have individual cost functions to minimize subject to subspace constraints that require the minimizers across the network to lie in low-dimensional subspaces. This constrained formulation includes consensus or single-task optimization as special cases, and allows for more general task relatedness models such as multitask smoothness and coupled optimization. In order to cope with communication constraints, we propose and study an adaptive decentralized strategy where the agents employ differential randomized quantizers to compress their estimates before communicating with their neighbors. The analysis shows that, under some general conditions on the quantization noise, and for sufficiently small step-sizes $\mu$, the strategy is stable both in terms of mean-square error and average bit rate: by reducing $\mu$, it is possible to keep the estimation errors small (on the order of $\mu$) without increasing indefinitely the bit rate as $\mu\rightarrow 0$. Simulations illustrate the theoretical findings and the effectiveness of the proposed approach, revealing that decentralized learning is achievable at the expense of only a few bits.