Quantization for Decentralized Learning Under Subspace Constraints

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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">differential randomized quantizers</i> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> , the strategy is stable both in terms of mean-square error and average bit rate: by reducing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> , it is possible to keep the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">estimation errors small (on the order of <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula>) without increasing indefinitely the bit rate as</i> <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu \rightarrow 0$</tex-math></inline-formula> <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">when variable-rate quantizers are used</i> . 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.