New Bounds For Distributed Mean Estimation and Variance Reduction

Davies, Peter; Gurunathan, Vijaykrishna; Moshrefi, Niusha; Ashkboos, Saleh; Alistarh, Dan

Abstract

We consider the problem of distributed mean estimation (DME), in which n machines are each given a local d-dimensional vector x v ∈ R d , and must cooperate to estimate the mean of their inputs µ = 1 n n v=1 x v , while minimizing total communication cost. DME is a fundamental construct in distributed machine learning, and there has been considerable work on variants of this problem, especially in the context of distributed variance reduction for stochastic gradients in parallel SGD. Previous work typically assumes an upper bound on the norm of the input vectors, and achieves an error bound in terms of this norm. However, in many real applications, the input vectors are concentrated around the correct output µ, but µ itself has large norm. In such cases, previous output error bounds perform poorly. In this paper, we show that output error bounds need not depend on input norm. We provide a method of quantization which allows distributed mean estimation to be performed with solution quality dependent only on the distance between inputs, not on input norm, and show an analogous result for distributed variance reduction. The technique is based on a new connection with lattice theory. We also provide lower bounds showing that the communication to error trade-off of our algorithms is asymptotically optimal. As the lattices achieving optimal bounds under 2-norm can be computationally impractical, we also present an extension which leverages easy-to-use cubic lattices, and is loose only up to a logarithmic factor in d. We show experimentally that our method yields practical improvements for common applications, relative to prior approaches.

Citation

Davies, P., Gurunathan, V., Moshrefi, N., Ashkboos, S., & Alistarh, D. (2021, May). New Bounds For Distributed Mean Estimation and Variance Reduction. Presented at 9th International Conference on Learning Representations (ICLR), Vienna, Austria

• Presentation Conference Type: Conference Paper (published)
• Conference Name: 9th International Conference on Learning Representations (ICLR)
• Start Date: May 3, 2021
• End Date: May 7, 2021
• Acceptance Date: Jan 7, 2021
• Online Publication Date: May 1, 2021
• Publication Date: May 1, 2021
• Deposit Date: Jan 10, 2025
• Publicly Available Date: Jan 10, 2025
• Peer Reviewed: Peer Reviewed
• Book Title: ICLR 2021 - The Ninth International Conference on Learning Representations
• Public URL: https://durham-repository.worktribe.com/output/3329433
• Related Public URLs: https://research-explorer.ista.ac.at/record/9543
• Other Repo URL: https://research-explorer.ista.ac.at/record/9543