A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent

Eduard Gorbunov, Filip Hanzely, Peter Richtarik
Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, PMLR 108:680-690, 2020.

Abstract

In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent (SGD) which so far have required different intuitions, convergence analyses, have different applications, and which have been developed separately in various communities. We show that our framework includes methods with and without the following tricks, and their combinations: variance reduction, importance sampling, mini-batch sampling, quantization, and coordinate sub-sampling. As a by-product, we obtain the first unified theory of SGD and randomized coordinate descent (RCD) methods, the first unified theory of variance reduced and non-variance-reduced SGD methods, and the first unified theory of quantized and non-quantized methods. A key to our approach is a parametric assumption on the iterates and stochastic gradients. In a single theorem we establish a linear convergence result under this assumption and strong-quasi convexity of the loss function. Whenever we recover an existing method as a special case, our theorem gives the best known complexity result. Our approach can be used to motivate the development of new useful methods, and offers pre-proved convergence guarantees. To illustrate the strength of our approach, we develop five new variants of SGD, and through numerical experiments demonstrate some of their properties.

Cite this Paper


BibTeX
@InProceedings{pmlr-v108-gorbunov20a, title = {A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent}, author = {Gorbunov, Eduard and Hanzely, Filip and Richtarik, Peter}, booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics}, pages = {680--690}, year = {2020}, editor = {Silvia Chiappa and Roberto Calandra}, volume = {108}, series = {Proceedings of Machine Learning Research}, month = {26--28 Aug}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v108/gorbunov20a/gorbunov20a.pdf}, url = { http://proceedings.mlr.press/v108/gorbunov20a.html }, abstract = {In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent (SGD) which so far have required different intuitions, convergence analyses, have different applications, and which have been developed separately in various communities. We show that our framework includes methods with and without the following tricks, and their combinations: variance reduction, importance sampling, mini-batch sampling, quantization, and coordinate sub-sampling. As a by-product, we obtain the first unified theory of SGD and randomized coordinate descent (RCD) methods, the first unified theory of variance reduced and non-variance-reduced SGD methods, and the first unified theory of quantized and non-quantized methods. A key to our approach is a parametric assumption on the iterates and stochastic gradients. In a single theorem we establish a linear convergence result under this assumption and strong-quasi convexity of the loss function. Whenever we recover an existing method as a special case, our theorem gives the best known complexity result. Our approach can be used to motivate the development of new useful methods, and offers pre-proved convergence guarantees. To illustrate the strength of our approach, we develop five new variants of SGD, and through numerical experiments demonstrate some of their properties. } }
Endnote
%0 Conference Paper %T A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent %A Eduard Gorbunov %A Filip Hanzely %A Peter Richtarik %B Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2020 %E Silvia Chiappa %E Roberto Calandra %F pmlr-v108-gorbunov20a %I PMLR %P 680--690 %U http://proceedings.mlr.press/v108/gorbunov20a.html %V 108 %X In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent (SGD) which so far have required different intuitions, convergence analyses, have different applications, and which have been developed separately in various communities. We show that our framework includes methods with and without the following tricks, and their combinations: variance reduction, importance sampling, mini-batch sampling, quantization, and coordinate sub-sampling. As a by-product, we obtain the first unified theory of SGD and randomized coordinate descent (RCD) methods, the first unified theory of variance reduced and non-variance-reduced SGD methods, and the first unified theory of quantized and non-quantized methods. A key to our approach is a parametric assumption on the iterates and stochastic gradients. In a single theorem we establish a linear convergence result under this assumption and strong-quasi convexity of the loss function. Whenever we recover an existing method as a special case, our theorem gives the best known complexity result. Our approach can be used to motivate the development of new useful methods, and offers pre-proved convergence guarantees. To illustrate the strength of our approach, we develop five new variants of SGD, and through numerical experiments demonstrate some of their properties.
APA
Gorbunov, E., Hanzely, F. & Richtarik, P.. (2020). A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 108:680-690 Available from http://proceedings.mlr.press/v108/gorbunov20a.html .

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