A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent
This work provides a foundational theory that unifies disparate SGD methods, benefiting researchers and practitioners in machine learning by simplifying analysis and enabling new algorithm development.
The paper introduces a unified theoretical framework for analyzing a wide range of stochastic gradient descent (SGD) variants, including methods with variance reduction, sampling, quantization, and coordinate descent, and demonstrates linear convergence under certain assumptions, recovering best-known complexity results for existing methods and enabling the development of five new variants.
In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent ({\tt 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 {\tt SGD} and randomized coordinate descent ({\tt RCD}) methods, the first unified theory of variance reduced and non-variance-reduced {\tt 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 {\tt SGD}, and through numerical experiments demonstrate some of their properties.