  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.
