Stochastic Approximation Beyond Gradient for Signal Processing and Machine Learning

Stochastic Approximation (
SA
) is a classical algorithm that has had since the early days a huge impact on signal processing, and nowadays on machine learning, due to the necessity to deal with a large amount of data observed with uncertainties. An exemplar special case of
SA
pertains to the popular stochastic (sub)gradient algorithm which is the working horse behind many important applications. A lesser-known fact is that the
SA
scheme also extends to non-stochastic-gradient algorithms such as compressed stochastic gradient, stochastic expectation-maximization, and a number of reinforcement learning algorithms. The aim of this article is to overview and introduce the non-stochastic-gradient perspectives of
SA
to the signal processing and machine learning audiences through presenting a design guideline of
SA
algorithms backed by theories. Our central theme is to propose a general framework that unifies existing theories of
SA
, including its non-asymptotic and asymptotic convergence results, and demonstrate their applications on popular non-stochastic-gradient algorithms. We build our analysis framework based on classes of Lyapunov functions that satisfy a variety of mild conditions. We draw connections between non-stochastic-gradient algorithms and scenarios when the Lyapunov function is smooth, convex, or strongly convex. Using the said framework, we illustrate the convergence properties of the non-stochastic-gradient algorithms using concrete examples. Extensions to the emerging variance reduction techniques for improved sample complexity will also be discussed.