  Stochastic approximation algorithms are iterative procedures which are used
to approximate a target value in an environment where the target is unknown and
direct observations are corrupted by noise. These algorithms are useful, for
instance, for root-finding and function minimization when the target function
or model is not directly known. Originally introduced in a 1951 paper by
Robbins and Monro, the field of Stochastic approximation has grown enormously
and has come to influence application domains from adaptive signal processing
to artificial intelligence. As an example, the Stochastic Gradient Descent
algorithm which is ubiquitous in various subdomains of Machine Learning is
based on stochastic approximation theory. In this paper, we give a formal proof
(in the Coq proof assistant) of a general convergence theorem due to Aryeh
Dvoretzky, which implies the convergence of important classical methods such as
the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, we build
a comprehensive Coq library of measure-theoretic probability theory and
stochastic processes.
