David Leslie (Bristol)
Applied abstract stochastic approximation

Stochastic approximation was introduced as a tool to find the zeroes of a function under only noisy observations of the function value. A classical statistical example is to find the zeroes of the score function when observations can only be processed sequentially. The method has since been developed and used mainly in the control theory, machine learning and economics literature to analyse iterative learning algorithms, but I contend that it is time for statistics to re-discover the power of stochastic approximation. I will introduce the main ideas of the method, and describe an extension; the parameter of interest is an element of a function space, and we wish to analyse its stochastic evolution through time. This extension allows the analysis of online nonparametric algorithms - we present an analysis of Newton's algorithm to estimate nonparametric mixing distributions. It also allows the investigation of learning in games with a continuous strategy set, where a mixed strategy is an arbitrary distribution on an interval.

(Joint work with Steven Perkins)