Uncertainty quantification for stochastic approximation limits using chaos expansion

The uncertainty quantification for the limit of a stochastic approximation (SA) algorithm is analyzed. In our setup, this limit ζ* is defined as a zero of an intractable function and is modeled as uncertain through a parameter θ. We aim at deriving the function ζ*, as well as the probabilistic distribution of ζ* (θ) given a probability distribution π for θ. We introduce the so-called uncertainty quantification for SA (UQSA) algorithm, an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of θ → ζ* (θ) on an orthogonal basis of a suitable Hilbert space. UQSA, run with a finite number of iterations K, returns a finite set of coefficients, providing an approximation ζ*K (∙) of ζ* (∙). We establish the almost-sure and L^p-convergences in the Hilbert space of the sequence of functions ζ*K (∙) when the number of iterations K tends to infinity. This is done under mild, tractable conditions, uncovered by the existing literature for convergence analysis of infinite dimensional SA algorithms. For a suitable choice of the Hilbert basis, the algorithm also provides an approximation of the expectation, of the variance-covariance matrix, and of higher order moments of the quantity ζ*K (θ) when θ is random with distribution π. UQSA is illustrated and the role of its design parameters is discussed numerically.