Uncertainty Quantification for Stochastic Approximation Limits Using Chaos Expansion
Résumé
We analyze the uncertainty quantification for the limit of a stochastic approximation (SA for short) algorithm. Typically, this limit φ is deter-ministic and given as the zero of a function written as an expectation. In our setup, the limit φ is modeled as uncertain through a parameter θ. We aim at deriving the probabilistic distribution of φ (θ), given a probability distribution π for θ. We introduce an SA algorithm in increasing dimension for computing the basis coefficients of a chaos expansion of φ on an orthogonal basis of a suitable Hilbert space. The procedure returns a series of estimated coefficients, the corresponding approximation φ (·) of φ (·), and simple approximations of the expectation and variance of { φ (θ); θ ∼ π} (as well as higher order moments when the basis is made of polynomials). The evaluation of more general statistics is possible using φ (·) and extra i.i.d. Monte-Carlo draws of θ ∼ π. Under explicit assumptions on stochastic approximation limits without uncertainty, we establish the almost sure convergence in the Hilbert space of the algorithm.
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