UCPH Statistics Seminar: Johan Segers

Title: Statistics for numerics: improving Monte Carlo integration by control variates

Abstract: Numerical integration of a function is a ubiquitous problem in applied mathematics, with applications in statistics, machine learning, finance, ... Monte Carlo methods have the advantage that they require very little regularity and provide a convergence rate that does not depend on the dimension of the integration domain. The use of control variates is one out of many methods to reduce the variance of the Monte Carlo estimate of the integral. In its simplest form, the control variate method can be cast as a multiple linear regression problem. Letting the number of control variates tend to infinity with the number of Monte Carlo particles can yield faster convergence rates but raises the problem of variable selection, which can be tackled by the lasso. Control variates can also be adapted to more sophisticated Monte Carlo schemes such as adaptive importance sampling. Finally, nearest neighbor estimates can act as control variates to speed up the convergence rate of the Monte Carlo procedure by an amount that depends on the intrinsic dimension of the domain and the regularity of the integrated.

Based on joint work with Rémi Leluc, Aymeric Dieuleveut, François Portier, and Aigerim Zhuman.