Adaptive importance sampling by kernel smoothing

This paper investigates a family of adaptive importance sampling algorithms for probability density function exploration. The proposed approach consists in modeling the sampling policy, the sequence of distributions used to generate the particles, as a mixture distribution between a flexible kernel density estimate (based on the previous particles), and a naive heavy tail density. When the share of samples generated according to the naive density goes to zero but not too quickly, two types of results are established: (i) uniform convergence rates are derived for the sampling policy estimate; (ii) a central limit theorem is obtained for the sampling policy estimate as well as for the resulting integral estimates. The fact that the asymptotic variance is the same as the variance of an oracle procedure, in which the sampling policy is chosen as the optimal one, illustrates the benefits of the approach. The practical behavior of the resulting algorithms is illustrated in a simulation study.