Asymptotic optimality of adaptive importance sampling
This work addresses a fundamental theoretical problem in adaptive importance sampling for researchers in computational statistics and machine learning, providing a rigorous asymptotic guarantee.
The paper tackles the theoretical analysis of adaptive importance sampling (AIS) by proving its asymptotic optimality, showing that AIS performs as well as an oracle strategy that knows the optimal sampling policy from the start, without assumptions on resource allocation between exploration and exploitation. It also introduces weighted AIS, a practical method to discard poor early samples.
Adaptive importance sampling (AIS) uses past samples to update the \textit{sampling policy} $q_t$ at each stage $t$. Each stage $t$ is formed with two steps : (i) to explore the space with $n_t$ points according to $q_t$ and (ii) to exploit the current amount of information to update the sampling policy. The very fundamental question raised in this paper concerns the behavior of empirical sums based on AIS. Without making any assumption on the allocation policy $n_t$, the theory developed involves no restriction on the split of computational resources between the explore (i) and the exploit (ii) step. It is shown that AIS is asymptotically optimal : the asymptotic behavior of AIS is the same as some "oracle" strategy that knows the targeted sampling policy from the beginning. From a practical perspective, weighted AIS is introduced, a new method that allows to forget poor samples from early stages.