Global convergence of optimized adaptive importance samplers
This work addresses a foundational challenge in computational statistics for researchers and practitioners using importance sampling, providing rigorous convergence guarantees for general proposals, though it is incremental in extending known convexity results to non-convex cases.
The paper tackles the problem of Monte Carlo integration with general proposals by developing an optimized adaptive importance sampler (OAIS) that globally optimizes the χ²-divergence between target and proposal, deriving nonasymptotic bounds for mean-squared error and achieving explicit uniform-in-time theoretical guarantees.
We analyze the optimized adaptive importance sampler (OAIS) for performing Monte Carlo integration with general proposals. We leverage a classical result which shows that the bias and the mean-squared error (MSE) of the importance sampling scales with the $χ^2$-divergence between the target and the proposal and develop a scheme which performs global optimization of $χ^2$-divergence. While it is known that this quantity is convex for exponential family proposals, the case of the general proposals has been an open problem. We close this gap by utilizing the nonasymptotic bounds for stochastic gradient Langevin dynamics (SGLD) for the global optimization of $χ^2$-divergence and derive nonasymptotic bounds for the MSE by leveraging recent results from non-convex optimization literature. The resulting AIS schemes have explicit theoretical guarantees that are uniform-in-time.