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 $\chi^2$-divergence between the target and the proposal and develop a scheme which performs global optimization of $\chi^2$-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 $\chi^2$-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.

翻译：我们分析了优化的自适应重要性采样器（OAIS），用于在一般提议分布下执行蒙特卡洛积分。我们利用一个经典结果，表明重要性采样的偏差和均方误差（MSE）与目标和提议之间的$\chi^2$散度成比例关系，并开发了一种执行$\chi^2$散度全局优化的方案。虽然已知对于指数族提议分布该量是凸的，但一般提议分布的情况一直是一个开放问题。我们通过利用随机梯度朗之万动力学（SGLD）的非渐近界对$\chi^2$散度进行全局优化，并借助非凸优化文献中的最新结果推导出MSE的非渐近界，从而填补了这一空白。由此产生的AIS方案具有显式的理论保证，且这些保证在时间上是一致的。