论文标题
优化自适应重要性采样器的全球收敛
Global convergence of optimized adaptive importance samplers
论文作者
论文摘要
我们分析了优化的自适应重要性采样器（OAI），以与一般建议进行蒙特卡洛整合。我们利用经典结果表明，重要性采样量表的偏差和均方误差（MSE）具有目标与提案之间的$χ^2 $ - 差异，并开发了一个方案，该方案执行$χ^2 $ didivergence的全局优化。尽管众所周知，此数量是指数级家庭建议的凸，但一般建议的情况一直是一个开放的问题。我们通过利用非反应界限来弥合这一差距，用于随机梯度Langevin Dynamics（SGLD），以全局优化$χ^2 $ divergence，并通过利用非convex优化文献的最新结果来为MSE提供非杂种范围。由此产生的AIS方案具有统一的明确理论保证。
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.
