On Nesting Monte Carlo Estimators
This work addresses a fundamental methodological challenge for researchers and practitioners in machine learning and statistics dealing with complex nested expectation problems, offering incremental improvements through new formulations and guidelines.
The paper tackles the problem of estimating nested expectations in machine learning and statistics, which conventional Monte Carlo methods cannot handle, by establishing convergence conditions and rates for nested Monte Carlo estimators and providing guidelines to avoid pitfalls. It also introduces novel methods to reformulate nested expectations into single expectations, leading to improved convergence rates, and demonstrates applicability with a new estimator for Bayesian experimental design and error bounds for variational objectives.
Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves involve calculation of a separate, nested, estimation. We investigate the statistical implications of nesting MC estimators, including cases of multiple levels of nesting, and establish the conditions under which they converge. We derive corresponding rates of convergence and provide empirical evidence that these rates are observed in practice. We further establish a number of pitfalls that can arise from naive nesting of MC estimators, provide guidelines about how these can be avoided, and lay out novel methods for reformulating certain classes of nested expectation problems into single expectations, leading to improved convergence rates. We demonstrate the applicability of our work by using our results to develop a new estimator for discrete Bayesian experimental design problems and derive error bounds for a class of variational objectives.