Unbiased Monte Carlo for optimization and functions of expectations via multi-level randomization

Abstract

We present general principles for the design and analysis of unbiased Monte Carlo estimators for quantities such as α = g (E (X)), where E (X) denotes the expectation of a (possibly multidimensional) random variable X, and g (·) is a given deterministic function. Our estimators possess finite work-normalized variance under mild regularity conditions such as local twice differentiability of g (·) and suitable growth and finite-moment assumptions. We apply our estimator to various settings of interest, such as optimal value estimation in the context of Sample Average Approximations, and unbiased steady-state simulation of regenerative processes. Other applications include unbiased estimators for particle filters and conditional expectations.

Semi‐supervised Learning Based on Distributionally Robust Optimization

“…The procedure developed in [4] proceeds as follows. First, define for a given h (W ), and n ≥ 0, the average over odd and even labels to bē…”

Stochastic Bias-Reduced Gradient Methods

“…Multilevel Monte-Carlo (MLMC) techniques originate from the literature on parametric integration for solving integral and differential equations [24]. Our approach is based on an MLMC variant put forth by Blanchet and Glynn [8] for estimating functionals of expectations. Among several applications, they propose [8, Section 5.2] an estimator for argmin x E S∼P f (x; S) where f (•; s) is convex for all s and assuming access to minimizers of empirical objectives of the form i∈[N ] f (x; s i ).…”

“…Our estimator is an instance of the multilevel Monte Carlo technique for de-biasing estimator sequences [24] and more specifically the method of Blanchet and Glynn [8]. Our key observation is that this method is readily applicable to strongly-convex variants of SGD, or indeed any stochastic optimization method with the same (optimal) rate of convergence.…”

“…

We develop a new primitive for stochastic optimization: a low-bias, low-cost estimator of the minimizer x of any Lipschitz strongly-convex function. In particular, we use a multilevel Monte-Carlo approach due to Blanchet and Glynn [8] to turn any optimal stochastic gradient method into an estimator of x with bias δ, variance O(log(1/δ)), and an expected sampling cost of O(log(1/δ)) stochastic gradient evaluations. As an immediate consequence, we obtain cheap and nearly unbiased gradient estimators for the Moreau-Yoshida envelope of any Lipschitz convex function, allowing us to perform dimension-free randomized smoothing.We demonstrate the potential of our estimator through four applications.

…”

Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI

“…The difference is that with EVPI all of the underlying random variables X and Y are inner variables; non are outer variables leading to a conditional expectation. Such an MLMC estimator for the maximum of an unconditional expectation has been introduced by Blanchet and Glynn (2015). As discussed in the introduction, however, EVPI can be estimated with O(ε 2 ) complexity by using standard Monte Carlo methods already, so that the benefit is that one could use a randomisation technique by Rhee and Glynn (2015) to obtain an unbiased estimator, which might be marginal in the current setting.…”