Unbiased Multilevel Monte Carlo methods for intractable distributions: MLMC meets MCMC

Wang, Guanyang; Wang, Tianze

doi:10.48550/arxiv.2204.04808

Cited by 2 publications

(2 citation statements)

References 53 publications

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“…It is worth studying the use of (unbiased) Markov chain Monte Carlo sampling within our unbiased gradient estimators. A recent work of [42] seems quite relevant to this research direction.…”

Section: Discussionmentioning

confidence: 99%

Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization

Goda¹,

Kitade²

2022

Preprint

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We study stochastic gradient descent for solving conditional stochastic optimization problems, in which an objective to be minimized is given by a parametric nested expectation with an outer expectation taken with respect to one random variable and an inner conditional expectation with respect to the other random variable. The gradient of such a parametric nested expectation is again expressed as a nested expectation, which makes it hard for the standard nested Monte Carlo estimator to be unbiased. In this paper, we show under some conditions that a multilevel Monte Carlo gradient estimator is unbiased and has finite variance and finite expected computational cost, so that the standard theory from stochastic optimization for a parametric (non-nested) expectation directly applies. We also discuss a special case for which yet another unbiased gradient estimator with finite variance and cost can be constructed.

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Section: Discussionmentioning

confidence: 99%

Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization

Goda¹,

Kitade²

2022

Preprint

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“…Designing unbiased estimators has recently attracted much interest in statistics, operations research, and machine learning communities for its potential for parallelization. Our methods add to the rich body of works of Glynn and Rhee [2014], Rhee and Glynn [2015], Blanchet and Glynn [2015], Jacob et al [2020], Biswas et al [2019], Wang et al [2021], Wang and Wang [2022], Kahale [2022].…”

mentioning

confidence: 99%

Optimal randomized multilevel Monte Carlo for repeatedly nested expectations

Syed¹,

Wang²

2023

Preprint

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The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any non-negative integer D for the total number of nestings. Standard Monte Carlo methods typically cost at least O(ε −(2+D) ) and sometimes O(ε −2(1+D) ) to obtain an estimator up to ε-error. More advanced methods, such as multilevel Monte Carlo, currently only exist for D = 1. In this paper, we propose a novel Monte Carlo estimator called READ, which stands for "Recursive Estimator for Arbitrary Depth." Our estimator has an optimal computational cost of O(ε −2 ) for every fixed D under suitable assumptions, and a nearly optimal computational cost of O(ε −2(1+δ) ) for any 0 < δ < 1 2 under much more general assumptions. Our estimator is also unbiased, which makes it easy to parallelize. The key ingredients in our construction are an observation of the problem's recursive structure and the recursive use of the randomized multilevel Monte Carlo method.

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Unbiased Multilevel Monte Carlo methods for intractable distributions: MLMC meets MCMC

Cited by 2 publications

References 53 publications

Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization

Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization

Optimal randomized multilevel Monte Carlo for repeatedly nested expectations

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