General multilevel adaptations for stochastic approximation algorithms of Robbins–Monro and Polyak–Ruppert type

Dereich, Steffen; Müller-Gronbach, Thomas

doi:10.1007/s00211-019-01024-y

Cited by 27 publications

(23 citation statements)

References 20 publications

(52 reference statements)

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“…Finding L η can be formulated as finding the root of f (L η ) = η−P[L > L η ]. To that end, we can use a stochastic root finding algorithm such as the Stochastic Approximation Method [1,15] and its multilevel extensions [5,6]. Instead, since X is one-dimensional and since P[L > L η ] is monotonically decreasing with respect to L η , we use in the current work the simplified algorithm listed in Algorithm 2.…”

Section: A Model Problemmentioning

confidence: 99%

Multilevel Nested Simulation for Efficient Risk Estimation

Giles¹,

Haji-Ali²

2019

SIAM/ASA J. Uncertainty Quantification

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We investigate the problem of computing a nested expectation of the formwhere H is the Heaviside function. This nested expectation appears, for example, when estimating the probability of a large loss from a financial portfolio. We present a method that combines the idea of using Multilevel Monte Carlo (MLMC) for nested expectations with the idea of adaptively selecting the number of samples in the approximation of the inner expectation, as proposed by (Broadie et al., 2011). We propose and analyse an algorithm that adaptively selects the number of inner samples on each MLMC level and prove that the resulting MLMC method with adaptive sampling has an O(ε −2 |log ε| 2 ) complexity to achieve a root mean-squared error ε. The theoretical analysis is verified by numerical experiments on a simple model problem. We also present a stochastic root-finding algorithm that, combined with our adaptive methods, can be used to compute other risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), with the latter being achieved with O(ε −2 ) complexity.

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Section: A Model Problemmentioning

confidence: 99%

Multilevel Nested Simulation for Efficient Risk Estimation

Giles¹,

Haji-Ali²

2019

SIAM/ASA J. Uncertainty Quantification

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“…Monte Carlo inference is straightforward with independent unbiased samples, allowing to construct confidence intervals in a reliable way (Glynn and Whitt, 1992b). Debiasing techniques may also be employed within a stochastic approximation algorithm (Dereich and Mueller-Gronbach, 2015). In particular, in a stochastic gradient descent type algorithm (Robbins and Monro, 1951) relevant for instance in maximum likelihood inference (Delyon et al, 1999), unbiased gradient estimate implies pure martingale noise, which is supported by a well-established theory (e.g.…”

Section: Introductionmentioning

confidence: 94%

“…The debiasing techniques involve balancing with cost and variance, which often boils down to similar methods and conditions as those that are used with MLMC. The connection between MLMC and debiasing techniques has been pointed out earlier at least by Rhee and Glynn (2015), Giles (2015) and Dereich and Mueller-Gronbach (2015), but this connection has not been fully explored yet. The purpose of this paper is to further clarify the connection of MLMC and debiasing techniques, within a general framework for unbiased estimators.…”

Section: Introductionmentioning

confidence: 99%

Unbiased Estimators and Multilevel Monte Carlo

Vihola

2018

Operations Research

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Multilevel Monte Carlo (MLMC) and unbiased estimators recently proposed by McLeish (Monte Carlo Methods Appl., 2011) and Rhee and Glynn (Oper. Res., 2015) are closely related. This connection is elaborated by presenting a new general class of unbiased estimators, which admits previous debiasing schemes as special cases. New lower variance estimators are proposed, which are stratified versions of earlier unbiased schemes. Under general conditions, essentially when MLMC admits the canonical square root Monte Carlo error rate, the proposed new schemes are shown to be asymptotically as efficient as MLMC, both in terms of variance and cost. The experiments demonstrate that the variance reduction provided by the new schemes can be substantial., and as in (6), E(Z m − Z ) 2 = v ,m + (EY m − EY ) 2 .

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“…Remark 11. The recent work [4] describes a similar algorithm employing a multilevel approach to directly find minima without reconstruction of the complete response surface using the Robbins-Monro algorithm [27].…”

Section: Remark 10 (Convergence In L ∞ )mentioning

confidence: 99%

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

2017

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We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.

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General multilevel adaptations for stochastic approximation algorithms of Robbins–Monro and Polyak–Ruppert type

Cited by 27 publications

References 20 publications

Multilevel Nested Simulation for Efficient Risk Estimation

Multilevel Nested Simulation for Efficient Risk Estimation

Unbiased Estimators and Multilevel Monte Carlo

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

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