Quantitative statistical robustness for tail-dependent law invariant risk measures

Wang, Wei; Xu, Huifu; Ma, Tiejun

doi:10.1080/14697688.2021.1892171

Cited by 11 publications

(3 citation statements)

References 34 publications

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“…Proof. The result is established in [31,Lemma 4.1] which is an extension of [13, Lemma 1] (which is presented when p = 1). Here we include a proof for self-containedness.…”

Section: Consequently We Havementioning

confidence: 80%

Statistical Robustness of Empirical Risks in Machine Learning

Guo¹,

Xu²,

Zhang³

2020

Preprint

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This paper studies convergence of empirical risks in reproducing kernel Hilbert spaces (RKHS). A conventional assumption in the existing research is that empirical training data do not contain any noise but this may not be satisfied in some practical circumstances. Consequently the existing convergence results do not provide a guarantee as to whether empirical risks based on empirical data are reliable or not when the data contain some noise. In this paper, we fill out the gap in a few steps. First, we derive moderate sufficient conditions under which the expected risk changes stably (continuously) against small perturbation of the probability distribution of the underlying random variables and demonstrate how the cost function and kernel affect the stability. Second, we examine the difference between laws of the statistical estimators of the expected optimal loss based on pure data and contaminated data using Prokhorov metric and Kantorovich metric and derive some qualitative and quantitative statistical robustness results. Third, we identify appropriate metrics under which the statistical estimators are uniformly asymptotically consistent. These results provide theoretical grounding for analysing asymptotic convergence and examining reliability of the statistical estimators in a number of well-known machine learning models.

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“…Proof. The result is established in [31,Lemma 4.1] which is an extension of [13, Lemma 1] (which is presented when p = 1). Here we include a proof for self-containedness.…”

Section: Consequently We Havementioning

confidence: 80%

Statistical Robustness of Empirical Risks in Machine Learning

Guo¹,

Xu²,

Zhang³

2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…It follows by [48,Lemma 4.4] that (6.75) ≤ dl F M,p (P ⊗N , Q ⊗N ) ≤ Ldl F M,p (P, Q) (6.79) and hence inequality (6.74).…”

Section: 71)mentioning

confidence: 97%

“…Let P ⊗N denote the probability measure on the measurable space (IR ⊗N , B(IR) ⊗N ) with marginal P on each (IR, B(IR)) and Q ⊗N with marginal Q. Now we can state the definition of statistical robustness of a statistic estimator, which is proposed in [18,48]. Definition 6.1 (Quantitative statistical robustness) Let M ⊂ P(IR) be a set of probability measures.…”

Section: 71)mentioning

confidence: 99%

Preference Robust Modified Optimized Certainty Equivalent

Wu¹,

Xu²

2022

Preprint

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Ben-Tal and Teboulle [6] introduce the concept of optimized certainty equivalent (OCE) of an uncertain outcome as the maximum present value of a combination of the cash to be taken out from the uncertain income at present and the expected utility value of the remaining uncertain income. In this paper, we consider two variations of the OCE. First, we introduce a modified OCE by maximizing the combination of the utility of the cash and the expected utility of the remaining uncertain income so that the combined quantity is in a unified utility value. Second, we consider a situation where the true utility function is unknown but it is possible to use partially available information to construct a set of plausible utility functions. To mitigate the risk arising from the ambiguity, we introduce a robust model where the modified OCE is based on the worst-case utility function from the ambiguity set. In the case when the ambiguity set of utility functions is constructed by a Kantorovich ball centered at a nominal utility function, we show how the modified OCE and the corresponding worst case utility function can be identified by solving two linear programs alternatively. We also show the robust modified OCE is statistically robust in a data-driven environment where the underlying data are potentially contaminated. Some preliminary numerical results are reported to demonstrate the performance of the modified OCE and the robust modified OCE model.

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Data perturbations in stochastic generalized equations: statistical robustness in static and sample average approximated models

Guo

2023

Math. Program.

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Quantitative statistical robustness for tail-dependent law invariant risk measures

Cited by 11 publications

References 34 publications

Statistical Robustness of Empirical Risks in Machine Learning

Statistical Robustness of Empirical Risks in Machine Learning

Preference Robust Modified Optimized Certainty Equivalent

Data perturbations in stochastic generalized equations: statistical robustness in static and sample average approximated models

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