A Wasserstein distance approach for concentration of empirical risk estimates
This paper presents a unified approach based on Wasserstein distance to derive concentration bounds for empirical estimates for two broad classes of risk measures defined in the paper. The classes of risk measures introduced include as special cases well known risk measures from the finance literature such as conditional value at risk (CVaR), optimized certainty equivalent risk, spectral risk measures, utility-based shortfall risk, cumulative prospect theory (CPT) value, rank dependent expected utility and distorted risk measures. Two estimation schemes are considered, one for each class of risk measures. One estimation scheme involves applying the risk measure to the empirical distribution function formed from a collection of i.i.d. samples of the random variable (r.v.), while the second scheme involves applying the same procedure to a truncated sample. The bounds provided apply to three popular classes of distributions, namely sub-Gaussian, sub-exponential and heavy-tailed distributions. The bounds are derived by first relating the estimation error to the Wasserstein distance between the true and empirical distributions, and then using recent concentration bounds for the latter. Previous concentration bounds are available only for specific risk measures such as CVaR and CPT-value. The bounds derived in this paper are shown to either match or improve upon previous bounds in cases where they are available. The usefulness of the bounds is illustrated through an algorithm and the corresponding regret bound for a stochastic bandit problem involving a general risk measure from each of the two classes introduced in the paper.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Uniform-in-time concentration in two-layer neural networks via transportation inequalities
We quantify, uniformly over time and with high probability, the discrepancy between the predictions of a two-layer neural network trained by stochastic gradient descent (SGD) and their mean-field limit, for quadratic los…
Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case
Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting …
A Distribution Optimization Framework for Confidence Bounds of Risk Measures
We present a distribution optimization framework that significantly improves confidence bounds for various risk measures compared to previous methods. Our framework encompasses popular risk measures such as the entropic …
Decision MakingConvergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces
We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Suc…
Gaussian ProcessesSpectral-transport stability and benign overfitting for minimum norm interpolation
Benign overfitting describes the ability of minimum norm interpolating estimators to generalize despite fitting noisy data exactly. Existing characterizations depend on delicate spectral functionals of the population cov…