We consider the problem of estimating a spectral risk measure (SRM) from
i.i.d. samples, and propose a novel method that is based on numerical
integration. We show that our SRM estimate concentrates exponentially, when the
underlying distribution has bounded support. Further, we also consider the case
when the underlying distribution is either Gaussian or exponential, and derive
a concentration bound for our estimation scheme. We validate the theoretical
findings on a synthetic setup, and in a vehicular traffic routing application.

提出一种基于数值积分的估算谱风险度量值的新方法，通过限定支撑集，实现估算浓度的指数级集中，并在合成和交通路由应用中验证理论发现。

频谱风险度量估计

Estimation of Spectral Risk Measures

This work studies applications and generalizations of a simple estimation
technique that provides exponential concentration under heavy-tailed
distributions, assuming only bounded low-order moments. We show that the
technique can be used for approximate minimization of smooth and strongly
convex losses, and specifically for least squares linear regression. For
instance, our $d$-dimensional estimator requires just
$\tilde{O}(d\log(1/\delta))$ random samples to obtain a constant factor
approximation to the optimal least squares loss with probability $1-\delta$,
without requiring the covariates or noise to be bounded or subgaussian. We
provide further applications to sparse linear regression and low-rank
covariance matrix estimation with similar allowances on the noise and covariate
distributions. The core technique is a generalization of the median-of-means
estimator to arbitrary metric spaces.

该论文研究了一种简单估计技术在重尾分布下提供指数集中性的应用和推广，证明该技术可用于平滑强凸损失函数的近似最小化，特别是在最小二乘线性回归、稀疏线性回归和低秩协方差矩阵估计中具有类似的特征。