We study the problem of designing minimax procedures in linear regression under the quantile risk. We start by considering the realizable setting with independent Gaussian noise, where for any given noise level and distribution of inputs, we obtain the exact minimax quantile risk for a rich family of error functions and establish the minimaxity of OLS. This improves on the known lower bounds for the special case of square error, and provides us with a lower bound on the minimax quantile risk over larger sets of distributions. Under the square error and a fourth moment assumption on the distribution of inputs, we show that this lower bound is tight over a larger class of problems. Specifically, we prove a matching upper bound on the worst-case quantile risk of a variant of the recently proposed min-max regression procedure, thereby establishing its minimaxity, up to absolute constants. We illustrate the usefulness of our approach by extending this result to all $p$-th power error functions for $p \in (2, \infty)$. Along the way, we develop a generic analogue to the classical Bayesian method for lower bounding the minimax risk when working with the quantile risk, as well as a tight characterization of the quantiles of the smallest eigenvalue of the sample covariance matrix.

翻译：我们研究了在分位数风险下设计线性回归极小极大程序的问题。首先考虑具有独立高斯噪声的可实现设置，其中对于任意给定的噪声水平和输入分布，我们针对一类丰富的误差函数获得了精确的极小极大分位数风险，并确立了普通最小二乘法（OLS）的极小极大性。这改进了平方误差这一特殊情况下已知的下界，并为我们提供了在更广泛分布集上的极小极大分位数风险下界。在平方误差和输入分布四阶矩假设下，我们证明该下界在更广泛的问题类别上是紧的。具体而言，我们证明了最近提出的极小极大回归方法变体在最坏情况分位数风险上的匹配上界，从而确立了其在绝对常数范围内的极小极大性。通过将该结果扩展到所有$p \in (2, \infty)$的$p$次幂误差函数，我们说明了所提出方法的实用性。在此过程中，我们发展了经典贝叶斯方法在分位数风险下求极小极大风险下界的通用类比方法，并对样本协方差矩阵最小特征值的分位数进行了紧致刻画。