标题： 重新审视$\ell_p$球上的均值估计：MLE 是否最优？ 显示英文标题

标题： Revisiting mean estimation over $\ell_p$ balls: Is the MLE optimal?

摘要： 我们重新研究了在具有$\ell_p$约束的高斯序列模型中的均值估计问题，其中$p \in [0, \infty]$。 我们展示了最大似然估计量（MLE）行为的两种现象，这些现象取决于噪声水平、(拟)范数约束的半径、维度以及范数指数$p$。 首先，如果$p$位于$0$和$1 + \Theta(\tfrac{1}{\log d})$之间（包括边界），或者大于等于$2$，则 MLE 对于所有噪声水平和所有约束半径都是最小最大速率最优的。 另一方面，对于其余的范数指标——即当$p$介于$1 + \Theta(\tfrac{1}{\log d})$和$2$之间时——这里表现出更为显著的行为：尽管其在观测值中是非线性的，但对于几乎所有需要非线性估计才能达到最小最大最优估计的噪声水平和约束半径，最大似然估计是极小极大率次优的。 我们的结果表明，当给定$n$个独立同分布的高斯样本时，最大似然估计在样本量上可能以多项式因子的差距表现次优。 我们的下界是构造性的：每当最大似然估计率次优时，我们都能提供明确的实例，其中最大似然估计可证明会带来次优的风险。 最后，在非凸情况下——即当$p < 1$时——我们开发了精确的局部高斯宽度界限，这可能具有独立的兴趣。 显示英文摘要

摘要： We revisit the problem of mean estimation in the Gaussian sequence model with $\ell_p$ constraints for $p \in [0, \infty]$. We demonstrate two phenomena for the behavior of the maximum likelihood estimator (MLE), which depend on the noise level, the radius of the (quasi)norm constraint, the dimension, and the norm index $p$. First, if $p$ lies between $0$ and $1 + \Theta(\tfrac{1}{\log d})$, inclusive, or if it is greater than or equal to $2$, the MLE is minimax rate-optimal for all noise levels and all constraint radii. On the other hand, for the remaining norm indices -- namely, if $p$ lies between $1 + \Theta(\tfrac{1}{\log d})$ and $2$ -- here is a more striking behavior: the MLE is minimax rate-suboptimal, despite its nonlinearity in the observations, for essentially all noise levels and constraint radii for which nonlinear estimates are necessary for minimax-optimal estimation. Our results imply that when given $n$ independent and identically distributed Gaussian samples, the MLE can be suboptimal by a polynomial factor in the sample size. Our lower bounds are constructive: whenever the MLE is rate-suboptimal, we provide explicit instances on which the MLE provably incurs suboptimal risk. Finally, in the non-convex case -- namely when $p < 1$ -- we develop sharp local Gaussian width bounds, which may be of independent interest.