基于凸规划的最优鲁棒均值和位置估计
Optimal robust mean and location estimation via convex programs with respect to any pseudo-norms
摘要：
我们考虑鲁棒均值和位置估计问题。形式为$x\in\mathbb{R}^d\to ||x||_S = \sup_{v\in S}<v,x>$的任何伪范数，其中$S$是$\mathbb{R}^d$的任何对称子集。我们证明了置信$1-δ$的偏差最优极小极大次高斯率是$$ \max\left(\frac{l^*(Σ^{1/2}S)}{\sqrt{N}}, \sup_{v\in S}||Σ^{1/2}v||_2\sqrt{\frac{\log(1/δ)}{N}}\right) $$，其中$l^*(Σ^{1/2}S)$是$Σ^{1/2}S$的高斯平均宽度，$Σ$是数据的协方差（在基准i.i.d. Gaussian case）。这改进了[Lugosi和Mendelson，2019]的熵极大极小下界，并缩小了该问题的熵与高斯平均宽度之间的Sudakov不等式所表征的差距。这表明均值估计问题的正确统计复杂性度量是高斯均值宽度。我们还表明，这一速度可以实现的解决方案的凸优化问题的对抗和$L_2$重尾设置考虑最小的一些Fenchel-Legendre变换构造使用中位数的平均值原则。我们最后表明，这种速率也可以实现的情况下，甚至没有一个第一时刻，但存在一个位置参数。
英文摘要：
We consider the problem of robust mean and location estimation w.r.t. any pseudo-norm of the form $x\in\mathbb{R}^d\to ||x||_S = \sup_{v\in S}<v,x>$ where $S$ is any symmetric subset of $\mathbb{R}^d$. We show that the deviation-optimal minimax subgaussian rate for confidence $1-δ$ is $$ \max\left(\frac{l^*(Σ^{1/2}S)}{\sqrt{N}}, \sup_{v\in S}||Σ^{1/2}v||_2\sqrt{\frac{\log(1/δ)}{N}}\right) $$where $l^*(Σ^{1/2}S)$ is the Gaussian mean width of $Σ^{1/2}S$ and $Σ$ the covariance of the data (in the benchmark i.i.d. Gaussian case). This improves the entropic minimax lower bound from [Lugosi and Mendelson, 2019] and closes the gap characterized by Sudakov's inequality between the entropy and the Gaussian mean width for this problem. This shows that the right statistical complexity measure for the mean estimation problem is the Gaussian mean width. We also show that this rate can be achieved by a solution to a convex optimization problem in the adversarial and $L_2$ heavy-tailed setup by considering minimum of some Fenchel-Legendre transforms constructed using the Median-of-means principle. We finally show that this rate may also be achieved in situations where there is not even a first moment but a location parameter exists.
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