具有不可信批次的高维鲁棒均值估计
High-Dimensional Robust Mean Estimation from Doubly Corrupted and Heterogeneous Batches
AI总结：
本文提出两种基于平方和的算法，解决了高维环境下存在不可信批次的鲁棒均值估计问题，实现了最小最大最优误差率。
摘要：
我们研究了在协作环境下高维均值估计的问题，其中数据由N个用户以批次大小n的方式提供。在这样的环境中，学习者试图从多个来源中恢复真实分布P的均值μ，这些来源在统计上异质且可能具有恶意行为。我们通过双重腐败景观来正式化这一挑战：ε比例的用户完全是敌对的，而剩下的“良好”用户提供的数据来自与P相关的分布，但通过接近参数α偏离。与现有不可信批次模型的工作不同，后者通常通过离散设置中的总变差距离来衡量这种偏差，我们针对连续、高维情形下的两种自然偏差变体进行了研究：(1)良好批次来自具有均值偏移√α的分布，或(2)每个良好批次中α比例的样本被敌对地破坏。特别是，第二种模型带来了新的挑战：在高维情况下，不同于离散设置，即使少量样本级破坏也能任意改变经验均值和协方差。我们提供了两种基于平方和（SoS）的算法来导航这种分层破坏。我们的算法实现了最小最大最优误差率O(√(ε/n) + √(d/nN) + √α)，证明了尽管异质性α代表了固有的统计困难，但敌对用户的影响被批次结构内部的平均所抑制，因子为1/√n。
英文摘要：
Modern data systems often require aggregating information from multiple sources: some might be entirely untrustworthy, while the remaining presumably reliable sources may still suffer from a small amount of corruption. We study robust mean estimation under this practical scenario, which is formulated through the double corruption model: samples are divided among $N$ users, each with $n$ samples. $\varepsilon$ fraction of the users could be arbitrarily corrupted, while each of the remaining ``good'' users still have $α$-fraction ($α>0$) of corrupted samples. The goal is to estimate the unknown mean $μ$ with small $\ell_2$ error. This is a challenging scenario because we must handle two types of adversarial corruption: user-level corruption and sample-level corruption. When the unknown distribution has covariance matrix bounded by the identity matrix, we design a sum-of-squares (SoS) algorithm that achieves an $\ell_2$ error of $\sqrtα$ with probability at least $2/3$ as long as the total number of samples $nN=\widetildeΩ(d/α)$. The estimation error and sample complexity are min-max optimal under the mild condition. For comparison, treating all $nN$ samples as $(\varepsilon+α)$-corrupted and applying standard robust mean estimation only achieves an error of $\sqrt{\varepsilon+α}$. Thus, our new algorithm removes the adverse effect of untrustworthy users and achieves the optimal error as if only sample-level corruption is present. Also, our SoS algorithm extends to the heterogeneous setting where each user's local mean could be additionally shifted by $\sqrtβ$ in $\ell_2$ distance. We show that an error of $O(\sqrt{α+β})$ can be achieved with probability at least $2/3$ when $nN=\widetildeΩ(d/(α+β))$.
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