We study fast algorithms for statistical regression problems under the strong contamination model, where the goal is to approximately optimize a generalized linear model (GLM) given adversarially corrupted samples. Prior works in this line of research were based on the robust gradient descent framework of Prasad et. al., a first-order method using biased gradient queries, or the Sever framework of Diakonikolas et. al., an iterative outlier-removal method calling a stationary point finder. We present nearly-linear time algorithms for robust regression problems with improved runtime or estimation guarantees compared to the state-of-the-art. For the general case of smooth GLMs (e.g. logistic regression), we show that the robust gradient descent framework of Prasad et. al. can be accelerated, and show our algorithm extends to optimizing the Moreau envelopes of Lipschitz GLMs (e.g. support vector machines), answering several open questions in the literature. For the well-studied case of robust linear regression, we present an alternative approach obtaining improved estimation rates over prior nearly-linear time algorithms. Interestingly, our method starts with an identifiability proof introduced in the context of the sum-of-squares algorithm of Bakshi and Prasad, which achieved optimal error rates while requiring large polynomial runtime and sample complexity. We reinterpret their proof within the Sever framework and obtain a dramatically faster and more sample-efficient algorithm under fewer distributional assumptions.

本文主要介绍了在强污染模型下进行统计回归问题的快速算法，其中使用了鲁棒梯度下降框架，并提出了更快、更有效的算法来加速最优化过程。

鲁棒回归再探：加速与改进的估计速率