On the Adversarial Robustness of Multivariate Robust Estimation
This work addresses adversarial robustness in statistical estimation, which is an incremental advancement in the field of robust statistics.
The paper tackles the problem of adversarial robustness in multivariate M-estimators, where an adversary can modify all data points to maximize inference errors, and characterizes the optimal M-estimator that minimizes the adversarial influence function (AIF) while balancing robustness against adversarial modifications and outliers.
In this paper, we investigate the adversarial robustness of multivariate $M$-Estimators. In the considered model, after observing the whole dataset, an adversary can modify all data points with the goal of maximizing inference errors. We use adversarial influence function (AIF) to measure the asymptotic rate at which the adversary can change the inference result. We first characterize the adversary's optimal modification strategy and its corresponding AIF. From the defender's perspective, we would like to design an estimator that has a small AIF. For the case of joint location and scale estimation problem, we characterize the optimal $M$-estimator that has the smallest AIF. We further identify a tradeoff between robustness against adversarial modifications and robustness against outliers, and derive the optimal $M$-estimator that achieves the best tradeoff.