  Motivated by recent data analytics applications, we study the adversarial
robustness of robust estimators. Instead of assuming that only a fraction of
the data points are outliers as considered in the classic robust estimation
setup, in this paper, we consider an adversarial setup in which an attacker can
observe the whole dataset and can modify all data samples in an adversarial
manner so as to maximize the estimation error caused by his attack. We
characterize the attacker's optimal attack strategy, and further introduce
adversarial influence function (AIF) to quantify an estimator's sensitivity to
such adversarial attacks. We provide an approach to characterize AIF for any
given robust estimator, and then design optimal estimator that minimizes AIF,
which implies it is least sensitive to adversarial attacks and hence is most
robust against adversarial attacks. From this characterization, we identify a
tradeoff between AIF (i.e., robustness against adversarial attack) and
influence function, a quantity used in classic robust estimators to measure
robustness against outliers, and design estimators that strike a desirable
tradeoff between these two quantities.
