  We investigate two important properties of M-estimator, namely, robustness
and tractability, in linear regression setting, when the observations are
contaminated by some arbitrary outliers. Specifically, robustness means the
statistical property that the estimator should always be close to the
underlying true parameters {\em regardless of the distribution of the
outliers}, and tractability indicates the computational property that the
estimator can be computed efficiently, even if the objective function of the
M-estimator is {\em non-convex}. In this article, by learning the landscape of
the empirical risk, we show that under mild conditions, many M-estimators enjoy
nice robustness and tractability properties simultaneously, when the percentage
of outliers is small. We further extend our analysis to the high-dimensional
setting, where the number of parameters is greater than the number of samples,
$p \gg n$, and prove that when the proportion of outliers is small, the
penalized M-estimators with {\em $L_1$} penalty will enjoy robustness and
tractability simultaneously. Our research provides an analytic approach to see
the effects of outliers and tuning parameters on the robustness and
tractability for some families of M-estimators. Simulation and case study are
presented to illustrate the usefulness of our theoretical results for
M-estimators under Welsch's exponential squared loss.
