  Numerous modern optimization and machine learning algorithms rely on
subgradient information being trustworthy and hence, they may fail to converge
when such information is corrupted. In this paper, we consider the setting
where subgradient information may be arbitrarily corrupted (with a given
probability) and study the robustness properties of the normalized subgradient
method. Under the probabilistic corruption scenario, we prove that the
normalized subgradient method, whose updates rely solely on directional
information of the subgradient, converges to a minimizer for convex, strongly
convex, and weakly-pseudo convex functions satisfying certain conditions.
Numerical evidence on linear regression and logistic classification problems
support our results.
