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Geometric median (\textsc{Gm}) is a classical method in statistics forachieving a robust estimation of the uncorrupted data; under gross corruption,it achieves the optimal breakdown point of 0.5. However, its computationalcomplexity makes it infeasible for robustifying stochastic gradient descent(SGD) for high-dimensional optimization problems. In this paper, we show thatby applying \textsc{Gm} to only a judiciously chosen block of coordinates at atime and using a memory mechanism, one can retain the breakdown point of 0.5for smooth non-convex problems, with non-asymptotic convergence ratescomparable to the SGD with \textsc{Gm}.

Robust Training in High Dimensions via Block Coordinate Geometric Median Descent