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Structured statistical estimation problems are often solved by ConditionalGradient (CG) type methods to avoid the computationally expensive projectionoperation. However, the existing CG type methods are not robust to datacorruption. To address this, we propose to robustify CG type methods againstHuber's corruption model and heavy-tailed data. First, we show that the twoPairwise CG methods are stable, i.e., do not accumulate error. Combined withrobust mean gradient estimation techniques, we can therefore guaranteerobustness to a wide class of problems, but now in a projection-freealgorithmic framework. Next, we consider high dimensional problems. Robust meanestimation based approaches may have an unacceptably high sample complexity.When the constraint set is a $\ell_0$ norm ball,Iterative-Hard-Thresholding-based methods have been developed recently. Yetextension is non-trivial even for general sets with $O(d)$ extreme points. Forsetting where the feasible set has $O(\text{poly}(d))$ extreme points, wedevelop a novel robustness method, based on a new condition we call the RobustAtom Selection Condition (RASC). When RASC is satisfied, our method convergeslinearly with a corresponding statistical error, with sample complexity thatscales correctly in the sparsity of the problem, rather than the ambientdimension as would be required by any approach based on robust mean estimation.

Robust Structured Statistical Estimation via Conditional Gradient Type Methods