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This paper analyzes $\ell_1$ regularized linear regression under thechallenging scenario of having only adversarially corrupted data for training.We use the primal-dual witness paradigm to provide provable performanceguarantees for the support of the estimated regression parameter vector tomatch the actual parameter. Our theoretical analysis shows thecounter-intuitive result that an adversary can influence sample complexity bycorrupting the irrelevant features, i.e., those corresponding to zerocoefficients of the regression parameter vector, which, consequently, do notaffect the dependent variable. As any adversarially robust algorithm has itslimitations, our theoretical analysis identifies the regimes under which thelearning algorithm and adversary can dominate over each other. It helps us toanalyze these fundamental limits and address critical scientific questions ofwhich parameters (like mutual incoherence, the maximum and minimum eigenvalueof the covariance matrix, and the budget of adversarial perturbation) play arole in the high or low probability of success of the LASSO algorithm. Also,the derived sample complexity is logarithmic with respect to the size of theregression parameter vector, and our theoretical claims are validated byempirical analysis on synthetic and real-world datasets.

A Theoretical Study of The Effects of Adversarial Attacks on Sparse Regression