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We study the canonical statistical estimation problem of linear regressionfrom $n$ i.i.d.~examples under $(\varepsilon,\delta)$-differential privacy whensome response variables are adversarially corrupted. We propose a variant ofthe popular differentially private stochastic gradient descent (DP-SGD)algorithm with two innovations: a full-batch gradient descent to improve samplecomplexity and a novel adaptive clipping to guarantee robustness. When there isno adversarial corruption, this algorithm improves upon the existingstate-of-the-art approach and achieves a near optimal sample complexity. Underlabel-corruption, this is the first efficient linear regression algorithm toguarantee both $(\varepsilon,\delta)$-DP and robustness. Synthetic experimentsconfirm the superiority of our approach.

Near Optimal Private and Robust Linear Regression