Abstract
We study the canonical statistical estimation problem of linear regression from n i.i.d.~examples under (ε,δ)-differential privacy when some response variables are adversarially corrupted. We propose a variant of the popular differentially private stochastic gradient descent (DP-SGD) algorithm with two innovations: a full-batch gradient descent to improve sample complexity and a novel adaptive clipping to guarantee robustness. When there is no adversarial corruption, this algorithm improves upon the existing state-of-the-art approach and achieves a near optimal sample complexity. Under label-corruption, this is the first efficient linear regression algorithm to guarantee both (ε,δ)-DP and robustness. Synthetic experiments confirm the superiority of our approach.