  We study the canonical statistical estimation problem of linear regression
from $n$ i.i.d.~examples under $(\varepsilon,\delta)$-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 $(\varepsilon,\delta)$-DP and robustness. Synthetic experiments
confirm the superiority of our approach.
