  We give the first polynomial time and sample $(\epsilon,
\delta)$-differentially private (DP) algorithm to estimate the mean, covariance
and higher moments in the presence of a constant fraction of adversarial
outliers. Our algorithm succeeds for families of distributions that satisfy two
well-studied properties in prior works on robust estimation: certifiable
subgaussianity of directional moments and certifiable hypercontractivity of
degree 2 polynomials. Our recovery guarantees hold in the "right
affine-invariant norms": Mahalanobis distance for mean, multiplicative spectral
and relative Frobenius distance guarantees for covariance and injective norms
for higher moments. Prior works obtained private robust algorithms for mean
estimation of subgaussian distributions with bounded covariance. For covariance
estimation, ours is the first efficient algorithm (even in the absence of
outliers) that succeeds without any condition-number assumptions.
  Our algorithms arise from a new framework that provides a general blueprint
for modifying convex relaxations for robust estimation to satisfy strong
worst-case stability guarantees in the appropriate parameter norms whenever the
algorithms produce witnesses of correctness in their run. We verify such
guarantees for a modification of standard sum-of-squares (SoS) semidefinite
programming relaxations for robust estimation. Our privacy guarantees are
obtained by combining stability guarantees with a new "estimate dependent"
noise injection mechanism in which noise scales with the eigenvalues of the
estimated covariance. We believe this framework will be useful more generally
in obtaining DP counterparts of robust estimators.
  Independently of our work, Ashtiani and Liaw [AL21] also obtained a
polynomial time and sample private robust estimation algorithm for Gaussian
distributions.
