9 research outputs found
Optimal robust mean and location estimation via convex programs with respect to any pseudo-norms
We consider the problem of robust mean and location estimation w.r.t. any
pseudo-norm of the form
where is any symmetric subset of . We show that the
deviation-optimal minimax subgaussian rate for confidence is where
is the Gaussian mean width of and
the covariance of the data (in the benchmark i.i.d. Gaussian case). This
improves the entropic minimax lower bound from [Lugosi and Mendelson, 2019] and
closes the gap characterized by Sudakov's inequality between the entropy and
the Gaussian mean width for this problem. This shows that the right statistical
complexity measure for the mean estimation problem is the Gaussian mean width.
We also show that this rate can be achieved by a solution to a convex
optimization problem in the adversarial and heavy-tailed setup by
considering minimum of some Fenchel-Legendre transforms constructed using the
Median-of-means principle. We finally show that this rate may also be achieved
in situations where there is not even a first moment but a location parameter
exists
ERM and RERM are optimal estimators for regression problems when malicious outliers corrupt the labels
We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk
Minimizers (RERM) for regression problems with convex and -Lipschitz loss
functions. We consider a setting where |\cO| malicious outliers contaminate
the labels. In that case, under a local Bernstein condition, we show that the
-error rate is bounded by r_N + AL |\cO|/N, where is the total
number of observations, is the -error rate in the non-contaminated
setting and is a parameter coming from the local Bernstein condition. When
is minimax-rate-optimal in a non-contaminated setting, the rate r_N +
AL|\cO|/N is also minimax-rate-optimal when |\cO| outliers contaminate the
label. The main results of the paper can be used for many non-regularized and
regularized procedures under weak assumptions on the noise. We present results
for Huber's M-estimators (without penalization or regularized by the
-norm) and for general regularized learning problems in reproducible
kernel Hilbert spaces when the noise can be heavy-tailed.Comment: 2 figure