  We study online learning when individual instances are corrupted by
adversarially chosen random noise. We assume the noise distribution is unknown,
and may change over time with no restriction other than having zero mean and
bounded variance. Our technique relies on a family of unbiased estimators for
non-linear functions, which may be of independent interest. We show that a
variant of online gradient descent can learn functions in any dot-product
(e.g., polynomial) or Gaussian kernel space with any analytic convex loss
function. Our variant uses randomized estimates that need to query a random
number of noisy copies of each instance, where with high probability this
number is upper bounded by a constant. Allowing such multiple queries cannot be
avoided: Indeed, we show that online learning is in general impossible when
only one noisy copy of each instance can be accessed.
