  We consider classification in the presence of class-dependent asymmetric
label noise with unknown noise probabilities. In this setting, identifiability
conditions are known, but additional assumptions were shown to be required for
finite sample rates, and so far only the parametric rate has been obtained.
Assuming these identifiability conditions, together with a measure-smoothness
condition on the regression function and Tsybakov's margin condition, we show
that the Robust kNN classifier of Gao et al. attains, the minimax optimal rates
of the noise-free setting, up to a log factor, even when trained on data with
unknown asymmetric label noise. Hence, our results provide a solid theoretical
backing for this empirically successful algorithm. By contrast the standard kNN
is not even consistent in the setting of asymmetric label noise. A key idea in
our analysis is a simple kNN based method for estimating the maximum of a
function that requires far less assumptions than existing mode estimators do,
and which may be of independent interest for noise proportion estimation and
randomised optimisation problems.
