  We establish a tight characterization of the worst-case rates for the excess
risk of agnostic learning with sample compression schemes and for uniform
convergence for agnostic sample compression schemes. In particular, we find
that the optimal rates of convergence for size-$k$ agnostic sample compression
schemes are of the form $\sqrt{\frac{k \log(n/k)}{n}}$, which contrasts with
agnostic learning with classes of VC dimension $k$, where the optimal rates are
of the form $\sqrt{\frac{k}{n}}$.
