  We study distribution-free nonparametric regression following a notion of
average smoothness initiated by Ashlagi et al. (2021), which measures the
"effective" smoothness of a function with respect to an arbitrary unknown
underlying distribution. While the recent work of Hanneke et al. (2023)
established tight uniform convergence bounds for average-smooth functions in
the realizable case and provided a computationally efficient realizable
learning algorithm, both of these results currently lack analogs in the general
agnostic (i.e. noisy) case.
  In this work, we fully close these gaps. First, we provide a
distribution-free uniform convergence bound for average-smoothness classes in
the agnostic setting. Second, we match the derived sample complexity with a
computationally efficient agnostic learning algorithm. Our results, which are
stated in terms of the intrinsic geometry of the data and hold over any totally
bounded metric space, show that the guarantees recently obtained for realizable
learning of average-smooth functions transfer to the agnostic setting. At the
heart of our proof, we establish the uniform convergence rate of a function
class in terms of its bracketing entropy, which may be of independent interest.
