  We propose to study the generalization error of a learned predictor $\hat h$
in terms of that of a surrogate (potentially randomized) predictor that is
coupled to $\hat h$ and designed to trade empirical risk for control of
generalization error. In the case where $\hat h$ interpolates the data, it is
interesting to consider theoretical surrogate classifiers that are partially
derandomized or rerandomized, e.g., fit to the training data but with modified
label noise. We also show that replacing $\hat h$ by its conditional
distribution with respect to an arbitrary $\sigma$-field is a convenient way to
derandomize. We study two examples, inspired by the work of Nagarajan and
Kolter (2019) and Bartlett et al. (2019), where the learned classifier $\hat h$
interpolates the training data with high probability, has small risk, and, yet,
does not belong to a nonrandom class with a tight uniform bound on two-sided
generalization error. At the same time, we bound the risk of $\hat h$ in terms
of surrogates constructed by conditioning and denoising, respectively, and
shown to belong to nonrandom classes with uniformly small generalization error.
