  We prove a new upper bound on the generalization gap of classifiers that are
obtained by first using self-supervision to learn a representation $r$ of the
training data, and then fitting a simple (e.g., linear) classifier $g$ to the
labels. Specifically, we show that (under the assumptions described below) the
generalization gap of such classifiers tends to zero if $\mathsf{C}(g) \ll n$,
where $\mathsf{C}(g)$ is an appropriately-defined measure of the simple
classifier $g$'s complexity, and $n$ is the number of training samples. We
stress that our bound is independent of the complexity of the representation
$r$. We do not make any structural or conditional-independence assumptions on
the representation-learning task, which can use the same training dataset that
is later used for classification. Rather, we assume that the training procedure
satisfies certain natural noise-robustness (adding small amount of label noise
causes small degradation in performance) and rationality (getting the wrong
label is not better than getting no label at all) conditions that widely hold
across many standard architectures. We show that our bound is non-vacuous for
many popular representation-learning based classifiers on CIFAR-10 and
ImageNet, including SimCLR, AMDIM and MoCo.
