  In this paper, we establish novel data-dependent upper bounds on the
generalization error through the lens of a "variable-size compressibility"
framework that we introduce newly here. In this framework, the generalization
error of an algorithm is linked to a variable-size 'compression rate' of its
input data. This is shown to yield bounds that depend on the empirical measure
of the given input data at hand, rather than its unknown distribution. Our new
generalization bounds that we establish are tail bounds, tail bounds on the
expectation, and in-expectations bounds. Moreover, it is shown that our
framework also allows to derive general bounds on any function of the input
data and output hypothesis random variables. In particular, these general
bounds are shown to subsume and possibly improve over several existing
PAC-Bayes and data-dependent intrinsic dimension-based bounds that are
recovered as special cases, thus unveiling a unifying character of our
approach. For instance, a new data-dependent intrinsic dimension-based bound is
established, which connects the generalization error to the optimization
trajectories and reveals various interesting connections with the
rate-distortion dimension of a process, the R\'enyi information dimension of a
process, and the metric mean dimension.
