Haghifam *et al.* provided an individual-sample strengthening of this result, as well as improvements through their use of disintegration. In all of these derivations, the loss function is required to be bounded, which is a stronger requirement than what is needed in the standard setting.

All of the information-theoretic results discussed so far pertain to bounds on the average generalization error. In [@bassily18-02a App. A.3], Bassily *et al.* provided a PAC-Bayesian bound in terms of mutual information. This bound is essentially a data-independent relaxation of a well-known data-dependent bound from the PAC-Bayesian literature [@guedj19-10a]. The dependence of the original data-dependent PAC-Bayesian bound on the confidence parameter $\delta$ is of order $\log(1/\delta)$, making it a high-probability bound. However, its mutual information relaxation in [@bassily18-02a] has a less benign $1/\delta$-dependence. PAC-Bayesian techniques have recently found some success in producing non-vacuous generalization bounds for (randomized) deep neural networks. In [@Dziugaite2017], Dziugaite and Roy optimized a PAC-Bayesian bound to get non-vacuous generalization estimates for a simple neural network setup. In [@zhou2018nonvacuous], Zhou *et al.* derived a bound for compressed networks, i.e., small neural networks that are formed by pruning larger ones, and illustrated numerically that the bound is non-trivial for realistic settings. An extensive 