  This paper presents an empirical study regarding training probabilistic
neural networks using training objectives derived from PAC-Bayes bounds. In the
context of probabilistic neural networks, the output of training is a
probability distribution over network weights. We present two training
objectives, used here for the first time in connection with training neural
networks. These two training objectives are derived from tight PAC-Bayes
bounds. We also re-implement a previously used training objective based on a
classical PAC-Bayes bound, to compare the properties of the predictors learned
using the different training objectives. We compute risk certificates for the
learnt predictors, based on part of the data used to learn the predictors. We
further experiment with different types of priors on the weights (both
data-free and data-dependent priors) and neural network architectures. Our
experiments on MNIST and CIFAR-10 show that our training methods produce
competitive test set errors and non-vacuous risk bounds with much tighter
values than previous results in the literature, showing promise not only to
guide the learning algorithm through bounding the risk but also for model
selection. These observations suggest that the methods studied here might be
good candidates for self-certified learning, in the sense of using the whole
data set for learning a predictor and certifying its risk on any unseen data
(from the same distribution as the training data) potentially without the need
for holding out test data.
