We focus on a stochastic learning model where the learner observes a finite set of training examples and the output of the learning process is a data-dependent distribution over a space of hypotheses. The learned data-dependent distribution is then used to make randomized predictions, and the high-level theme addressed here is guaranteeing the quality of predictions on examples that were not seen during training, i.e. generalization. In this setting the unknown quantity of interest is the expected risk of the data-dependent randomized predictor, for which upper bounds can be derived via a PAC-Bayes analysis, leading to PAC-Bayes bounds. Specifically, we present a basic PAC-Bayes inequality for stochastic kernels, from which one may derive extensions of various known PAC-Bayes bounds as well as novel bounds. We clarify the role of the requirement of fixed 'data-free' priors and illustrate the use of data-dependent priors. We also present a simple bound that is valid for a loss function with unbounded range. Our analysis clarifies that those two requirements were used to upper-bound an exponential moment term, while the basic PAC-Bayes inequality remains valid with those restrictions removed.

We explore the family of methods 'PAC-Bayes with Backprop' (PBB) to train probabilistic neural networks by minimizing PAC-Bayes bounds. We present two training objectives, one derived from a previously known PAC-Bayes bound, and a second one derived from a novel PAC-Bayes bound. Both training objectives are evaluated on MNIST and on various UCI data sets. Our experiments show two striking observations: we obtain competitive test set error estimates (∼ 1.4% on MNIST) and at the same time we compute non-vacuous bounds with much tighter values (∼ 2.3% on MNIST) than previous results. These observations suggest that neural nets trained by PBB may lead to self-bounding learning, where the available data can be used to simultaneously learn a predictor and certify its risk, with no need to follow a data-splitting protocol.

We present new PAC-Bayesian generalisation bounds for learning problems with unbounded loss functions. This extends the relevance and applicability of the PAC-Bayes learning framework, where most of the existing literature focuses on supervised learning problems with a bounded loss function (typically assumed to take values in the interval [0;1]). In order to relax this classical assumption, we propose to allow the range of the loss to depend on each predictor. This relaxation is captured by our new notion of HYPothesis-dependent rangE (HYPE). Based on this, we derive a novel PAC-Bayesian generalisation bound for unbounded loss functions, and we instantiate it on a linear regression problem. To make our theory usable by the largest audience possible, we include discussions on actual computation, practicality and limitations of our assumptions.

We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances from below. This allows us to consider matrices with null entries or, more generally, with entries having small variances. Our results do not assume identical distribution of the entries of a random matrix and help to clarify the role of the variances of the entries. We also show that it is enough to require boundedness from above of the r-th moment, r > 2, of the corresponding entries.

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