PAC-Bayes bounds for stable algorithms with instance-dependent priors

P AC-Bayes bounds have been proposed to get risk estimates based on a training sample. In this paper the P AC-Bayes approach is combined with stability of the hypothesis learned by a Hilbert space valued algorithm. The P AC-Bayes setting is used with a Gaussian prior centered at the expected output. Thus a novelty of our paper is using priors defined in terms of the data-generating distribution. Our main result estimates the risk of the randomized algorithm in terms of the hypothesis stability coefficients. We also provide a new bound for the SVM classifier, which is compared to other known bounds experimentally. Ours appears to be the first uniform hypothesis stability-based bound that evaluates to non-trivial values.