PAC-Bayesian Theory Meets Bayesian Inference

We exhibit a strong link between frequentist PAC-Bayesian risk bounds and the Bayesian marginal likelihood. That is, for the negative log-likelihood loss function, we show that the minimization of PAC-Bayesian generalization risk bounds maximizes the Bayesian marginal likelihood. This provides an alternative explanation to the Bayesian Occam's razor criteria, under the assumption that the data is generated by an i.i.d distribution. Moreover, as the negative log-likelihood is an unbounded loss function, we motivate and propose a PAC-Bayesian theorem tailored for the sub-gamma loss family, and we show that our approach is sound on classical Bayesian linear regression tasks.Comment: Published at NIPS 2015 (http://papers.nips.cc/paper/6569-pac-bayesian-theory-meets-bayesian-inference

PAC-Bayes-empirical-Bernstein inequality

We present PAC-Bayes-Empirical-Bernstein inequality. The inequality is based on combination of PAC-Bayesian bounding technique with Empirical Bernstein bound. It allows to take advantage of small empirical variance and is especially useful in regression. We show that when the empirical variance is significantly smaller than the empirical loss PAC-Bayes-Empirical-Bernstein inequality is significantly tighter than PAC-Bayes-kl inequality of Seeger (2002) and otherwise it is comparable. PAC-Bayes-Empirical-Bernstein inequality is an interesting example of application of PAC-Bayesian bounding technique to self-bounding functions. We provide empirical comparison of PAC-Bayes-Empirical-Bernstein inequality with PAC-Bayes-kl inequality on a synthetic example and several UCI datasets

PAC-Bayesian Inequalities for Martingales

We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. Our results extend the PAC-Bayesian analysis in learning theory from the i.i.d. setting to martingales opening the way for its application to importance weighted sampling, reinforcement learning, and other interactive learning domains, as well as many other domains in probability theory and statistics, where martingales are encountered. We also present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the [0,1] interval by the expectation of the same function of independent Bernoulli variables. This inequality is applied to derive a tighter analog of Hoeffding-Azuma's inequality

Online Clustering of Bandits

We introduce a novel algorithmic approach to content recommendation based on adaptive clustering of exploration-exploitation ("bandit") strategies. We provide a sharp regret analysis of this algorithm in a standard stochastic noise setting, demonstrate its scalability properties, and prove its effectiveness on a number of artificial and real-world datasets. Our experiments show a significant increase in prediction performance over state-of-the-art methods for bandit problems.Comment: In E. Xing and T. Jebara (Eds.), Proceedings of 31st International Conference on Machine Learning, Journal of Machine Learning Research Workshop and Conference Proceedings, Vol.32 (JMLR W&CP-32), Beijing, China, Jun. 21-26, 2014 (ICML 2014), Submitted by Shuai Li (https://sites.google.com/site/shuailidotsli

A Primer on PAC-Bayesian Learning

International audienc

BOF-UCB: A Bayesian-Optimistic Frequentist Algorithm for Non-Stationary Contextual Bandits

We propose a novel Bayesian-Optimistic Frequentist Upper Confidence Bound (BOF-UCB) algorithm for stochastic contextual linear bandits in non-stationary environments. This unique combination of Bayesian and frequentist principles enhances adaptability and performance in dynamic settings. The BOF-UCB algorithm utilizes sequential Bayesian updates to infer the posterior distribution of the unknown regression parameter, and subsequently employs a frequentist approach to compute the Upper Confidence Bound (UCB) by maximizing the expected reward over the posterior distribution. We provide theoretical guarantees of BOF-UCB's performance and demonstrate its effectiveness in balancing exploration and exploitation on synthetic datasets and classical control tasks in a reinforcement learning setting. Our results show that BOF-UCB outperforms existing methods, making it a promising solution for sequential decision-making in non-stationary environments