PAC-Bayesian Analysis of the Exploration-Exploitation Trade-off

We develop a coherent framework for integrative simultaneous analysis of the exploration-exploitation and model order selection trade-offs. We improve over our preceding results on the same subject (Seldin et al., 2011) by combining PAC-Bayesian analysis with Bernstein-type inequality for martingales. Such a combination is also of independent interest for studies of multiple simultaneously evolving martingales.Comment: On-line Trading of Exploration and Exploitation 2 - ICML-2011 workshop. http://explo.cs.ucl.ac.uk/workshop

A General Framework for Updating Belief Distributions

We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered under the special case of using self information loss. Modern application areas make it is increasingly challenging for Bayesians to attempt to model the true data generating mechanism. Moreover, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our proposed framework uses loss-functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known, yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.Comment: This is the pre-peer reviewed version of the article "A General Framework for Updating Belief Distributions", which has been accepted for publication in the Journal of Statistical Society - Series B. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archivin

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

An Improvement to the Domain Adaptation Bound in a PAC-Bayesian context

This paper provides a theoretical analysis of domain adaptation based on the PAC-Bayesian theory. We propose an improvement of the previous domain adaptation bound obtained by Germain et al. in two ways. We first give another generalization bound tighter and easier to interpret. Moreover, we provide a new analysis of the constant term appearing in the bound that can be of high interest for developing new algorithmic solutions.Comment: NIPS 2014 Workshop on Transfer and Multi-task learning: Theory Meets Practice, Dec 2014, Montr{\'e}al, Canad

Domain adaptation of weighted majority votes via perturbed variation-based self-labeling

In machine learning, the domain adaptation problem arrives when the test (target) and the train (source) data are generated from different distributions. A key applied issue is thus the design of algorithms able to generalize on a new distribution, for which we have no label information. We focus on learning classification models defined as a weighted majority vote over a set of real-val ued functions. In this context, Germain et al. (2013) have shown that a measure of disagreement between these functions is crucial to control. The core of this measure is a theoretical bound--the C-bound (Lacasse et al., 2007)--which involves the disagreement and leads to a well performing majority vote learning algorithm in usual non-adaptative supervised setting: MinCq. In this work, we propose a framework to extend MinCq to a domain adaptation scenario. This procedure takes advantage of the recent perturbed variation divergence between distributions proposed by Harel and Mannor (2012). Justified by a theoretical bound on the target risk of the vote, we provide to MinCq a target sample labeled thanks to a perturbed variation-based self-labeling focused on the regions where the source and target marginals appear similar. We also study the influence of our self-labeling, from which we deduce an original process for tuning the hyperparameters. Finally, our framework called PV-MinCq shows very promising results on a rotation and translation synthetic problem

Explicit Learning Curves for Transduction and Application to Clustering and Compression Algorithms

Inductive learning is based on inferring a general rule from a finite data set and using it to label new data. In transduction one attempts to solve the problem of using a labeled training set to label a set of unlabeled points, which are given to the learner prior to learning. Although transduction seems at the outset to be an easier task than induction, there have not been many provably useful algorithms for transduction. Moreover, the precise relation between induction and transduction has not yet been determined. The main theoretical developments related to transduction were presented by Vapnik more than twenty years ago. One of Vapnik's basic results is a rather tight error bound for transductive classification based on an exact computation of the hypergeometric tail. While tight, this bound is given implicitly via a computational routine. Our first contribution is a somewhat looser but explicit characterization of a slightly extended PAC-Bayesian version of Vapnik's transductive bound. This characterization is obtained using concentration inequalities for the tail of sums of random variables obtained by sampling without replacement. We then derive error bounds for compression schemes such as (transductive) support vector machines and for transduction algorithms based on clustering. The main observation used for deriving these new error bounds and algorithms is that the unlabeled test points, which in the transductive setting are known in advance, can be used in order to construct useful data dependent prior distributions over the hypothesis space