The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent.

Neural Information Processing Systems http://nips.cc/

Furthermore, the computational efficiency of pruning is maintained at both training and testing time despite having to aggregate over an exponential number of subtrees. We believe this is the first subtree aggregation approach with such guarantees.

Neural Information Processing Systems http://nips.cc/

We propose new PAC-Bayes bounds for the risk of the weighted majority vote that depend on the mean and variance of the error of its associated Gibbs classifier. We show that these bounds can be smaller than the risk of the Gibbs classifier and can be arbitrarily close to zero even if the risk of the Gibbs classifier is close to 1/2. Moreover, we show that these bounds can be uniformly estimated on the training data for all possible posteriors Q. Moreover, they can be improved by using a large sample of unlabelled data.

The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by viewing Gibbs classifiers as linear operators. The goal of this paper is to extend this bridge between Rademacher complexity and state-of-the-art PAC-Bayesian theory. We first demonstrate that one can match the fast rate of Catoni's PAC-Bayes bounds (Catoni, 2007) using shifted Rademacher processes (Wegkamp, 2003; Lecu\'{e} and Mitchell, 2012; Zhivotovskiy and Hanneke, 2018). We then derive a new fast-rate PAC-Bayes bound in terms of the "flatness" of the empirical risk surface on which the posterior concentrates. Our analysis establishes a new framework for deriving fast-rate PAC-Bayes bounds and yields new insights on PAC-Bayesian theory.

We present a weighted-majority classification approach over subtrees of a fixed tree, which provably achieves excess-risk of the same order as the best tree-pruning. Furthermore, the computational efficiency of pruning is maintained at both training and testing time despite having to aggregate over an exponential number of subtrees. We believe this is the first subtree aggregation approach with such guarantees. The guarantees are obtained via a simple combination of insights from PAC-Bayes theory, which we believe should be of independent interest, as it generically implies consistency for weighted-voting classifiers w.r.t. Bayes - while, in contrast, usual PAC-bayes approaches only establish consistency of Gibbs classifiers.

We present a weighted-majority classification approach over subtrees of a fixed tree, which provably achieves excess-risk of the same order as the best tree-pruning. Furthermore, the computational efficiency of pruning is maintained at both training and testing time despite having to aggregate over an exponential number of subtrees. We believe this is the first subtree aggregation approach with such guarantees.

Existing guarantees in terms of rigorous upper bounds on the generalization error for the original random forest algorithm, one of the most frequently used machine learning methods, are unsatisfying. We discuss and evaluate various PAC-Bayesian approaches to derive such bounds. The bounds do not require additional hold-out data, because the out-of-bag samples from the bagging in the training process can be exploited. A random forest predicts by taking a majority vote of an ensemble of decision trees. The first approach is to bound the error of the vote by twice the error of the corresponding Gibbs classifier (classifying with a single member of the ensemble selected at random). However, this approach does not take into account the effect of averaging out of errors of individual classifiers when taking the majority vote. This effect provides a significant boost in performance when the errors are independent or negatively correlated, but when the correlations are strong the advantage from taking the majority vote is small. The second approach based on PAC-Bayesian C-bounds takes dependencies between ensemble members into account, but it requires estimating correlations between the errors of the individual classifiers. When the correlations are high or the estimation is poor, the bounds degrade. In our experiments, we compute generalization bounds for random forests on various benchmark data sets. Because the individual decision trees already perform well, their predictions are highly correlated and the C-bounds do not lead to satisfactory results. For the same reason, the bounds based on the analysis of Gibbs classifiers are typically superior and often reasonably tight. Bounds based on a validation set coming at the cost of a smaller training set gave better performance guarantees, but worse performance in most experiments.

We study a two-level multiview learning with more than two views under the PAC-Bayesian framework. This approach, sometimes referred as late fusion, consists in learning sequentially multiple view-specific classifiers at the first level, and then combining these view-specific classifiers at the second level. Our main theoretical result is a generalization bound on the risk of the majority vote which exhibits a term of diversity in the predictions of the view-specific classifiers. From this result it comes out that controlling the trade-off between diversity and accuracy is a key element for multiview learning, which complements other results in multiview learning. Finally, we experiment our principle on multiview datasets extracted from the Reuters RCV1/RCV2 collection.