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Bounding the generalization error of learning algorithms has a long history,which yet falls short in explaining various generalization successes includingthose of deep learning. Two important difficulties are (i) exploiting thedependencies between the hypotheses, (ii) exploiting the dependence between thealgorithm's input and output. Progress on the first point was made with thechaining method, originating from the work of Kolmogorov, and used in theVC-dimension bound. More recently, progress on the second point was made withthe mutual information method by Russo and Zou '15. Yet, these two methods arecurrently disjoint. In this paper, we introduce a technique to combine thechaining and mutual information methods, to obtain a generalization bound thatis both algorithm-dependent and that exploits the dependencies between thehypotheses. We provide an example in which our bound significantly outperformsboth the chaining and the mutual information bounds. As a corollary, we tightenDudley's inequality when the learning algorithm chooses its output from a smallsubset of hypotheses with high probability.

Chaining Mutual Information and Tightening Generalization Bounds