In recent years, machine learning has shown great promise in a variety of real-world applications.

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We still need to demonstrate that the properties in P AC-Bayes analysis hold for both the margin operator and the robust margin operator. Then we complete the proof of Lemma 6.1. The proof of Lemma 7.1 and 7.2 is similar. We provide the proof of Lemma 7.2 below. Lemma 7.1 follows the proof of Lemma 7.2 by replacing the robust margin operator by the margin Since the above bound holds for any x in the domain X, we can get the following a.s.: R The second inequality is the tail bound above.

Inprinciple, onecandesign Lipschitz constrained architectures using the composition property of Lipschitz functions, but Anil et al.[2] recently identified a key obstacle to this approach: gradient norm attenuation.