This paper presents a margin-based multiclass generalization bound for neural networks that scales with their margin-normalized spectral complexity: their Lipschitz constant, meaning the product of the spectral norms of the weight matrices, times a certain correction factor. This bound is empirically investigated for a standard AlexNet network trained with SGD on the MNIST and CIFAR10 datasets, with both original and random labels; the bound, the Lipschitz constants, and the excess risks are all in direct correlation, suggesting both that SGD selects predictors whose complexity scales with the difficulty of the learning task, and secondly that the presented bound is sensitive to this complexity.

Deep neural networks are notorious for being sensitive to small well-chosen perturbations, and estimating the regularity of such architectures is of utmost importance for safe and robust practical applications. In this paper, we investigate one of the key characteristics to assess the regularity of such methods: the Lipschitz constant of deep learning architectures. First, we show that, even for two layer neural networks, the exact computation of this quantity is NP-hard and state-of-art methods may significantly overestimate it. Then, we both extend and improve previous estimation methods by providing AutoLip, the first generic algorithm for upper bounding the Lipschitz constant of any automatically differentiable function. We provide a power method algorithm working with automatic differentiation, allowing efficient computations even on large convolutions.

Lemma A.2. Algorithm ϕ gives a lower bound on the query γ . That is, ϕ (x, R) γ (x, R).

By exploiting students' response logs, SCM aims to predict their exercise

From the three-points identity of the Bregman divergence (Lemma 3.1 of [9]), KL (x y) KL ( x y) = KL (x x) + ln x ln y,x x (12) The first term in (12) can be bounded by KL (x x) = By the Hölder's inequality, the second term in (12) is bounded as ln x ln y,x x ln x ln y Lemma 5. Consider a block diagonal matrix We prove the lemma via induction on N . This completes the induction proof.Lemma 6. We introduce one more notation before presenting the proof. This leads us to the initialization-dependent convergence rate of Algorithm 1, which we re-state and prove as follows. In addition, if we initialize the players' policies to be uniform policies, i.e., The rest of the proof follows by putting all the aforementioned results together.