GloVE [25] is a 300-dimensional word embedding space. It is an dimensionality-representation representation of word-word co-occurrence statistics. BERT-E [10] is a 3072-dimensional contextualized word embedding space extracted from BERT. We used the Flair NLP [1] implementation of BERT embeddings. FLAIR [1] is a 4096-dimensional contextualized character level word embedding space.

For the theoretical exposition, we first establish the following Lemmas. Lemma A.1 proves that the derivative of the function φis bounded in the `2-norm when the domain is restricted to the support of P. Lemma A.1. Lemma A.3 proves that the function fΘ, as a function of Θ, is Lipschitz with respect to the k k norm. Joint first authors contributed equally Corresponding author 35th Conference on Neural Information Processing Systems (NeurIPS 2021). Thus, from equation (1), h φ(PC(θ)) φ(θ),x PC(θ)i 0. (2) We now observe that, dφ(x,θ) dφ(x,PC(θ)) dφ(PC(θ),θ) = h φ(PC(θ)) φ(θ),x PC(θ)i 0. Hence the result.

The SC stands for the spectral complexity defined in [4]. We use the empirical estimation of k-variance and Lipschitz constant defined in section 5 to calculate kV-Margin and kV-GN-Margin. B.2 Variance of Empirical Estimation In Table 1, we show the average scores over 4 random sampled subsets. We now show the standard deviation in Table 4. Overall, the standard deviation of the estimation is fairly small, consistent to the observation in Theorem 7.

Understanding the generalization of deep neural networks is one of the most important tasks in deep learning. Although much progress has been made, theoretical error bounds still often behave disparately from empirical observations. In this work, we develop margin-based generalization bounds, where the margins are normalized with optimal transport costs between independent random subsets sampled from the training distribution. In particular, the optimal transport cost can be interpreted as a generalization of variance which captures the structural properties of the learned feature space. Our bounds robustly predict the generalization error, given training data and network parameters, on large scale datasets. Theoretically, we demonstrate that the concentration and separation of features play crucial roles in generalization, supporting empirical results in the literature.