Statistical Guarantees of Generative Adversarial Networks for Distribution Estimation

Generative Adversarial Networks (GANs) have achieved great success in unsupervised learning. Despite the remarkable empirical performance, there are limited theoretical understandings on the statistical properties of GANs. This paper provides statistical guarantees of GANs for the estimation of data distributions which have densities in a H\"{o}lder space. Our main result shows that, if the generator and discriminator network architectures are properly chosen (universally for all distributions with H\"{o}lder densities), GANs are consistent estimators of the data distributions under strong discrepancy metrics, such as the Wasserstein distance. To our best knowledge, this is the first statistical theory of GANs for H\"{o}lder densities. In comparison with existing works, our theory requires minimum assumptions on data distributions. Our generator and discriminator networks utilize general weight matrices and the non-invertible ReLU activation function, while many existing works only apply to invertible weight matrices and invertible activation functions. In our analysis, we decompose the error into a statistical error and an approximation error by a new oracle inequality, which may be of independent interest

Towards Understanding Hierarchical Learning: Benefits of Neural Representations

Deep neural networks can empirically perform efficient hierarchical learning, in which the layers learn useful representations of the data. However, how they make use of the intermediate representations are not explained by recent theories that relate them to "shallow learners" such as kernels. In this work, we demonstrate that intermediate neural representations add more flexibility to neural networks and can be advantageous over raw inputs. We consider a fixed, randomly initialized neural network as a representation function fed into another trainable network. When the trainable network is the quadratic Taylor model of a wide two-layer network, we show that neural representation can achieve improved sample complexities compared with the raw input: For learning a low-rank degree-$p$ polynomial ($p \geq 4$) in $d$ dimension, neural representation requires only $\tilde{O}(d^{\lceil p/2 \rceil})$ samples, while the best-known sample complexity upper bound for the raw input is $\tilde{O}(d^{p-1})$. We contrast our result with a lower bound showing that neural representations do not improve over the raw input (in the infinite width limit), when the trainable network is instead a neural tangent kernel. Our results characterize when neural representations are beneficial, and may provide a new perspective on why depth is important in deep learning.Comment: 41 pages, published in NeurIPS 202