Although gradient descent (GD) almost always escapes saddle points asymptotically [Lee et al., 2016], this paper shows that even with fairly natural random initialization schemes and non-pathological functions, GD can be significantly slowed down by saddle points, taking exponential time to escape. On the other hand, gradient descent with perturbations [Ge et al., 2015, Jin et al., 2017] is not slowed down by saddle points--it can find an approximate local minimizer in polynomial time. This result implies that GD is inherently slower than perturbed GD, and justifies the importance of adding perturbations for efficient non-convex optimization. While our focus is theoretical, we also present experiments that illustrate our theoretical findings.

Generative moment matching network (GMMN) is a deep generative model that differs from Generative Adversarial Network (GAN) by replacing the discriminator in GAN with a two-sample test based on kernel maximum mean discrepancy (MMD). Although some theoretical guarantees of MMD have been studied, the empirical performance of GMMN is still not as competitive as that of GAN on challenging and large benchmark datasets. The computational efficiency of GMMN is also less desirable in comparison with GAN, partially due to its requirement for a rather large batch size during the training. In this paper, we propose to improve both the model expressiveness of GMMN and its computational efficiency by introducing adversarial kernel learning techniques, as the replacement of a fixed Gaussian kernel in the original GMMN. The new approach combines the key ideas in both GMMN and GAN, hence we name it MMD GAN.

We study minimax convergence rates of nonparametric density estimation under a large class of loss functions called "adversarial losses", which, besides classical L

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

While graph kernels (GKs) are easy to train and enjoy provable theoretical guarantees, their practical performances are limited by their expressive power, as the kernel function often depends on hand-crafted combinatorial features of graphs. Compared to graph kernels, graph neural networks (GNNs) usually achieve better practical performance, as GNNs use multi-layer architectures and non-linear activation functions to extract high-order information of graphs as features. However, due to the large number of hyper-parameters and the non-convex nature of the training procedure, GNNs are harder to train. Theoretical guarantees of GNNs are also not well-understood. Furthermore, the expressive power of GNNs scales with the number of parameters, and thus it is hard to exploit the full power of GNNs when computing resources are limited. The current paper presents a new class of graph kernels, Graph Neural Tangent Kernels (GNTKs), which correspond to infinitely wide multi-layer GNNs trained by gradient descent. GNTKs enjoy the full expressive power of GNNs and inherit advantages of GKs. Theoretically, we show GNTKs provably learn a class of smooth functions on graphs. Empirically, we test GNTKs on graph classification datasets and show they achieve strong performance.

While graph kernels (GKs) are easy to train and enjoy provable theoretical guarantees, their practical performances are limited by their expressive power, as the kernel function often depends on hand-crafted combinatorial features of graphs. Compared to graph kernels, graph neural networks (GNNs) usually achieve better practical performance, as GNNs use multi-layer architectures and non-linear activation functions to extract high-order information of graphs as features. However, due to the large number of hyper-parameters and the non-convex nature of the training procedure, GNNs are harder to train. Theoretical guarantees of GNNs are also not well-understood. Furthermore, the expressive power of GNNs scales with the number of parameters, and thus it is hard to exploit the full power of GNNs when computing resources are limited. The current paper presents a new class of graph kernels, Graph Neural Tangent Kernels (GNTKs), which correspond to infinitely wide multi-layer GNNs trained by gradient descent. GNTKs enjoy the full expressive power of GNNs and inherit advantages of GKs. Theoretically, we show GNTKs provably learn a class of smooth functions on graphs. Empirically, we test GNTKs on graph classification datasets and show they achieve strong performance.

Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, due to a lack of practical estimators.

Stochastic gradient-based Monte Carlo methods such as stochastic gradient Langevin dynamics are useful tools for posterior inference on large scale datasets in many machine learning applications. These methods scale to large datasets by using noisy gradients calculated using a mini-batch or subset of the dataset. However, the high variance inherent in these noisy gradients degrades performance and leads to slower mixing. In this paper, we present techniques for reducing variance in stochastic gradient Langevin dynamics, yielding novel stochastic Monte Carlo methods that improve performance by reducing the variance in the stochastic gradient. We show that our proposed method has better theoretical guarantees on convergence rate than stochastic Langevin dynamics. This is complemented by impressive empirical results obtained on a variety of real world datasets, and on four different machine learning tasks (regression, classification, independent component analysis and mixture modeling). These theoretical and empirical contributions combine to make a compelling case for using variance reduction in stochastic Monte Carlo methods.

We provide finite-sample analysis of a general framework for using k-nearest neighbor statistics to estimate functionals of a nonparametric continuous probability density, including entropies and divergences. Rather than plugging a consistent density estimate (which requires k as the sample size n) into the functional of interest, the estimators we consider fix k and perform a bias correction. This is more efficient computationally, and, as we show in certain cases, statistically, leading to faster convergence rates. Our framework unifies several previous estimators, for most of which ours are the first finite sample guarantees.