Learning with Spectral Kernels and Heavy-Tailed Data

Two ubiquitous aspects of large-scale data analysis are that the data often have heavytailed properties and that diffusion-based or spectral-based methods are often used to identify and extract structure of interest. Perhaps surprisingly, popular distribution-independent methods such as those based on the VC dimension fail to provide nontrivial results for even simple learning problems such as binary classification in these two settings. In this paper, we develop distribution-dependent learning methods that can be used to provide dimensionindependent sample complexity bounds for the binary classification problem in these two popular settings. In particular, we provide bounds on the sample complexity of maximum margin classifiers when the magnitude of the entries in the feature vector decays according to a power law and also when learning is performed with the so-called Diffusion Maps kernel. Both of these results rely on bounding the annealed entropy of gap-tolerant classifiers in a Hilbert space. We provide such a bound, and we demonstrate that our proof technique generalizes to the case when the margin is measured with respect to more general Banach space norms. The latter result is of potential interest in cases where modeling the relationship between data elements as a dot product in a Hilbert space is too restrictive. Two ubiquitous aspects of large-scale data analysis are that the data often have heavy-tailed properties and that diffusion-based or spectral-based methods are often used to identify and extract structure of interest. In the absence of strong assumptions on the data, popular distributionindependent methods such as those based on the VC dimension fail to provide nontrivial results for even simple learning problems such as binary classification in these two settings. At root, the reason is that in both of these situations the data are formally very high dimensional and that (without additional regularity assumptions on the data) there may be a small number of “very outlying” data points. In this paper, we develop distribution-dependent learning methods that can be used to provide dimension-independent sample complexity bounds for the maximum margin version of the binary classification problem in these two popular settings. In both cases, we are able to obtain nearly optimal linear classification hyperplanes since the distribution-dependent tools we employ are able to control the aggregate effect of the “outlying” data points. In particular, our results will hold even though the data may be infinite-dimensional and unbounded.