Data-dependent compression of random features for large-scale kernel approximation
Author(s)Agrawal, Raj; Campbell, Trevor David; Huggins, Jonathan H.; Broderick, Tamara A
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Kernel methods offer the flexibility to learn complex relationships in modern, large data sets while enjoying strong theoretical guarantees on quality. Unfortunately, these methods typically require cubic running time in the data set size, a prohibitive cost in the large-data setting. Random feature maps (RFMs) and the Nyström method both consider low-rank approximations to the kernel matrix as a potential solution. But, in order to achieve desirable theoretical guarantees, the former may require a prohibitively large number of features J+, and the latter may be prohibitively expensive for high-dimensional problems. We propose to combine the simplicity and generality of RFMs with a data-dependent feature selection scheme to achieve desirable theoretical approximation properties of Nyström with just O(log J+) features. Our key insight is to begin with a large set of random features, then reduce them to a small number of weighted features in a data-dependent, computationally efficient way, while preserving the statistical guarantees of using the original large set of features. We demonstrate the efficacy of our method with theory and experiments-including on a data set with over 50 million observations. In particular, we show that our method achieves small kernel matrix approximation error and better test set accuracy with provably fewer random features than state-of-the-art methods.
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS)
Agrawal, Raj et al. “Data-dependent compression of random features for large-scale kernel approximation.” Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), 89, (April 2019) © 2019 The Author(s)