How Negative Dependence Broke the Quadratic Barrier for Learning with Graphs and Kernels

Jun 14, 2019



As we advance with resources, we move from reasoning on entities to reasoning on pairs and groups. We have beautiful frameworks: graphs, kernels, DPPs. However, the methods that work with pairs, relationships, and similarity are just slow. Kernel regression, or second-order gradient methods, or sampling from DPPs do not scale to large data, because of the costly construction and storing of matrix K_n. Prior work showed that sampling points according to their ridge leverage scores (RLS) generates small dictionaries with strong spectral approximation guarantees for K_n. However, computing exact RLS requires building and storing the whole kernel matrix. In this talk, we start with SQUEAK, a new online approach for kernel approximations based on RLS sampling that sequentially processes the data, storing a dictionary with a number of points that only depends on the effective dimension d_eff(gamma) of the dataset. The beauty of negative dependence, that we estimate on the fly, makes it possible to exclude huge portions of dictionary. With the small dictionary, SQUEAK never constructs the whole matrix K_n, runs in linear time O(n*d_eff(gamma)^3) w.r.t. n, and requires only a single pass over the dataset. A distributed version of SQUEAK runs in as little as O(log(n)*d_eff(gamma)^3) time. This online tool opens out a range of possibilities to finally have scalable, adaptive, and provably accurate kernel methods: semi-supervised learning or Laplacian smoothing on large graphs, scalable GP-UCB, efficient second-order kernel online learning, and even fast DPP sampling, some of these being featured in this workshop.



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The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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