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

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 Kn. Prior work showed that sampling points according to their ridge leverage scores (RLS) generates small dictionaries with strong spectral approximation guarantees for Kn. 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 deff(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 Kn, runs in linear time O(nd_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.