Online kernel selection via tensor sketching

Abstract

Online kernel selection is a more complex problem compared with offline kernel selection, which intermixes training and selection at each round and requires a sublinear regret and low computational complexities. But existing online kernel selection approaches have at least linear time and space complexities at each round with respect to the number of rounds, or lack sublinear regret guarantees for an uncountably infinite number of candidate kernels. To address these issues, we propose a novel online kernel selection approach using tensor sketching, which has constant computational complexities at each round and enjoys a sublinear regret bound for an uncountably infinite number of candidate kernels. We represent the data using the tensor products and construct data sketches using the Taylor series and the Count Sketch matrices, which yields a sketched reproducing kernel Hilbert space (SRKHS). Then we update the optimal kernels and the hypotheses using online gradient descent in SRKHS. We prove that the kernel corresponding to SRKHS satisfies the reproducing property, the hypotheses in SRKHS are convex with respect to the kernel parameter, and the proposed online kernel selection approach in SRKHS enjoys a regret bound of order O(√T) for an uncountably infinite number of candidate kernels, which is optimal for a convex loss function, where T is the number of rounds. By the fast Fourier transform, the hypotheses in SRKHS can be computed in a quasilinear time complexity and a logarithmic space complexity with respect to the sketch size at each round, where the sketch size is a constant. Experimental results demonstrate that our online kernel selection approach is more accurate and efficient for online kernel learning on high dimension data.