Conditional mean embeddings as regressors

Grunewalder, S. and Lever, G. and Gretton, A. and Baldassarre, L. and Patterson, S. and Pontil, M. (2012) Conditional mean embeddings as regressors. In: Proceedings of the 29th International Conference on Machine Learning, Edinburgh, Scotland, 2012 :. UNSPECIFIED.

Abstract

We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vector-valued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embeddings and a justification for their use. Furthermore, the equivalence allows the application of vector-valued regression methods and results to the problem of learning conditional distributions. Using this link we derive a sparse version of the embedding by considering alternative formulations. Further, by applying convergence results for vector-valued regression to the embedding problem we derive minimax convergence rates which are O(log(n)=n) – compared to current state of the art rates of O(n􀀀1=4) – and are valid under milder and more intuitive assumptions. These minimax upper rates coincide with lower rates up to a logarithmic factor, showing that the embedding method achieves nearly optimal rates. We study our sparse embedding algorithm in a reinforcement learning task where the algorithm shows significant improvement in sparsity over an incomplete Cholesky decomposition.