Analysis of Nyström method with sequential ridge leverage scores

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix Kt. To avoid storing the entire matrix Kt, Nyström methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed K̃t. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, [16, 1] show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for K̃t. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INKESTIMATE maintains a small sketch of Kt, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance ‖Kt− K̃t‖2, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.