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Random features provide a practical framework for large-scale kernelapproximation and supervised learning. It has been shown that data-dependentsampling of random features using leverage scores can significantly reduce thenumber of features required to achieve optimal learning bounds. Leverage scoresintroduce an optimized distribution for features based on aninfinite-dimensional integral operator (depending on input distribution), whichis impractical to sample from. Focusing on empirical leverage scores in thispaper, we establish an out-of-sample performance bound, revealing aninteresting trade-off between the approximated kernel and the eigenvalue decayof another kernel in the domain of random features defined based on datadistribution. Our experiments verify that the empirical algorithm consistentlyoutperforms vanilla Monte Carlo sampling, and with a minor modification themethod is even competitive to supervised data-dependent kernel learning,without using the output (label) information.

On Sampling Random Features From Empirical Leverage Scores: Implementation and Theoretical Guarantees