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In this paper we first identify a basic limitation in gradient descent-basedoptimization methods when used in conjunctions with smooth kernels. An analysisbased on the spectral properties of the kernel demonstrates that only avanishingly small portion of the function space is reachable after a polynomialnumber of gradient descent iterations. This lack of approximating powerdrastically limits gradient descent for a fixed computational budget leading toserious over-regularization/underfitting. The issue is purely algorithmic,persisting even in the limit of infinite data.  To address this shortcoming in practice, we introduce EigenPro iteration,based on a preconditioning scheme using a small number of approximatelycomputed eigenvectors. It can also be viewed as learning a new kernel optimizedfor gradient descent. It turns out that injecting this small (computationallyinexpensive and SGD-compatible) amount of approximate second-order informationleads to major improvements in convergence. For large data, this translatesinto significant performance boost over the standard kernel methods. Inparticular, we are able to consistently match or improve the state-of-the-artresults recently reported in the literature with a small fraction of theircomputational budget.  Finally, we feel that these results show a need for a broader computationalperspective on modern large-scale learning to complement more traditionalstatistical and convergence analyses. In particular, many phenomena oflarge-scale high-dimensional inference are best understood in terms ofoptimization on infinite dimensional Hilbert spaces, where standard algorithmscan sometimes have properties at odds with finite-dimensional intuition. Asystematic analysis concentrating on the approximation power of such algorithmswithin a budget of computation may lead to progress both in theory andpractice.

Diving into the shallows: a computational perspective on large-scale shallow learning