Abstract:

Given a matrix A∈ℝn×d and a vector b∈ℝd, we show how to compute an ϵ-approximate solution to the regression problem minx∈ℝd12‖Ax−b‖22 in time Õ ((n+d⋅κsum‾‾‾‾‾‾‾√)⋅s⋅logϵ−1) where κsum=tr(A⊤A)/λmin(ATA) and s is the maximum number of non-zero entries in a row of A. Our algorithm improves upon the previous best running time of Õ ((n+n⋅κsum‾‾‾‾‾‾‾√)⋅s⋅logϵ−1). We achieve our result through a careful combination of leverage score sampling techniques, proximal point methods, and accelerated coordinate descent. Our method not only matches the performance of previous methods, but further improves whenever leverage scores of rows are small (up to polylogarithmic factors). We also provide a non-linear generalization of these results that improves the running time for solving a broader class of ERM problems.

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