Fast Regression with an $\ell_\infty$ Guarantee

Sketching has emerged as a powerful technique for speeding up problems in numerical linear algebra, such as regression. In the overconstrained regression problem, one is given an $n \times d$ matrix $A$, with $n \gg d$, as well as an $n \times 1$ vector $b$, and one wants to find a vector $\hat{x}$ so as to minimize the residual error $|Ax-b|2$. Using the sketch and solve paradigm, one first computes $S \cdot A$ and $S \cdot b$ for a randomly chosen matrix $S$, then outputs $x' = (SA)^{\dagger} Sb$ so as to minimize $|SAx' - Sb|2$. The sketch-and-solve paradigm gives a bound on $|x'-x^*|_2$ when $A$ is well-conditioned. Our main result is that, when $S$ is the subsampled randomized Fourier/Hadamard transform, the error $x' - x^*$ behaves as if it lies in a "random" direction within this bound: for any fixed direction $a\in \mathbb{R}^d$, we have with $1 - d^{-c}$ probability that [ \langle a, x'-x^*\rangle \lesssim \frac{|a|2|x'-x^*|2}{d^{{\frac{1}{2}-\gamma}},} \quad (1) ] where $c, \gamma > 0$ are arbitrary constants. This implies $|x'-x^*|_{\infty}$ is a factor $d^{{\frac{1}{2}-\gamma}$} smaller than $|x'-x^*|_2$. It also gives a better bound on the generalization of $x'$ to new examples: if rows of $A$ correspond to examples and columns to features, then our result gives a better bound for the error introduced by sketch-and-solve when classifying fresh examples. We show that not all oblivious subspace embeddings $S$ satisfy these properties. In particular, we give counterexamples showing that matrices based on Count-Sketch or leverage score sampling do not satisfy these properties. We also provide lower bounds, both on how small $|x'-x^*|_2$ can be, and for our new guarantee (1), showing that the subsampled randomized Fourier/Hadamard transform is nearly optimal.