Abstract
We consider the sparsification of sums $F : \R^n \to \R_+$ where $F(x) = f_1(<a_1,x>) + ... + f_m(<a_m,x>)$ for vectors $a_1, ..., a_m \in \R^n$ and functions $f_1, ... ,f_m : \R \to \R_+$. We show that $(1+\eps)$-approximate sparsifiers of $F$ with support size $\frac{n}{\eps^2} (\log \frac{n}{\eps})^{O(1)}$ exist whenever the functions $f_1, ... ,f_m$ are symmetric, monotone, and satisfy natural growth bounds. Additionally, we give efficient algorithms to compute such a sparsifier assuming each $f_i$ can be evaluated efficiently.
Our results generalize the classic case of $\ell_p$ sparsification, where $f_i(z) = |z|^p$, for $p \in (0, 2]$, and give the first near-linear size sparsifiers in the well-studied setting of the Huber loss function and its generalizations, e.g., $f_i(z) = \min\{|z|^p, |z|^2\}$ for $0 < p \leq 2$. Our sparsification algorithm can be applied to give near-optimal reductions for optimizing a variety of generalized linear models including $\ell_p$ regression for $p \in (1, 2]$ to high accuracy, via solving $(\log n)^{O(1)}$ sparse regression instances with $m \le n(\log n)^{O(1)}$, plus runtime proportional to the number of nonzero entries in the vectors $a_1, \dots, a_m$.
Based on joint work with Arun Jambulapati, James Lee, and Aaron Sidford.