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**Robust Regression on MapReduce**

**Authors:** *Xiangrui Meng* and *Michael Mahoney*

**Conference:** Proceedings of the 30th International Conference on Machine Learning (ICML-13)

**Year:** 2013

**Pages:** 888-896

**Abstract:** Although the MapReduce framework is now the \emph{de facto} standard for analyzing massive data sets, many algorithms (in particular, many iterative algorithms popular in machine learning, optimization, and linear algebra) are hard to fit into MapReduce. Consider, \emph{e.g.}, the $\ell_p$ regression problem: given a matrix $A \in \mathbb{R}^{m \times n}$ and a vector $b \in \mathbb{R}^m$, find a vector $x^* \in \mathbb{R}^n$ that minimizes $f(x) = \|A x - b\|_p$. The widely-used $\ell_2$ regression, \emph{i.e.}, linear least-squares, is known to be highly sensitive to outliers; and choosing $p \in [1, 2)$ can help improve robustness. In this work, we propose an efficient algorithm for solving strongly over-determined $(m \gg n)$ robust $\ell_p$ regression problems to moderate precision on MapReduce. Our empirical results on data up to the terabyte scale demonstrate that our algorithm is a significant improvement over traditional iterative algorithms on MapReduce for $\ell_1$ regression, even for a fairly small number of iterations. In addition, our proposed interior-point cutting-plane method can also be extended to solving more general convex problems on MapReduce.

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