The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators

Since the time of Gauss, it has been generally accepted that ₂-methods of combining observations by minimizing sums of squared errors have significant computational advantages over earlier ₁-methods based on minimization of absolute errors advocated by Boscovich, Laplace and others. However, ₁-methods are known to have significant robustness advantages over ₂-methods in many applications, and related quantile regression methods provide a useful, complementary approach to classical least-squares estimation of statistical models. Combining recent advances in interior point methods for solving linear programs with a new statistical preprocessing approach for ₁-type problems, we obtain a 10- to 100-fold improvement in computational speeds over current (simplex-based) ₁-algorithms in large problems, demonstrating that ₁-methods can be made competitive with ₂-methods in terms of computational speed throughout the entire range of problem sizes. Formal complexity results suggest that ₁-regression can be made faster than least-squares regression for n sufficiently large and p modest.