`1 Regression using Lewis Weights Preconditioning

We consider the `1 minimization problem minx ‖Ax − b‖1 in the overconstrained case, commonly known as the Least Absolute Deviations problem, where there are far more constraints than variables. More specifically, we have A ∈ Rn×d for n d. Many important combinatorial problems, such as minimum cut and shortest path, can be formulated as `1 regression problems [CMMP13]. We follow the general paradigm of preconditioning the matrix and solving the resulting problem with gradient descent techniques, and our primary insight will be that these methods are actually interdependent in the context of this problem. The key idea will be that preconditioning from [CP15] allows us to obtain an isotropic matrix with fewer rows and strong upper bounds on all row norms. We leverage these conditions to find a careful initialization, which we use along with smoothing reductions in [AH16] and the accelerated stochastic gradient descent algorithms in [All17] to achieve relative error in about nnz(A)+ndω−1+ √ nd −1 time with high probability. Moreover, we can also assume n ≤ O(d −2 log n) from preconditioning. This improves over the previous best result using gradient descent for `1 regression [YCRM16], and is comparable to the best known running times for interior point methods [LS15]. Finally, we also show that if our original matrix A is approximately isotropic and the row norms are approximately equal, we can avoid using fast matrix multiplication and prove a running time of about nnz(A)+sd −2 +d −2, where s is the maximum number of non-zeros in a row of A. ∗Georgia Institute of Technology. email:[email protected] †Georgia Institute of Technology. email:[email protected] ‡Georgia Institute of Technology. email:[email protected]