Minimum Cost Flows, MDPs, and $\ell_1$-Regression in Nearly Linear Time for Dense Instances

    In the case of $\ell_1$-regression in a matrix with $n$-columns and $m$-rows we obtain a randomized method which computes an $\epsilon$-approximate solution in $\tilde{O}(mn + n^{2.5})$ time. This yields a randomized method which computes an $\epsilon$-optimal policy of a discounted Markov Decision Process with $S$ states and, $A$ actions per state in time $\tilde{O}(S^2 A + S^{2.5})$. These methods improve upon the previous best runtimes of methods which depend polylogarithmically on problem parameters, which were $\tilde{O}(mn^{1.5})$ \cite{LeeS15} and $\tilde{O}(S^{2.5} A)$ \cite{ls14,SidfordWWY18} respectively.

    To obtain this result we introduce two new algorithmic tools of possible independent interest. First, we design a new general interior point method for solving linear programs with two sided constraints which combines techniques from \cite{lsz19} and \cite{BrandLN+20} to obtain a robust stochastic method with iteration count nearly the square root of the smaller dimension.  Second, to implement this method we provide dynamic data structures for efficiently maintaining approximations to variants of Lewis-weights, a fundamental importance measure for matrices which generalize leverage scores and effective resistances.