  We propose a novel randomized linear programming algorithm for approximating
the optimal policy of the discounted Markov decision problem. By leveraging the
value-policy duality and binary-tree data structures, the algorithm adaptively
samples state-action-state transitions and makes exponentiated primal-dual
updates. We show that it finds an $\epsilon$-optimal policy using nearly-linear
run time in the worst case. When the Markov decision process is ergodic and
specified in some special data formats, the algorithm finds an
$\epsilon$-optimal policy using run time linear in the total number of
state-action pairs, which is sublinear in the input size. These results provide
a new venue and complexity benchmarks for solving stochastic dynamic programs.
