Title: Lagrangian Method for Episodic Learning. Abstract: This paper considers the problem of learning optimal value functions for finite-time decision tasks via saddle-point optimization of a nonlinear Lagrangian function that is derived from the $Q$-form Bellman optimality equation. Despite a long history of research on this topic in the literature, previous works on this general approach have been almost exclusively focusing on a linear special case known as the linear programming approach to RL/MDP. Our paper brings new perspectives to this general approach in the following aspects: 1) Inspired by the usually-used linear $V$-form Lagrangian, we proposed a nonlinear $Q$-form Lagrangian function and proved that it enjoys strong duality property in spite of its nonlinearity. The Lagrangian duality property immediately leads to a new imitation learning algorithm, which we applied to Machine Translation and obtained favorable performance on standard MT benchmark. 2) We pointed out a fundamental limit of existing works, which seeks to find minimax-type saddle points of the Lagrangian function. We proved that another class of saddle points, the maximin-type ones, turn out to have better optimality property. 3) In contrast to most previous works, our theory and algorithm are oriented to the undiscounted episode-wise reward, which is practically more relevant than the usually considered discounted-MDP setting, thus have filled a gap between theory and practice on the topic.