  In this paper, we consider the linear programming (LP) formulation for deep
reinforcement learning. The number of the constraints depends on the size of
state and action spaces, which makes the problem intractable in large or
continuous environments. The general augmented Lagrangian method suffers the
double-sampling obstacle in solving the LP. Namely, the conditional
expectations originated from the constraint functions and the quadratic
penalties in the augmented Lagrangian function impose difficulties in sampling
and evaluation. Motivated from the updates of the multipliers, we overcome the
obstacles in minimizing the augmented Lagrangian function by replacing the
intractable conditional expectations with the multipliers. Therefore, a deep
parameterized augment Lagrangian method is proposed. Furthermore, the
replacement provides a promising breakthrough to integrate the two steps in the
augmented Lagrangian method into a single constrained problem. A general
theoretical analysis shows that the solutions generated from a sequence of the
constrained optimizations converge to the optimal solution of the LP if the
error is controlled properly. A theoretical analysis on the quadratic penalty
algorithm under neural tangent kernel setting shows the residual can be
arbitrarily small if the parameter in network and optimization algorithm is
chosen suitably. Preliminary experiments illustrate that our method is
competitive to other state-of-the-art algorithms.
