Implicitly and explicitly constrained optimization problems for training of recurrent neural networks

Training of recurrent neural networks is typically formulated as unconstrained optimization problems. There is, however, an implicit constraint stating that the equations of state must be satisfied at every iteration in the optimization process. Such constraints can make a problem highly non-linear and thus difficult to solve. A potential remedy is to reformulate the problem into one in which the parameters and state are treated as independent variables and all constraints appear explicitly. In this paper we compare an implicitly and an explicitly constrained formulation of the same problem. Reported numerical results suggest that the latter is in some respects superior.