  Deep declarative networks and other recent related works have shown how to
differentiate the solution map of a (continuous) parametrized optimization
problem, opening up the possibility of embedding mathematical optimization
problems into end-to-end learnable models. These differentiability results can
lead to significant memory savings by providing an expression for computing the
derivative without needing to unroll the steps of the forward-pass optimization
procedure during the backward pass. However, the results typically require
inverting a large Hessian matrix, which is computationally expensive when
implemented naively. In this work we study two applications of deep declarative
networks -- robust vector pooling and optimal transport -- and show how problem
structure can be exploited to obtain very efficient backward pass computations
in terms of both time and memory. Our ideas can be used as a guide for
improving the computational performance of other novel deep declarative nodes.
