We explore conditions for when the gradient of a deep declarative node can be approximated by ignoring constraint terms and still result in a descent direction for the global loss function. This has important practical application when training deep learning models since the approximation is often computationally much more efficient than the true gradient calculation. We provide theoretical analysis for problems with linear equality constraints and normalization constraints, and show examples where the approximation works well in practice as well as some cautionary tales for when it fails.

We introduce a new class of end-to-end learnable models wherein data processing nodes (or network layers) are defined in terms of desired behavior rather than an explicit forward function. Specifically, the forward function is implicitly defined as the solution to a mathematical optimization problem. Consistent with nomenclature in the programming languages community, we name our models deep declarative networks. Importantly, we show that the class of deep declarative networks subsumes current deep learning models. Moreover, invoking the implicit function theorem, we show how gradients can be back-propagated through declaratively defined data processing nodes thereby enabling end-to-end learning. We show how these declarative processing nodes can be implemented in the popular PyTorch deep learning software library allowing declarative and imperative nodes to co-exist within the same network. We provide numerous insights and illustrative examples of declarative nodes and demonstrate their application for image and point cloud classification tasks.