Smooth Learning with Hard Constraints via Legendre-Regularized Policies

What the math gives to ML

The paper's central transferable construction is a policy layer whose output is the solution of a strictly convex, Legendre-regularized optimization problem over the original feasible set. Unlike penalty-based constraint handling, this gives exact feasibility while retaining a single-valued differentiable map from network logits to decisions; the inverse regularizer Hessian supplies an explicit Jacobian for backpropagation. This is particularly useful for neural policies with polyhedral resource, budget, flow, or allocation constraints, where projection layers can be nondifferentiable at active-set changes. The most practical adaptation is a trainable constrained decision head using a Legendre or log-barrier layer, with the paper's approximation result motivating a sufficiently expressive context encoder upstream.