Many practical problems need the output of a machine learning model to satisfy a set of constraints, K. There are, however, no known guarantees that classical neural networks can exactly encode constraints while simultaneously achieving universality. We provide a quantitative constrained universal approximation theorem which guarantees that for any convex or non-convex compact set K and any continuous function f : R n → K, there is a probabilistic transformer F whose randomized outputs all lie in K and whose expected output uniformly approximates f . Our second main result is a "deep neural version" of Berge (1963)'s Maximum Theorem. The result guarantees that given an objective function L, a constraint set K, and a family of soft constraint sets, there is a probabilistic transformer F that approximately minimizes L and whose outputs belong to K; moreover, F approximately satisfies the soft constraints. Our results imply the first universal approximation theorem for classical transformers with exact convex constraint satisfaction, and a chart-free universal approximation theorem for Riemannian manifold-valued functions subject to geodesically-convex constraints.

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