Universal Approximation Under Constraints is Possible with Transformers

• Main Authors: Kratsios, Anastasis, Zamanlooy, Behnoosh, Liu, Tianlin, Dokmanić, Ivan
• Format: Journal Article
• Language: English
• Published: 07.10.2021
• Subjects: Computer Science - Artificial Intelligence; Computer Science - Learning; Computer Science - Neural and Evolutionary Computing; Mathematics - Functional Analysis; Mathematics - Metric Geometry
• DOI: 10.48550/arxiv.2110.03303

ICLR 2022 (Spotlight) Many practical problems need the output of a machine learning model to satisfy a set of constraints, $K$. Nevertheless, there is no known guarantee that classical neural network architectures can exactly encode constraints while simultaneously achieving universality. We provide a quantitative constrained universal approximation theorem which guarantees that for any non-convex compact set $K$ and any continuous function $f:\mathbb{R}^n\rightarrow K$, there is a probabilistic transformer $\hat{F}$ 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's Maximum Theorem (1963). 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 $\hat{F}$ that approximately minimizes $L$ and whose outputs belong to $K$; moreover, $\hat{F}$ approximately satisfies the soft constraints. Our results imply the first universal approximation theorem for classical transformers with exact convex constraint satisfaction. They also yield that a chart-free universal approximation theorem for Riemannian manifold-valued functions subject to suitable geodesically convex constraints.