Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity
Authors: Anastasis Kratsios, Gregory Cousins, Haitz Sáez de Ocáriz Borde, Bum Jun Kim, Simone Brugiapaglia
Organizations: Department of Mathematics & Statistics, McMaster University, Hamilton, ON, Canada · University of Cambridge, United Kingdom · Graduate School of Engineering, The University of Tokyo, Tokyo, Japan · Department of Mathematics & Statistics, Concordia University, Montréal, QC, Canada
We show that, in a precise sense, a broad class of feedforward neural networks learn (have finite sample complexity) in the PAC model: every fixed finite feedforward architecture whose layers are definable in an o-minimal structure has finite sample complexity in the agnostic PAC setting, even with unbounded parameters. This covers standard fixed-size MLPs, CNNs, GNNs, and transformers with fixed sequence length, together with the operations and layers typically used in such architectures, including linear projections, residual connections, attention mechanisms, pooling layers, normalization layers, and admissible positional encodings. Hence, distribution-free learnability for modern non-recurrent architectures is not an exceptional property of particular activations or architecture-specific VC arguments, but a consequence of tame feedforward computation. Our results reposition finite-sample PAC learnability as a baseline rather than a differentiator: they shift the focus of architectural comparison toward inductive biases, symmetries and geometric priors, scalability, and optimization behaviour.