Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity

TL;DR

This paper proves that a broad class of feedforward neural networks, definable in an o-minimal structure, have finite sample complexity in the PAC learning framework, encompassing many modern architectures.

Contribution

It establishes that modern non-recurrent neural networks are PAC learnable due to their definability in o-minimal structures, not just specific activation functions or VC arguments.

Findings

Fixed architectures in o-minimal structures have finite sample complexity.

Modern architectures like CNNs, GNNs, and transformers are PAC learnable.

Finite-sample PAC learnability is a general property of tame feedforward computation.

Abstract

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.