Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformers

A common lens to theoretically study neural net architectures is to analyze the functions they can approximate. However, constructions from approximation theory may be unrealistic and therefore less meaningful. For example, a common unrealistic trick is to encode target function values using infinite precision. To address these issues, this work proposes a formal definition of statistically meaningful (SM) approximation which requires the approximating network to exhibit good statistical learnability. We study SM approximation for two function classes: boolean circuits and Turing machines. We show that overparameterized feedforward neural nets can SM approximate boolean circuits with sample complexity depending only polynomially on the circuit size, not the size of the network. In addition, we show that transformers can SM approximate Turing machines with computation time bounded by $T$ with sample complexity polynomial in the alphabet size, state space size, and $\log (T)$. We also introduce new tools for analyzing generalization which provide much tighter sample complexities than the typical VC-dimension or norm-based bounds, which may be of independent interest.