Expressivity of Neural Networks via Chaotic Itineraries beyond Sharkovsky's Theorem

Given a target function f, how large must a neural network be in order to approximate f? Recent works examine this basic question on neural network textitexpressivity from the lens of dynamical systems and provide novel ``depth-vs-width'' tradeoffs for a large family of functions f. They suggest that such tradeoffs are governed by the existence of textitperiodic points or cycles in f. Our work, by further deploying dynamical systems concepts, illuminates a more subtle connection between periodicity and expressivity: we prove that periodic points alone lead to suboptimal depth-width tradeoffs and we improve upon them by demonstrating that certain ``chaotic itineraries'' give stronger exponential tradeoffs, even in regimes where previous analyses only imply polynomial gaps. Contrary to prior works, our bounds are nearly-optimal, tighten as the period increases, and handle strong notions of inapproximability (e.g., constant L1 error). More broadly, we identify a phase transition to the textitchaotic regime that exactly coincides with an abrupt shift in other notions of function complexity, including VC-dimension and topological entropy.