Sample Complexity of Scientific Discovery: PAC Learnability of Compositional Function Trees

2026-06-28

AI Summary

This work addresses the statistical unlearnability of symbolic regression caused by combinatorial explosion in the hypothesis space by analyzing composite function trees built from finitely many smooth operators within the PAC learning framework. For the first time, it establishes a theoretical foundation for their learnability, proving a generalization error bound of 𝒪(Lᵈ/√n) via Rademacher complexity analysis, a Maurer-type vector contraction inequality, and finite union bounding techniques, where L denotes the operator Lipschitz constant and d the tree depth. The result demonstrates that model complexity is governed by depth and Lipschitz constants rather than the exponential growth of symbolic structures. Empirical validation further confirms that the generalization gap correlates strongly with (𝐿̂ᵈ)/√n, challenging conventional understanding of the statistical complexity inherent in symbolic regression.

Abstract

Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function trees built from a finite vocabulary of smooth operators (e.g., $\{+,\times,\sin,\exp\}$ and affine maps). We prove that the relevant generalization quantity, Rademacher complexity, hence the excess risk, does not necessarily blow up exponentially with the number of distinct symbolic structures, but is controlled by (i) the depth $d$ and (ii) the Lipschitz constants of the base operators along the composed computation graph. Concretely, under mild Lipschitz conditions on operators and bounded affine leaves, a finite-union bound over a vocabulary of size $K=|\mathcal{H}_{\mathrm{base}}|$ together with Maurer-type vector contraction yields $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{d}) \leq (Kb\sqrt{2}L)^{d-1}\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})$ with arity bound $b$; corresponding high-probability risk bounds scale as $\mathcal{O}(L^{d}/\sqrt{n})$ when $K,b=O(1)$ and $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})=O(n^{-1/2})$. We complement the theory with a modular codebase that trains differentiable operator trees (not MLPs) on synthetic "physics-like" targets of controlled depth and shows that the empirical generalization gap correlates positively with the predicted complexity term $(\widehat{L}^{d})/\sqrt{n}$.

Problem

• symbolic regression
• PAC learnability
• compositional function trees
• Rademacher complexity
• sample complexity

Innovation

• PAC learnability
• compositional function trees
• Rademacher complexity
• symbolic regression
• Lipschitz operators