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

Şuayp Talha Kocabay, Talha Rüzgar Akkuş, Kerem Yalçın

Keywords: Sample Complexity, PAC Learning, Rademacher Complexity, Symbolic Regression, Compositional Models

TL;DR: Generalization for compositional symbolic models scales with tree depth and how stable the operators are, not with the exponentially large expression count—and simple synthetic fits line up with that picture.

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, depth-wise vector contraction yields $\mathfrak{R}^{(n)}(\mathcal{H}^{\mathrm{comp},d}) \le (bL)^{d-1}\mathfrak{R}^{(n)}(\mathcal{H}^{\mathrm{comp},1})$ with arity bound $b$, in particular $(2L)^{d-1}$ for binary trees; corresponding high-probability risk bounds scale as $\mathcal{O}(L^{d}/\sqrt{n})$ when $b=O(1)$ and $\mathfrak{R}^{(n)}(\mathcal{H}^{\mathrm{comp},1})=O(n^{-1/2})$.
Here $\mathfrak{R}^{(n)}(\cdot)$ denotes the usual (empirical) Rademacher complexity at sample size $n$, and $\mathcal{H}^{\mathrm{comp},d}$ denotes the depth-$d$ compositional class built from the operator vocabulary.
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 $(\widehat{L}^{d})/\sqrt{n}$.

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Submission Number: 138