Understanding the Out-of-Distribution Generalization of Chain-of-Thought Reasoning in LLMs

Chain-of-Thought (CoT) reasoning substantially improves Large Language Models (LLMs), yet its out-of-distribution (OOD) generalization mechanism remains theoretically underexplored. We study CoT as compositional OOD generalization: the target task lacks complete reasoning trajectories or direct input-output pairs, while training provides only in-distribution (ID) atomic step data. We show that CoT generalizes by repeatedly applying a shared single-step predictor to compose reusable ID transitions. Theoretically, let $T$ denote the number of reasoning steps in the target task, $n$ the number of training samples per sub-task, and $k$ the number of output tokens per step; then the OOD task risk is controlled by the sum of the ID sub-task risks, scaled by a coefficient of order $\mathcal{O}(k)$ that is independent of $T$, although the summed risk itself may still grow with $T$. A Rademacher-complexity analysis further shows that the estimation term is of order $\mathcal{O}(\sqrt{T/n})$, rather than linear in $T$, due to predictor sharing under a uniform capacity control on the shared class. For autoregressive Next-Token Prediction, the final-task risk is bounded by the best-in-class step-level risk scaled by the same coefficient of order $\mathcal{O}(k)$, plus an estimation term of order $\widetilde{\mathcal{O}}\big(k\sqrt{T/n}\big)$. A parity-coupled construction further shows that, for the sub-task empirical risk minimizer, the expected final-task risk is at least the best in-class risk plus $Ω(\sqrt{T/n})$ when $T\le n$. Experiments on Atom-Task and GSM8K support the theory: CoT enables short-to-long and compositional generalization, while explicit state tracking is crucial. Our code is available at https://github.com/maxliang114514/CoT-OOD.