Sample Complexity of Autoregressive Reasoning: Chain-of-Thought vs. End-to-End

Autoregressive generation lies at the heart of the mechanism of large language models. It can be viewed as the repeated application of a next-token generator: starting from an input string (prompt), the generator is applied for $M$ steps, and the last generated token is taken as the final output. [Joshi et al., 2025] proposed a PAC model for studying the learnability of the input-output maps arising from this process. We develop an online analogue of this framework, focusing on the mistake bound of learning the final output induced by an unknown next-token generator. We distinguish between two forms of feedback. In the End-to-End model, after each round the learner observes only the final token produced after $M$ autoregressive steps. In the Chain-of-Thought model, the learner is additionally shown the entire $M$-step trajectory. Our goal is to understand how the optimal mistake bound depends on the generation horizon $M$, and to what extent observing intermediate tokens can reduce this dependence. Our main results show that the online theory of autoregressive learning exhibits a qualitative picture analogous to the statistical one found by [Hanneke et al., 2026], but with a different scale of dependence on the generation horizon. In the End-to-End model, we prove a taxonomy of possible mistake-bound growth rates in the generation horizon $M$: essentially any rate between constant and logarithmic can arise. We further show that this logarithmic ceiling is unavoidable. In the Chain-of-Thought model, we show that access to the full generated trajectory eliminates the dependence on $M$ altogether. We also analyze autoregressive linear threshold classes, and prove optimal mistake bounds, as well as a new lower bound for the statistical setting. Along the way, our results resolve several questions left open by [Joshi et al., 2025].

Motivated by LLMs, which generate outputs by iteratively sampling from next-token distributions, we introduce a PAC-learning model for binary stochastic autoregressive learning. This generalizes the deterministic autoregressive learning framework of Joshi et al., COLT 2025. In our model, one fixed generator assigns a Bernoulli next-token distribution to every prompt string. Starting from an input prompt, a token is sampled and appended to the prompt; the same generator is then applied again to this expanded prompt; this procedure is repeated for $M$ steps. Three forms of supervision are considered: base one-step samples, chain-of-thought (CoT) samples that reveal full random trajectories of length $M$, and end-to-end (e2e) samples that reveal only the final token of length $M$ trajectories. For a generator class, we study the minimum number of samples $m_{base}(\varepsilon),m_{CoT}(\varepsilon), m_{e2e}(\varepsilon)$, resp., required to learn the one-step probabilities in the base model, and the final-token probability in the CoT and e2e models, under squared loss error~$\varepsilon$. We show that stochastic autoregressive learning fundamentally differs from the deterministic theory. At scale $\varepsilon$, there is no universal comparison between the three learning tasks: both $m_{CoT}/m_{base}$ and $m_{e2e}/m_{CoT}$ can be made simultaneously arbitrarily larger than $M/\varepsilon$, the natural analogue for the existing deterministic results. Nevertheless, after altering scales, for every class, CoT learning at scale $\varepsilon$ is upper-bounded by base learning at scale $\varepsilon/M^2$, whereas e2e learning at scale $\varepsilon$ is upper-bounded, up to logarithmic factors, by $(M/\varepsilon) m_{CoT}(Θ(\varepsilon))$. These dependencies and scales are essentially tight. We complement these bounds by studying dimension $d$ logistic functions in our model.

We introduce a family of synthetic languages with hierarchical structure -- generated by a broadcast process on trees -- for which the role of context length and reasoning in autoregressive generation can be analyzed precisely. At the heart of our analytic approach is an \emph{exact $k$-gram ansatz} in place of transformers with context length $k$, a substitution we then validate empirically. Using this ansatz we derive explicit asymptotic predictions for distributional statistics of the sequences produced by a trained model, instantiated in two settings. For the Ising broadcast process (a soft-constrained language), we prove that the variance of the generated sum scales log-linearly in the context depth and its kurtosis converges to that of a Gaussian -- both deviating from the true language for any sublinear context. For the coloring broadcast process (a hard-constrained language) in the freezing regime, bounded-context autoregression produces sequences that, with high probability, are inconsistent with any valid coloring of the underlying tree. Together these results imply an $Ω(n)$ lower bound on the context length required to faithfully sample length-$n$ sequences. In contrast, we prove that an autoregressive reasoning model with only $Θ(\log n)$ working memory can sample exactly from the true language -- an exponential improvement. We confirm both the lower-bound predictions and the reasoning-based upper bound empirically with transformers trained on the synthetic language; the trained models track our asymptotic predictions quantitatively across a wide range of context sizes.