The Role of Generator Access in Autoregressive Post-Training

Autoregressive pretraining increasingly draws on heterogeneous data, making it important to understand how a model learns from an individual token. The next-token prediction objective naturally identifies a token's contribution with its own loss. However, each token is not only a prediction target but also context for what follows. Using controlled corruption, we decouple these two roles and find a reversal: making a noisy token easier to predict reduces its damage as a target but increases it as context. The same decoupling helps explain text generated by language models: generation selects each token by its fit to the prefix, while its role as context is never tested against an independently determined continuation, because that continuation is generated to fit it. At known corrupted positions, acting through the context can reduce damage that removing the token's own loss does not. Understanding and controlling what a model learns from a token therefore requires decoupling its roles.

Modern large language models generate text autoregressively, producing tokens one at a time. To study the learnability of such systems, Joshi et al. (COLT 2025) introduced a PAC-learning framework for next-token generators, the primitive underlying autoregressive models. In this framework, an unknown next-token generator maps a sequence of tokens to the next token and is iteratively applied for $T$ steps, producing a chain of tokens whose final token constitutes the model's output. The learning task is to learn the input-output mapping induced by this autoregressive process. Depending on the available supervision, training examples may reveal only the final output (End-to-End supervision) or the entire generated chain (Chain-of-Thought supervision). This raises two natural questions: how the sample complexity depends on the generation length $T$, and how much Chain-of-Thought supervision can reduce this dependence. In this work we give a nearly complete answer to both questions by uncovering a taxonomy of how the sample complexity scales with $T$. For End-to-End learning, we show that the landscape is remarkably rich: subject to mild conditions, essentially any growth rate $r(T)$ between constant and linear can arise as the sample complexity, and combined with the linear upper bound of Joshi et al., this yields an essentially complete characterization. In contrast, under Chain-of-Thought supervision we show that the sample complexity is independent of $T$, demonstrating that access to intermediate reasoning steps can eliminate the dependence on the generation length altogether. Our analysis introduces new combinatorial tools, and as corollaries we resolve several open questions posed by Joshi et al. regarding the dependence of learnability on the generation length and the role of Chain-of-Thought supervision.

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.