--- title: PAC Learning for Next-Token Generators url: https://www.emergentmind.com/topics/pac-learning-framework-for-next-token-generators type: topic --- # PAC Learning for Next-Token Generators Searching arXiv for the specified paper and closely related work on PAC-style/autoregressive next-token learning. to=arxiv_search.query 北京赛车开奖ված 下载彩神争霸 {"query":"all:\"Sample Complexity of Autoregressive Reasoning: Chain-of-Thought vs. End-to-End\" OR id:2604.12013","max_results":5,"sort_by":"relevance"} The PAC-learning framework for next-token generators studies autoregressive text generation as a statistical learning problem in which an unknown next-token rule is iteratively applied for \(T\) steps, and the learning target is the input-output map induced by that rollout rather than merely the one-step predictor. In the formulation introduced by Joshi et al. and sharply extended in "Sample Complexity of Autoregressive Reasoning: Chain-of-Thought vs. End-to-End," a next-token generator maps a sequence to its next token, rollout length \(T\) becomes an explicit complexity parameter, and the supervision model—observing only the final token or the entire intermediate trajectory—determines whether sample complexity can scale linearly in \(T\), logarithmically in \(T\), or not at all [2604.12013]. ## 1. Formal autoregressive PAC model The basic object is a next-token generator \[ f:\Sigma^\star \to \Sigma, \] where \(\Sigma\) is a finite alphabet and \(\Sigma^\star\) is the set of finite strings over \(\Sigma\). One autoregressive step is represented by the apply-and-append map \[ \bar f:\Sigma^\star\to\Sigma^\star,\qquad \bar f(x)=x\circ f(x), \] and iterating this map for \(T\) steps produces a generated chain whose length-\(T\) suffix is denoted \(f^{\mathsf{CoT},T}(x)\in\Sigma^T\), while its last token is denoted \(f^{\mathsf{e2e},T}(x)\in\Sigma\) [2604.12013]. This formulation makes the PAC target explicitly rollout-based. For a base class \(\mathcal F\subseteq \Sigma^{\Sigma^\star}\), the induced classes are \[ \mathcal F^{\mathsf{CoT},T} = \{f^{\mathsf{CoT},T}:f\in\mathcal F\}, \qquad \mathcal F^{\mathsf{e2e},T} = \{f^{\mathsf{e2e},T}:f\in\mathcal F\}. \] Accordingly, the learning task is not necessarily to identify the one-step mechanism \(f\), but to learn the map from prompt \(x\) to the final token obtained after \(T\) rounds of autoregressive generation. The paper studies primarily the binary case \(\Sigma=\{0,1\}\), while stating that the ideas extend to any finite alphabet. The learning setup is realizable: an unknown distribution \(D\) over \(\Sigma^\star\) supplies i.i.d. prompts \(x\sim D\), and labels are generated by some unknown \(f_\star\in F\). Test-time error is always the end-to-end error \[ \Pr_{x\sim D}\bigl[h(x)\neq f_\star^{\mathsf{e2e},T}(x)\bigr]. \] For finite-VC classes, the usual PAC sample complexity notation \(m_{\mathcal H}(\epsilon,\delta)\) is used, with \[ m_{\mathcal H}(\epsilon,\delta) = \Theta\!\left(\frac{d+\log(1/\delta)}{\epsilon}\right), \qquad d=VC(\mathcal H). \] Thus, in the end-to-end regime, the problem is exactly PAC learning of \(F^{\mathsf{e2e},T}\), governed by \(VC(F^{\mathsf{e2e},T})\). A common misconception is that next-token PAC theory concerns only the one-step predictor. In this framework, the relevant hypothesis class is the class induced by \(T\) rounds of rollout, so the learnability question is intrinsically sequential rather than pointwise. ## 2. Supervision models: end-to-end and Chain-of-Thought The framework formalizes two supervision models. Under **End-to-End supervision**, a training example reveals only the final token, \[ \bigl(x, f_\star^{\mathsf{e2e},T}(x)\bigr). \] Under **Chain-of-Thought supervision**, a training example reveals the full generated trajectory, \[ \bigl(x, f_\star^{\mathsf{CoT},T}(x)\bigr), \] where \(f_\star^{\mathsf{CoT},T}(x)\in\Sigma^T\) [2604.12013]. The distinction is asymmetric. Test-time evaluation remains end-to-end even in the CoT regime: the learner is still judged only on the final token. The paper is explicit that CoT supervision is a training-time side-information model, not an ordinary symmetric PAC classification task. This point matters because observing the entire intermediate chain supplies \(T\) local constraints on the latent one-step generator, whereas ordinary end-to-end supervision exposes only the terminal outcome. The central positive theorem for CoT supervision states that for every class \(F\) of binary next-token generators with \(VC(F)<\infty\), there exists a constant \(c=c(F)\) such that for every \(T\ge 1\), \[ m_{F}^{\mathsf{CoT},T}(\epsilon,\delta) \le \frac{c}{\epsilon} \left( \log\frac{1}{\epsilon} + \log\frac{1}{\delta} \right). \] More precisely, the proof yields \(c=\tilde O(dd^\star)\), where \(d=VC(F)\) and \(d^\star=VC^\star(F)\) is the dual VC dimension. The stronger multiclass statement is that if \(VC(F)<\infty\), then \(F^{\mathsf{CoT},T}\) is learnable for all \(T\) with \[ m(\epsilon,\delta)=\tilde O\!\left(\frac{VC(F)\,VC^\star(F)+\log(1/\delta)}{\epsilon}\right), \] while if \(VC(F)=\infty\), then \(F^{\mathsf{CoT},T}\) is not PAC learnable for any \(T\) [2604.12013]. The lower bound is already present at \(T=1\): \[ \Omega\!\left(\frac{d+\log(1/\delta)}{\epsilon}\right). \] For classes with stable compression, the paper gives the sharper bound \[ m_{F}^{\mathsf{CoT},T}(\epsilon,\delta) = O\!\left(\frac{k+\log(1/\delta)}{\epsilon}\right) \] whenever \(F\) admits a stable sample compression scheme of size \(k\). This applies in particular to the linear autoregressor class \[ f_{w,\theta}(x) = \mathbf 1\!\left[\langle w,\mathrm{tail}_d(x)\rangle \ge \theta\right], \] yielding \[ m = O\!\left(\frac{d+\log(1/\delta)}{\epsilon}\right). \] The paper’s conceptual interpretation is direct: CoT supervision turns one difficult \(T\)-step autoregressive problem into \(T\) ordinary one-step supervision constraints on the base class. This suggests that observing intermediate computation can remove the statistical burden created by hidden latent rollouts. ## 3. Sample-complexity taxonomy as a function of rollout length The most distinctive result of the framework is a taxonomy of how sample complexity scales with generation length \(T\). In the CoT regime, the dependence on \(T\) disappears entirely. In the end-to-end regime, by contrast, the space of possibilities is “remarkably rich”: subject to mild conditions, essentially any growth rate between constant and linear can occur [2604.12013]. Joshi et al. had established the general upper bound \[ VC(F^{e2e,T}) = O(VC(F)\cdot T), \] and examples with linear growth, while also showing that finite Littlestone dimension implies logarithmic growth. The later theory proves that intermediate behaviors are not artifacts but genuinely realizable. A function \(r:\mathbb N_+\to\mathbb N_+\) is called a monotone-subadditive rate if it is monotone non-decreasing and satisfies \[ r(T_1+T_2)\le r(T_1)+r(T_2). \] Then for every such rate \(r\), there exists a class \(F\) of binary next-token generators such that \[ r(T) \le VC(F^{e2e,T}) \le r(T)+r(1)\le 2r(T) \qquad\text{for all }T\ge 1. \] In the normalized case \(r(1)=1\), the exact identity \[ VC(F^{e2e,T})=r(T) \qquad\text{for all }T \] is obtained. The realizable rates include constant growth, \(\log T\), polylogarithmic growth, power laws \(T^c\) for \(0