On the VC Dimension of Autoregressive Linear Threshold Functions

We prove that, in the realizable PAC setting, the sample complexity of exact-trace learning for full autoregressive Chain-of-Thought traces is upper bounded by the standard multiclass rate of the local next-token class, where this rate is governed by the Daniely--Shalev-Shwartz dimension. Under exact-trace loss, one wrong action makes the whole trace incorrect; nevertheless, for every stopping rule $\mathtt{halt}$ and every pointwise $\mathtt{halt}$-halting local class $\mathrm{H}$, $n_{\mathrm{PAC}}^{\varepsilon,δ}(\operatorname{Roll}_{\mathtt{halt}}(\mathrm{H}))=O((\operatorname{DSdim}(\mathrm{H})+\log(1/δ))/\varepsilon)$, with no dependence on rollout length. The dependence on $\operatorname{DSdim}(\mathrm{H})$ is worst-case optimal, since one-step stopping recovers ordinary multiclass learning of $\mathrm{H}$. The proof introduces parity dimension, a rollout-stable refinement of DS dimension based on even pseudo-cubes. It controls one-inclusion density via a low-coordinate spanning theorem on finite restrictions and, unlike DS dimension itself, does not increase under autoregressive rollout. We also show why this detour is necessary: DS dimension can increase under rollout.

We study the contradiction graphs associated with binary concept classes. For a class $H \subseteq \{0,1\}^X$, the order-$m$ contradiction graph $G_m(H)$ has as vertices the $H$-realizable labeled sequences of length $m$, with two vertices adjacent when the two sequences assign opposite labels to some common domain point. Our main result is that the single graph $G_m(H)$ determines the threshold predicate $\mathrm{VCdim}(H)\ge m$. Consequently, the full sequence $(G_m(H))_{m \ge 1}$ determines the exact VC dimension and, in particular, detects finite versus infinite VC dimension, answering a question posed by Alon et al. (2024).

We revisit the problem of learning predictors robust to adversarial examples at test-time. We prove that VC classes are adversarially robustly learnable with sample complexity linear in the VC dimension $d$, providing an exponential improvement over the previous upper bound of Montasser, Hanneke, and Srebro (2019). Remarkably, this result is achieved with a simple improper algorithm that combines the classic heuristic bagging (bootstrap aggregation) of Breiman (1996) with robust empirical risk minimization (RERM). Our algorithm computes RERMs on $O(d^\star)$ independent bootstrap samples and outputs their majority vote, where $d^\star$ denotes the dual VC dimension. We complement this result with a lower bound showing that this is unavoidable: in general, any learner in this oracle model requires $Ω(d^\star)$ calls to an RERM oracle, even when given arbitrarily many training examples.