Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity

In spite of the fundamental role of neural networks in contemporary machine learning research, our understanding of the computational complexity of optimally training neural networks remains incomplete even when dealing with the simplest kinds of activation functions. Indeed, while there has been a number of very recent results that establish ever-tighter lower bounds for the problem under linear and ReLU activation functions, less progress has been made towards the identification of novel polynomial-time tractable network architectures. In this article we obtain novel algorithmic upper bounds for training linear- and ReLU-activated neural networks to optimality which push the boundaries of tractability for these problems beyond the previous state of the art. In particular, for ReLU networks we establish the polynomial-time tractability of all architectures where hidden neurons have an out-degree of $1$, improving upon the previous algorithm of Arora, Basu, Mianjy and Mukherjee. On the other hand, for networks with linear activation functions we identify the first non-trivial polynomial-time solvable class of networks by obtaining an algorithm that can optimally train network architectures satisfying a novel data throughput condition.

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.

Infinite-width limits are a standard way to reason about neural networks, but it is not automatic that the limiting learner has the same complexity-theoretic inductive bias as large finite networks. We study this question for Bayesian neural networks at the mean-field, or critical feature-learning, scaling. The central quantity is the \emph{reduced entropy} $$s_\infty(y,\varepsilon)=\limsup_N -\frac{1}{N}\log π_N^0(L\le \varepsilon),$$ the intensive prior cost of representing a target function $y$ to population mean-squared error $\varepsilon$. Our main result is a width-robust learnability theorem. At fixed depth, a family of Boolean-cube targets is learnable from polynomially many samples at infinite width if and only if it is learnable at polynomial width, if and only if its reduced entropy is polynomially bounded. Equivalently, up to polynomial slack in accuracy, the Bayesian mean-field learner generalizes exactly on the targets that can be represented by polynomial-size networks. The forward direction is proved by a form of subsampling: from the infinitely many hidden neurons in the mean-field solution, one can select polynomially many representatives and still preserve the learned function on every input simultaneously. At the critical scaling this subsampling has both an active'' component, which keeps the data-dependent low-dimensional statistics, and a lazy'' component, which resamples the entropy-dominated directions from the prior. Thus the infinite-width mean-field limit gives a clean analytic description of learning without introducing spurious width-dependent generalization power.