Robust Regression of General ReLUs with Queries

We continue the study of relatively smart learning, introduced by Dughmi and Pour (2026), which asks a supervised learner to compete, marginal by marginal, with every distribution-fixed error guarantee soundly certifiable from unlabeled data. They showed that the One-Inclusion Graph (OIG) learner is relatively smart with a quadratic sample-complexity blowup, and that no relatively smart learner can do better, leaving open whether ERM or another natural or tractable learner achieves comparable guarantees. They also left open whether the blowup can be restricted to unlabeled data. Our firs results shows that ERM---and in fact any proper consistent learner---is relatively smart for binary classification in the distribution-free setting. We show that a small certifiable error with $m$ samples implies a similarly small error on the uniform distribution over a random sample of size $O(m^2)$, yielding a cover of size at most $2^{m+1}$ on that sample. This suffices to control the error of proper consistent learners with $O(m^2)$ samples. We then show that semi-supervised relatively smart learning is information-theoretically possible with a quadratic blowup only in unlabeled sample complexity and no blowup in labeled sample complexity. The learner uses a natural generalization of OIG to a leave-most-out transductive problem, where labels of part of a finite pool are revealed and the remaining labels are predicted. Finally, this label efficiency comes at a cost in simplicity and tractability. If the hypothesis class is accessed only through an agnostic ERM oracle, any semi-supervised relatively smart learner with substantially sub-quadratic labeled-sample blowup requires super-polynomially many oracle calls. This holds even when the marginal is given explicitly, and thus also yields an intractability result for distribution-fixed learning that may be of independent interest.

We study the complexity of smoothed agnostic learning of halfspaces on $\{\pm 1\}^n$ under uniform marginals in the model of~\cite{KM25}, where each input coordinate is independently flipped with probability $σ\in (0, {1}/{2})$. We show that $L^1$ polynomial regression achieves runtime and sample complexity $\tilde{O}(n^{O(\log(1/\varepsilon)/σ)})$, and prove a nearly matching Statistical Query complexity lower bound of $n^{Ω(\log(1+σ/\varepsilon^2)/σ)}$. This complements the recent work of~\cite{DK26}, which established analogous bounds in the continuous setting under Gaussian marginals.

We prove that, on finite simple undirected graphs equipped with a single Boolean node feature, the Boolean queries expressible in $Σ$-MPLang, for any collection $Σ$ of eventually constant activation functions and with arbitrary real coefficients, form a strict subclass of the Boolean queries expressible in ReLU-MPLang. We thereby settle a recently posed open problem: whether ReLU-MPLang is more powerful than trReLU-MPLang when it comes to Boolean queries. In particular, this implies that ReLU-GNNs are strictly more expressive than {TrReLU,id}-GNNs with respect to Boolean queries on Boolean-featured graphs.