Abstract
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(m2), yielding a cover of size at most 2m+1 on that sample. This suffices to control the error of proper consistent learners with O(m2) 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.
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Mar 2, 2026cs.LG
We revisit the framework of Smart PAC learning, which seeks supervised learners which compete with semi-supervised learners that are provided full knowledge of the marginal distribution on unlabeled data. Prior work has shown that such marginal-by-marginal guarantees are possible for "most" marginals, with respect to an arbitrary fixed and known measure, but not more generally. We discover that this failure can be attributed to an "indistinguishability" phenomenon: There are marginals which cannot be statistically distinguished from other marginals that require different learning approaches. In such settings, semi-supervised learning cannot certify its guarantees from unlabeled data, rendering them arguably non-actionable. We propose relatively smart learning, a new framework which demands that a supervised learner compete only with the best "certifiable" semi-supervised guarantee. We show that such modest relaxation suffices to bypass the impossibility results from prior work. In the distribution-free setting, we show that the One-Inclusion Graph learner is relatively smart up to squaring the sample complexity, and show that no supervised learning algorithm can do better. For distribution-family settings, we show that relatively smart learning can be impossible or can require idiosyncratic learning approaches, and its difficulty can be non-monotone in the inclusion order on distribution families.
Shaddin Dughmi, Alireza F. Pour
Sep 15, 2026cs.DS
In an important recent work, Blanc (2026) gave an algorithm for robustly learning Boolean concept classes with respect to a fixed distribution that outputs a (randomized) classifier achieving the optimal error of
η+ε where
η is the noise rate. In contrast, it is well known that deterministic hypotheses cannot achieve error less than
2η+ε. Blanc's algorithm is computationally inefficient, and the main problem left open in his work is to find a polynomial-time algorithm given access to an oracle for empirical risk minimization (ERM). In this paper, we resolve this problem and give such an algorithm. Perhaps surprisingly, our techniques make crucial use of various types of no-regret learners. Additionally, we give an efficient algorithm (no ERM oracle required) for robustly learning any function class that admits sandwiching polynomials with respect to hypercontractive distributions. As one consequence, we give the first polynomial-time algorithm for robustly learning a halfspace with respect to Gaussian marginals that achieves error
η+ε for any constant
ε.
Adam R. Klivans, Konstantinos Stavropoulos, Sergei Tikhonov +1
May 8, 2026cs.LG
Semi-supervised learning faces significant challenges in realistic scenarios where labeled data is scarce and unlabeled data follows unknown, arbitrary distributions. We formalize this critical yet under-explored paradigm as Universal Semi-supervised Learning (UniSSL). Existing methods typically leverage unlabeled data via pseudo-labeling. However, they often rely on the idealized assumption of a uniform unlabeled data distribution or require sufficient labeled data to estimate it. In the UniSSL setting, such dependencies lead to numerous erroneous pseudo-labels, thereby triggering representation confusion. Fortunately, we observe that inter-sample relations captured by representations are more reliable than pseudo-labels. Leveraging this insight, we shift our focus to representation-level structural inference to bypass distribution estimation. Accordingly, we propose Simplex Anchored Graph-state Equipartition (SAGE), which captures high-order inter-sample dependencies to establish structural consensus for guiding representation learning. Meanwhile, to mitigate representation confusion, we employ vectors that satisfy a simplex equiangular tight frame to serve as a coordinate frame for guiding inter-class representation separation. Finally, we introduce a weighting strategy based on distribution-agnostic metrics to prioritize reliable pseudo-labels and an auxiliary branch to isolate potentially erroneous pseudo-labels. Evaluations on five standard benchmarks show that SAGE consistently outperforms state-of-the-art methods, with an average accuracy gain of \textbf{8.52%}.
Yaxin Hou, Jun Ma, Hanyang Li +3