Robust Learning with Optimal Error

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 $η+ \varepsilon$ where $η$ is the noise rate. In contrast, it is well known that deterministic hypotheses cannot achieve error less than $2η+ \varepsilon.$ 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 $η+ \varepsilon$ for any constant $\varepsilon$.

Probably Approximately Correct (PAC) learning [Val84] is a fundamental learning model that has been extensively investigated. In this model, $\mathcal{H} \subseteq \{0,1\}^{\mathcal{X}}$ is a concept class, and $h^*\in\mathcal{H}$ is the target concept to be learned. Having access to i.i.d. labeled examples from a distribution $\mathcal{D}$ over $\mathcal{X}\times\{0,1\}$, which admits $h^*$ as the best concept in $\mathcal{H}$, the goal is to design a learning algorithm that outputs a hypothesis having low error competitive to $h^{*}$ with high probability. This model was initially studied under the realizable setting, which assumes that $h^*$ has no error. A natural relaxation is to allow label noise, that is, the true label can be flipped with probability $η\in(0,1/2)$. In reality, certain labels might be extremely noisy, especially for those points near the decision boundary. Hence, it is natural to allow very noisy points, though only rarely. This is quantified by a noise model introduced by [MT99] and [Tsy04], now known as Tsybakov noise. For learning general concept classes, [MN06] gave the general upper and lower bounds for error guarantees under Tsybakov noise. However, their upper and lower bounds differ by a logarithmic factor. Resolving this gap has remained a well-known open question for the past twenty years. In this work, we resolve this open question by improving the upper bound to match the best known lower bound, thus establishing the optimal error guarantee for learning under Tsybakov noise. Our learning algorithm operates by adaptively partitioning the instance space into regions, roughly corresponding to different noise levels, and returning a hypothesis in the concept class satisfying a specific error constraint for each region. Our technique shares a conceptual foundation with several recent advances in non-realizable learning, such as [HLZ24] and [Han25].

We study the problem of adversarially robust PAC learning. In this framework, the learner observes independent samples from an unknown distribution over $\mathcal{X} \times \{0,1\}$, as in classical PAC learning. However, given a perturbation map $\mathcal{U} : \mathcal{X} \to 2^{\mathcal{X}}$ known to the learner, the goal is to output, with high probability, a predictor that correctly classifies every perturbation $z \in \mathcal{U}(x)$ of most future examples $(x,y)$ drawn from the same underlying distribution. We determine the optimal $\mathcal{U}$-independent sample complexity of this problem in both the realizable and agnostic settings. More specifically, for every concept class $\mathcal{H}$ of $\operatorname{VC}$ dimension $d$, we prove upper bounds of $\mathcal{O} \big( d/ε+ \log(1/δ)/ε\big)$ in the realizable setting and $\mathcal{O} \big( d/ε^2 + \log(1/δ)/ε^2 \big)$ in the agnostic setting, together with an optimal first-order refinement of the latter. These bounds match the corresponding lower bounds for classical PAC learning. Consequently, and perhaps surprisingly, adversarial robustness incurs no additional distribution-free statistical cost, uniformly over all perturbation maps. Our bounds improve exponentially on those of [Montasser, Hanneke, and Srebro; COLT '19]. On the technical side, we present short and elementary proofs based on a new algorithmic principle that we call binomial-bagging. We believe that binomial-bagging and its analysis may be of independent interest.