Efficient Robust Learning at the Information-Theoretic Limit

RESUMO

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