Adam R. Klivans, Konstantinos Stavropoulos, Sergei Tikhonov, Arsen Vasilyan
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). Here we resolve this problem and give such an algorithm. Additionally, we give an efficient algorithm (no ERM oracle required) for robustly learning any function class that admits low-degree polynomial approximators with respect to hypercontractive distributions. This captures basic settings including halfspaces with respect to Gaussian marginals. Our techniques make new connections to various types of online convex optimization algorithms. As a surprising consequence, we show that efficient SQ learnability in the agnostic setting implies efficient learning with contamination (with respect to Gaussians or uniform over the hypercube).
Gautam Chandrasekaran, Jason Gaitonde, Ankur Moitra, Arsen Vasilyan
The low-degree algorithm is a cornerstone in computational learning theory. For product or Gaussian distributions, it shows that efficiently learnability follows from Fourier concentration. However, a longstanding obstacle towards building such a theory over nonproduct distributions is that we can no longer rely on having closed-form expressions for a good basis of orthogonal functions. This hints that the analysis of the low-degree algorithm might be brittle to the underlying distributional assumptions. We refute this intuition by showing that the low-degree algorithm provably succeeds for a large class of graphical models under mild high-temperature conditions. As an application, for nearly every Ising model that is known to rapidly mix, we obtain learning algorithms for every function class for which the Fourier approach succeeds for product distributions (e.g. low-depth circuits, halfspaces, intersections, monotone, among others) with no significant loss. Our framework is based on a strikingly clean connection with the coordinate-by-coordinate localization scheme of Chen and Eldan introduced for establishing rapid mixing in high-temperature models, as well as other analytical transference principles to product distributions.