← 研踪 PaperTrailarXiv:2609.17655 · v3
Efficient Robust Learning at the Information-Theoretic Limit
首次提交 2026-09-16 01:56
版本更新 2026-11-05 00:23
雷达观测 2026-11-05 12:13
摘要
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).
作者备注：Corrected the arXiv abstract to match the pdf
观测到的版本
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v3 · 2026-11-05 00:23 · 最新观测版本
Efficient Robust Learning at the Information-Theoretic Limit
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).
查看此版本解读 →查看此版本原文 ↗v2 · 2026-11-04 00:14
Efficient Robust Learning at the Information-Theoretic Limit
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 $\varepsilon$.
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