We study the problem of learning Single-Index Models under the $L_2^2$ loss in the agnostic model. We give an efficient learning algorithm, achieving a constant factor approximation to the optimal loss, that succeeds under a range of distributions (including log-concave distributions) and a broad class of monotone and Lipschitz link functions. This is the first efficient constant factor approximate agnostic learner, even for Gaussian data and for any nontrivial class of link functions. Prior work for the case of unknown link function either works in the realizable setting or does not attain constant factor approximation. The main technical ingredient enabling our algorithm and analysis is a novel notion of a local error bound in optimization that we term alignment sharpness and that may be of broader interest.

翻译：我们研究在不可知模型下基于$L_2^2$损失学习单指标模型的问题。我们提出一种高效的学习算法，能在多种分布（包括对数凹分布）及广泛单调Lipschitz连接函数类上实现最优损失的常数因子近似。这是首个高效的常数因子近似不可知学习器，即使对于高斯数据及任何非平凡连接函数类也是如此。此前针对未知连接函数的研究要么工作在可实现设定下，要么未能达到常数因子近似。支撑我们算法与分析的核心技术要素是优化中一种新颖的局部误差界概念——我们称之为对齐锐度，该概念可能具有更广泛的应用价值。