We study the problem of agnostically learning homogeneous halfspaces in the
distribution-specific PAC model. For a broad family of structured
distributions, including log-concave distributions, we show that non-convex SGD
efficiently converges to a solution with misclassification error
$O(\opt)+\eps$, where $\opt$ is the misclassification error of the best-fitting
halfspace. In sharp contrast, we show that optimizing any convex surrogate
inherently leads to misclassification error of $\omega(\opt)$, even under
Gaussian marginals.

研究学习同质半空间的非说服学派问题，证明对于一类结构化分布，包括对数凹分布，在误分类误差 O（opt）+eps 下，非凸 SGD 有效地收敛到解决方案。相比之下，证明优化任何凸代理本质上会导致 ω（opt）的误分类误差，即使在高斯边际分布下。

非凸 SGD 学习带有对抗标签噪声的半空间

Non-Convex SGD Learns Halfspaces with Adversarial Label Noise

We present a PTAS for agnostically learning halfspaces w.r.t. the uniform
distribution on the $d$ dimensional sphere. Namely, we show that for every
$\mu>0$ there is an algorithm that runs in time
$\mathrm{poly}(d,\frac{1}{\epsilon})$, and is guaranteed to return a classifier
with error at most $(1+\mu)\mathrm{opt}+\epsilon$, where $\mathrm{opt}$ is the
error of the best halfspace classifier. This improves on Awasthi, Balcan and
Long [ABL14] who showed an algorithm with an (unspecified) constant
approximation ratio. Our algorithm combines the classical technique of
polynomial regression (e.g. [LMN89, KKMS05]), together with the new
localization technique of [ABL14].

本文提出了一个以多项式回归和定位技术相结合的算法， 用于在 d - 球上均匀分布的情况下，实现对零时最佳半空间分类器的确定性多项式近似方案（PTAS），误差保证为 opt 的 (1+μ)+ε 倍， 并提供了比以前使用定位技术的算法更加优越的针对全局的误差近似解决方案。