We present a simple noise-robust margin-based active learning algorithm to
find homogeneous (passing the origin) linear separators and analyze its error
convergence when labels are corrupted by noise. We show that when the imposed
noise satisfies the Tsybakov low noise condition (Mammen, Tsybakov, and others
1999; Tsybakov 2004) the algorithm is able to adapt to unknown level of noise
and achieves optimal statistical rate up to poly-logarithmic factors. We also
derive lower bounds for margin based active learning algorithms under Tsybakov
noise conditions (TNC) for the membership query synthesis scenario (Angluin
1988). Our result implies lower bounds for the stream based selective sampling
scenario (Cohn 1990) under TNC for some fairly simple data distributions. Quite
surprisingly, we show that the sample complexity cannot be improved even if the
underlying data distribution is as simple as the uniform distribution on the
unit ball. Our proof involves the construction of a well separated hypothesis
set on the d-dimensional unit ball along with carefully designed label
distributions for the Tsybakov noise condition. Our analysis might provide
insights for other forms of lower bounds as well.

提出了一个简单的噪声 - 鲁棒性基于边缘的主动学习算法来查找同质 (通过原点) 的线性分离器，并分析了标签受噪声干扰时的误差收敛性。我们证明了当所施加的噪声满足 Tsybakov 低噪声条件时，算法能够适应未知的噪声水平，并在多对数因子上实现最优统计速率。同时，我们为基于边缘的主动学习算法在会员查询合成场景下的 Tsybakov 噪音条件 (TNC) 导出了下限。我们的分析可能为其他形式的下限提供了见解。