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We present a simple noise-robust margin-based active learning algorithm tofind homogeneous (passing the origin) linear separators and analyze its errorconvergence when labels are corrupted by noise. We show that when the imposednoise satisfies the Tsybakov low noise condition (Mammen, Tsybakov, and others1999; Tsybakov 2004) the algorithm is able to adapt to unknown level of noiseand achieves optimal statistical rate up to poly-logarithmic factors. We alsoderive lower bounds for margin based active learning algorithms under Tsybakovnoise conditions (TNC) for the membership query synthesis scenario (Angluin1988). Our result implies lower bounds for the stream based selective samplingscenario (Cohn 1990) under TNC for some fairly simple data distributions. Quitesurprisingly, we show that the sample complexity cannot be improved even if theunderlying data distribution is as simple as the uniform distribution on theunit ball. Our proof involves the construction of a well separated hypothesisset on the d-dimensional unit ball along with carefully designed labeldistributions for the Tsybakov noise condition. Our analysis might provideinsights for other forms of lower bounds as well.

Noise-adaptive Margin-based Active Learning and Lower Bounds under Tsybakov Noise Condition