Noise-adaptive Margin-based Active Learning and Lower Bounds under Tsybakov Noise Condition
This work addresses noise robustness in active learning for machine learning practitioners, providing theoretical guarantees and lower bounds, but it is incremental as it builds on existing noise conditions and scenarios.
The paper tackles the problem of active learning for linear separators under Tsybakov noise, presenting a noise-adaptive algorithm that achieves optimal statistical rates up to poly-logarithmic factors and deriving lower bounds showing sample complexity cannot be improved even for simple distributions like the uniform unit ball.
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.