Linear Classification and Selective Sampling Under Low Noise Conditions

We provide a new analysis of an efficient margin-based algorithm for selective sampling in classification problems. Using the so-called Tsybakov low noise condition to parametrize the instance distribution, we show bounds on the convergence rate to the Bayes risk of both the fully supervised and the selective sampling versions of the basic algorithm. Our analysis reveals that, excluding logarithmic factors, the average risk of the selective sampler converges to the Bayes risk at rate N^{-(1+\alpha)(2+\alpha)/2(3+\alpha)} where N denotes the number of queried labels, and \alpha > 0 is the exponent in the low noise condition. For all \alpha > \sqrt{3} - 1 \approx 0.73 this convergence rate is asymptotically faster than the rate N^{-(1+\alpha)/(2+\alpha)} achieved by the fully supervised version of the same classifier, which queries all labels, and for \alpha\to\infty the two rates exhibit an exponential gap. Experiments on textual data reveal that simple variants of the proposed selective sampler perform much better than popular and similarly efficient competitors.