An Analysis of Active Learning with Uniform Feature Noise.

In active learning, the user sequentially chooses values for feature X and an oracle returns the corresponding label Y. In this paper, we consider the effect of feature noise in active learning, which could arise either because X itself is being measured, or it is corrupted in transmission to the oracle, or the oracle returns the label of a noisy version of the query point. In statistics, feature noise is known as "errors in variables" and has been studied extensively in non-active settings. However, the effect of feature noise in active learning has not been studied before. We consider the well-known Berkson errors-in-variables model with additive uniform noise of width sigma. Our simple but revealing setting is that of one-dimensional binary classification setting where the goal is to learn a threshold (point where the probability of a + label crosses half). We deal with regression functions that are antisymmetric in a region of size a around the threshold and also satisfy Tsybakov's margin condition around the threshold. We prove minimax lower and upper bounds which demonstrate that when a is smaller than the minimiax active/passive noiseless error derived in Castro & Nowak (2007), then noise has no effect on the rates and one achieves the same noiseless rates. For larger sigma, the unfiattening of the regression function on convolution with uniform noise, along with its local antisymmetry around the threshold, together yield a behaviour where noise appears to be beneficial. Our key result is that active learning can buy significant improvement over a passive strategy even in the presence of feature noise.