  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 $\sigma$ around the threshold and
also satisfy Tsybakov's margin condition around the threshold. We prove minimax
lower and upper bounds which demonstrate that when $\sigma$ is smaller than the
minimiax active/passive noiseless error derived in \cite{CN07}, then noise has
no effect on the rates and one achieves the same noiseless rates. For larger
$\sigma$, the \textit{unflattening} of the regression function on convolution
with uniform noise, along with its local antisymmetry around the threshold,
together yield a behaviour where noise \textit{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.
