  In the covariate shift learning scenario, the training and test covariate
distributions differ, so that a predictor's average loss over the training and
test distributions also differ. In this work, we explore the potential of
extreme dimension reduction, i.e. to very low dimensions, in improving the
performance of importance weighting methods for handling covariate shift, which
fail in high dimensions due to potentially high train/test covariate divergence
and the inability to accurately estimate the requisite density ratios. We first
formulate and solve a problem optimizing over linear subspaces a combination of
their predictive utility and train/test divergence within. Applying it to
simulated and real data, we show extreme dimension reduction helps sometimes
but not always, due to a bias introduced by dimension reduction.
