  In linear regression we wish to estimate the optimum linear least squares
predictor for a distribution over $d$-dimensional input points and real-valued
responses, based on a small sample. Under standard random design analysis,
where the sample is drawn i.i.d. from the input distribution, the least squares
solution for that sample can be viewed as the natural estimator of the optimum.
Unfortunately, this estimator almost always incurs an undesirable bias coming
from the randomness of the input points, which is a significant bottleneck in
model averaging. In this paper we show that it is possible to draw a non-i.i.d.
sample of input points such that, regardless of the response model, the least
squares solution is an unbiased estimator of the optimum. Moreover, this sample
can be produced efficiently by augmenting a previously drawn i.i.d. sample with
an additional set of $d$ points, drawn jointly according to a certain
determinantal point process constructed from the input distribution rescaled by
the squared volume spanned by the points. Motivated by this, we develop a
theoretical framework for studying volume-rescaled sampling, and in the process
prove a number of new matrix expectation identities. We use them to show that
for any input distribution and $\epsilon>0$ there is a random design consisting
of $O(d\log d+ d/\epsilon)$ points from which an unbiased estimator can be
constructed whose expected square loss over the entire distribution is bounded
by $1+\epsilon$ times the loss of the optimum. We provide efficient algorithms
for generating such unbiased estimators in a number of practical settings and
support our claims experimentally.
