  We consider the problem of estimating how well a model class is capable of
fitting a distribution of labeled data. We show that it is often possible to
accurately estimate this "learnability" even when given an amount of data that
is too small to reliably learn any accurate model. Our first result applies to
the setting where the data is drawn from a $d$-dimensional distribution with
isotropic covariance (or known covariance), and the label of each datapoint is
an arbitrary noisy function of the datapoint. In this setting, we show that
with $O(\sqrt{d})$ samples, one can accurately estimate the fraction of the
variance of the label that can be explained via the best linear function of the
data. In contrast to this sublinear sample size, finding an approximation of
the best-fit linear function requires on the order of $d$ samples. Our
sublinear sample results and approach also extend to the non-isotropic setting,
where the data distribution has an (unknown) arbitrary covariance matrix: we
show that, if the label $y$ of point $x$ is a linear function with independent
noise, $y = \langle x , \beta \rangle + noise$ with $\|\beta \|$ bounded, the
variance of the noise can be estimated to error $\epsilon$ with
$O(d^{1-1/\log{1/\epsilon}})$ if the covariance matrix has bounded condition
number, or $O(d^{1-\sqrt{\epsilon}})$ if there are no bounds on the condition
number. We also establish that these sample complexities are optimal, to
constant factors. Finally, we extend these techniques to the setting of binary
classification, where we obtain analogous sample complexities for the problem
of estimating the prediction error of the best linear classifier, in a natural
model of binary labeled data. We demonstrate the practical viability of our
approaches on several real and synthetic datasets.
