  Modern algorithms for binary classification rely on an intermediate
regression problem for computational tractability. In this paper, we establish
a geometric distinction between classification and regression that allows risk
in these two settings to be more precisely related. In particular, we note that
classification risk depends only on the direction of the regressor, and we take
advantage of this scale invariance to improve existing guarantees for how
classification risk is bounded by the risk in the intermediate regression
problem. Building on these guarantees, our analysis makes it possible to
compare algorithms more accurately against each other and suggests viewing
classification as unique from regression rather than a byproduct of it. While
regression aims to converge toward the conditional expectation function in
location, we propose that classification should instead aim to recover its
direction.
