  Given a loss function $F:\mathcal{X} \rightarrow \R^+$ that can be written as
the sum of losses over a large set of inputs $a_1,\ldots, a_n$, it is often
desirable to approximate $F$ by subsampling the input points. Strong
theoretical guarantees require taking into account the importance of each
point, measured by how much its individual loss contributes to $F(x)$.
Maximizing this importance over all $x \in \mathcal{X}$ yields the
\emph{sensitivity score} of $a_i$. Sampling with probabilities proportional to
these scores gives strong guarantees, allowing one to approximately minimize of
$F$ using just the subsampled points.
  Unfortunately, sensitivity sampling is difficult to apply since (1) it is
unclear how to efficiently compute the sensitivity scores and (2) the sample
size required is often impractically large. To overcome both obstacles we
introduce \emph{local sensitivity}, which measures data point importance in a
ball around some center $x_0$. We show that the local sensitivity can be
efficiently estimated using the \emph{leverage scores} of a quadratic
approximation to $F$ and that the sample size required to approximate $F$
around $x_0$ can be bounded. We propose employing local sensitivity sampling in
an iterative optimization method and analyze its convergence when $F$ is smooth
and convex.
