  We study the acceleration of the Local Polynomial Interpolation-based
Gradient Descent method (LPI-GD) recently proposed for the approximate solution
of empirical risk minimization problems (ERM). We focus on loss functions that
are strongly convex and smooth with condition number $\sigma$. We additionally
assume the loss function is $\eta$-H\"older continuous with respect to the
data. The oracle complexity of LPI-GD is $\tilde{O}\left(\sigma m^d
\log(1/\varepsilon)\right)$ for a desired accuracy $\varepsilon$, where $d$ is
the dimension of the parameter space, and $m$ is the cardinality of an
approximation grid. The factor $m^d$ can be shown to scale as
$O((1/\varepsilon)^{d/2\eta})$. LPI-GD has been shown to have better oracle
complexity than gradient descent (GD) and stochastic gradient descent (SGD) for
certain parameter regimes. We propose two accelerated methods for the ERM
problem based on LPI-GD and show an oracle complexity of
$\tilde{O}\left(\sqrt{\sigma} m^d \log(1/\varepsilon)\right)$. Moreover, we
provide the first empirical study on local polynomial interpolation-based
gradient methods and corroborate that LPI-GD has better performance than GD and
SGD in some scenarios, and the proposed methods achieve acceleration.
