  In the (special) smoothing spline problem one considers a variational problem
with a quadratic data fidelity penalty and Laplacian regularisation. Higher
order regularity can be obtained via replacing the Laplacian regulariser with a
poly-Laplacian regulariser. The methodology is readily adapted to graphs and
here we consider graph poly-Laplacian regularisation in a fully supervised,
non-parametric, noise corrupted, regression problem. In particular, given a
dataset $\{x_i\}_{i=1}^n$ and a set of noisy labels
$\{y_i\}_{i=1}^n\subset\mathbb{R}$ we let $u_n:\{x_i\}_{i=1}^n\to\mathbb{R}$ be
the minimiser of an energy which consists of a data fidelity term and an
appropriately scaled graph poly-Laplacian term. When $y_i = g(x_i)+\xi_i$, for
iid noise $\xi_i$, and using the geometric random graph, we identify (with high
probability) the rate of convergence of $u_n$ to $g$ in the large data limit
$n\to\infty$. Furthermore, our rate, up to logarithms, coincides with the known
rate of convergence in the usual smoothing spline model.
