  We consider the global minimization of smooth functions based solely on
function evaluations. Algorithms that achieve the optimal number of function
evaluations for a given precision level typically rely on explicitly
constructing an approximation of the function which is then minimized with
algorithms that have exponential running-time complexity. In this paper, we
consider an approach that jointly models the function to approximate and finds
a global minimum. This is done by using infinite sums of square smooth
functions and has strong links with polynomial sum-of-squares hierarchies.
Leveraging recent representation properties of reproducing kernel Hilbert
spaces, the infinite-dimensional optimization problem can be solved by
subsampling in time polynomial in the number of function evaluations, and with
theoretical guarantees on the obtained minimum.
  Given $n$ samples, the computational cost is $O(n^{3.5})$ in time, $O(n^2)$
in space, and we achieve a convergence rate to the global optimum that is
$O(n^{-m/d + 1/2 + 3/d})$ where $m$ is the degree of differentiability of the
function and $d$ the number of dimensions. The rate is nearly optimal in the
case of Sobolev functions and more generally makes the proposed method
particularly suitable for functions that have a large number of derivatives.
Indeed, when $m$ is in the order of $d$, the convergence rate to the global
optimum does not suffer from the curse of dimensionality, which affects only
the worst-case constants (that we track explicitly through the paper).
