Finding Global Minima via Kernel Approximations
Résumé
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).
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