Global Optimization via Schr{ö}dinger-F{ö}llmer Diffusion

We study the problem of finding global minimizers of $V(x):\mathbb{R}^d\rightarrow\mathbb{R}$ approximately via sampling from a probability distribution $μ_σ$ with density $p_σ(x)=\dfrac{\exp(-V(x)/σ)}{\int_{\mathbb R^d} \exp(-V(y)/σ) dy }$ with respect to the Lebesgue measure for $σ\in (0,1]$ small enough. We analyze a sampler based on the Euler-Maruyama discretization of the Schr{ö}dinger-F{ö}llmer diffusion processes with stochastic approximation under appropriate assumptions on the step size $s$ and the potential $V$. We prove that the output of the proposed sampler is an approximate global minimizer of $V(x)$ with high probability at cost of sampling $\mathcal{O}(d^{3})$ standard normal random variables. Numerical studies illustrate the effectiveness of the proposed method and its superiority to the Langevin method.

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