Tail probability estimates of continuous-time simulated annealing processes

Abstract

<p style='text-indent:20px;'>We study the convergence rate of a continuous-time simulated annealing process <inline-formula><tex-math id="M1">\begin{document}$ (X_t; \, t \ge 0) $\end{document}</tex-math></inline-formula> for approximating the global optimum of a given function <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula>. We prove that the tail probability <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{P}(f(X_t) > \min f +\delta) $\end{document}</tex-math></inline-formula> decays polynomial in time with an appropriately chosen cooling schedule of temperature, and provide an explicit convergence rate through a non-asymptotic bound. Our argument applies recent development of the Eyring-Kramers law on functional inequalities for the Gibbs measure at low temperatures.</p>