Simulated annealing from continuum to discretization: a convergence analysis via the Eyring--Kramers law

• URL: http://arxiv.org/abs/2102.02339v1
• Date: Wed, 3 Feb 2021 23:45:39 GMT
• Title: Simulated annealing from continuum to discretization: a convergence analysis via the Eyring--Kramers law
• Authors: Wenpin Tang and Xun Yu Zhou
• Abstract summary: We study the convergence rate of continuous-time simulated annealing $(X_t;, t ge 0)$ and its discretization $(x_k;, k =0,1, ldots)$ We prove that the tail probability $mathbbP(f(X_t) > min f +delta)$ (resp. $mathP(f(x_k) > min f +delta)$) decays in time (resp. in cumulative step size)
• Score: 10.406659081400354
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the convergence rate of continuous-time simulated annealing $(X_t; \, t \ge 0)$ and its discretization $(x_k; \, k =0,1, \ldots)$ for approximating the global optimum of a given function $f$. We prove that the tail probability $\mathbb{P}(f(X_t) > \min f +\delta)$ (resp. $\mathbb{P}(f(x_k) > \min f +\delta)$) decays polynomial in time (resp. in cumulative step size), and provide an explicit rate as a function of the model parameters. Our argument applies the recent development on functional inequalities for the Gibbs measure at low temperatures -- the Eyring-Kramers law. In the discrete setting, we obtain a condition on the step size to ensure the convergence.

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