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

Abstract

We study the convergence rate of continuous-time simulated annealing (Xt; t ≥ 0) and its discretization (x k ; k = 0, 1, . . .) for approximating the global optimum of a given function f . We prove that the tail probability P(f (Xt) > min f + δ) (resp. P(f (x k ) > min f +δ)) 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.

Convergence of Simulated Annealing Using Kinetic Langevin Dynamics

“…(ii) The convergence rate in Theorem 2.5 is the same to that in Miclo [21] and Tang and Zhou [30] for the simulated annealing using overdamped Langevin dynamic, and also to that in Chak, Kantas and Pavliotis [3] for the simulated annealing using generalized Langevin process. While higher order Langevin dynamics are often used in MCMC as accelerated versions compare to the overdamped Langevin dynamic (see for instance [18,10,23]), this is not the case in the simulated annealing problem.…”

“…Remark 2.4. To obtain the convergence of the simulated annealing using overdamped Langevin dynamic in (1.1), it is standard to consider the cooling schedule ε t = O 1 log t as t −→ ∞; see, e.g., [11,5,28,14,21,30]. For the kinetic simulated annealing process (1.2), this cooling schedule is also assumed in Journel and Monmarché [15] to obtain a convergence result without convergence rate.…”

“…Since the introduction of the simulated annealing algorithm by Kirkpatrick, Gellatt and Vecchi [16], many works have been devoted to the convergence analysis of (1.1); see e.g., Geman and Hwang [11], Chiang, Hwang and Sheu [5], Royer [28], Holley, Kusuoka and Stroock [14], Miclo [21], Zitt [32], Fournier and Tardif [9], Tang and Zhou [30], etc. It has been shown that, the cooling schedule ε t should be at least of the order E log t as t −→ ∞ for some constant E > 0, in order to ensure the convergence of X t to the global minimum of U as t −→ ∞.…”

“…Intuitively, this cooling schedule allows the diffusion process to have enough time to escape from the local minima and at the same time to explore the whole space in order to find the global minimum of U ; finally, the annealing process will "freeze" at the global minimum of U as ε t −→ 0. We would like to mention in particular the recent paper by Tang and Zhou [30], where the authors derived a convergence rate result of (1.1), where a fine estimation of the log-Sobolev inequality in Menz and Schlichting [20] for invariant measure µ * ε,o with low-temperature (small ε > 0) has been crucially used. Moreover, they have also analyzed a corresponding discrete time scheme of (1.1) and obtained a convergence rate result.…”

Mean-Field Langevin Dynamics: Exponential Convergence and Annealing

“…The linear case when G(µ) = V dµ has been considered by a numerous works (e.g. Geman and Hwang [1986], Miclo [1992], Raginsky et al [2017], Tang and Zhou [2021]). It is known in particular [Miclo, 1992] that under suitable coercivity assumptions for V and if τ t = C/ log(t) for some C > 0 large enough, then G(µ t ) converges to min µ∈P 2 (R d ) G(µ) = min x∈R d V (x).…”