Asymptotics of the spectral gap with applications to the theory of simulated annealing

Holley, Richard A.; Kusuoka, Shigeo; Stroock, Daniel W.

doi:10.1016/0022-1236(89)90023-2

Cited by 153 publications

(168 citation statements)

References 4 publications

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“…To understand how the above example fits into our framework, consider the following auxiliary system: (19) One can check that the compatibility condition (5) holds for (19) with the proportionality constant as , which reveals the meaning of the constant in the formula (18). If we compute the steady-state density for (19) using the formula (13), and then take the limit as , we arrive precisely at (18).…”

Section: Examplesmentioning

confidence: 99%

“…If we compute the steady-state density for (19) using the formula (13), and then take the limit as , we arrive precisely at (18).…”

Section: Examplesmentioning

confidence: 99%

“…For a discussion along these lines, see [4]. In the paper [19], and more recently in [5] and [6], Fokker-Planck operators and their spectral properties were studied with the view toward applications to function minimization. The analysis given below is based on the techniques employed in those references (the subsequent calculations are given in greater detail in [22]).…”

Section: Existence Uniqueness and Convergencementioning

confidence: 99%

“…The problem of estimating the spectral gap plays a role in the theory of simulated annealing [19]. In view of the results obtained in [21] which lead to an explicit characterization of the spectral gap for linear systems, it might be useful to relate to the value of the spectral gap in the case when is linear.…”

Section: Existence Uniqueness and Convergencementioning

confidence: 99%

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Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

Liberzon

Brockett

2000

IEEE Trans. Automat. Contr.

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Abstract-In this paper we describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steady-state probability densities can be found. We show that this class includes what has been called monotemperaturic systems in earlier work and establish relationships with Lyapunov functions for the corresponding deterministic systems. We also treat a number of stochastic optimal control problems in the case of quantized feedback, with performance criteria formulated in terms of the steady-state probability density.Index Terms-Nonlinear feedback system, quantizer, steady-state probability density, stochastic optimal control, white noise.

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Section: Examplesmentioning

confidence: 99%

“…If we compute the steady-state density for (19) using the formula (13), and then take the limit as , we arrive precisely at (18).…”

Section: Examplesmentioning

confidence: 99%

Section: Existence Uniqueness and Convergencementioning

confidence: 99%

Section: Existence Uniqueness and Convergencementioning

confidence: 99%

See 2 more Smart Citations

Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

Liberzon

Brockett

2000

IEEE Trans. Automat. Contr.

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“…Most frequently, people use a linearly or geometrically decreasing cooling schedule, which can no longer guarantee the global energy minimum to be reached. Holley et al (1989) showed that no cooling schedule faster than the logarithmic rate can be guaranteed to find the global energy minimum.…”

Section: Introductionmentioning

confidence: 99%

Annealing stochastic approximation Monte Carlo algorithm for neural network training

Liang

2007

Mach Learn

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We propose a general-purpose stochastic optimization algorithm, the so-called annealing stochastic approximation Monte Carlo (ASAMC) algorithm, for neural network training. ASAMC can be regarded as a space annealing version of the stochastic approximation Monte Carlo (SAMC) algorithm. Under mild conditions, we show that ASAMC can converge weakly at a rate of Ω(1/ √ t) toward a neighboring set (in the space of energy) of the global minimizers. ASAMC is compared with simulated annealing, SAMC, and the BFGS algorithm for training MLPs on a number of examples. The numerical results indicate that ASAMC outperforms the other algorithms in both training and test errors. Like other stochastic algorithms, ASAMC requires longer training time than do the gradient-based algorithms. It provides, however, an efficient approach to train MLPs for which the energy landscape is rugged.

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References

2010

Advanced Markov Chain Monte Carlo Methods

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Asymptotics of the spectral gap with applications to the theory of simulated annealing

Cited by 153 publications

References 4 publications

Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control

Annealing stochastic approximation Monte Carlo algorithm for neural network training

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