doi: 10.3934/naco.2022015
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Tail probability estimates of continuous-time simulated annealing processes

Department of Industrial Engineering and Operations Research, Columbia University, USA

*Corresponding author: Xun Yu Zhou

The paper is handled by Song Yao as guest editor

Received  December 2021 Revised  May 2022 Early access May 2022

We study the convergence rate of a continuous-time simulated annealing process $ (X_t; \, t \ge 0) $ for approximating the global optimum of a given function $ f $. We prove that the tail probability $ \mathbb{P}(f(X_t) > \min f +\delta) $ 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.

Citation: Wenpin Tang, Xun Yu Zhou. Tail probability estimates of continuous-time simulated annealing processes. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022015
References:
[1]

D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der Mathematischen Wissenschaften, Springer, 2014. doi: 10.1007/978-3-319-00227-9.

[2]

Y. Bengio, J. Louradour, R. Collobert and J. Weston, Curriculum learning, In ICML, (2009), 41–48.

[3]

A. BovierM. EckhoffV. Gayrard and M. Klein, Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc., 6 (2004), 399-424.  doi: 10.4171/JEMS/14.

[4]

A. BovierV. Gayrard and M. Klein, Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc., 7 (2005), 69-99.  doi: 10.4171/JEMS/22.

[5]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[6]

V. Cerny, Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm, J. Optim. Theory Appl., 45 (1985), 41-51.  doi: 10.1007/BF00940812.

[7]

X. Chen, S. S. Du and X. T. Tong, On stationary-point hitting time and ergodicity of stochastic gradient Langevin dynamics, J. Mach. Learn. Res., 21 (2020), Paper No. 68.

[8]

A. S. Cherny and H.-J. Engelbert, Singular Stochastic Differential Equations, volume 1858 of Lecture Notes in Mathematics, Springer-Verlag, 2005. doi: 10.1007/b104187.

[9]

T.-S. ChiangC.-R. Hwang and S. J. Sheu, Diffusion for global optimization in Rn, SIAM J. Control Optim., 25 (1987), 737-753.  doi: 10.1137/0325042.

[10]

D. Delahaye, S. Chaimatanan and M. Mongeau, Simulated annealing: from basics to applications, In Handbook of Metaheuristics, volume 272 of Internat. Ser. Oper. Res. Management Sci., (2019), 1–35.

[11]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., 1986. doi: 10.1002/9780470316658.

[12]

H. Eyring, The activated complex in chemical reactions, J. Chem. Phys., 3 (1935), 107-115. 

[13]

H. FangM. Qian and G. Gong, An improved annealing method and its large-time behavior, Stochastic Process. Appl., 71 (1997), 55-74.  doi: 10.1016/S0304-4149(97)00069-0.

[14]

N. Fournier and C. Tardif, On the simulated annealing in $\mathbb{R}^d$, arXiv: 2003.06360, 2020. doi: 10.1016/j.jfa.2021.109086.

[15]

X. Gao, Z. Q. Xu and X. Y. Zhou, State-dependent temperature control for Langevin diffusions, arXiv: 2005.04507, 2020. doi: 10.1137/21M1429424.

[16]

R. Ge, F. Huang, C. Jin and Y. Yuan., Escaping from saddle points – online stochastic gradient for tensor decomposition, In COLT, (2015), 797–842.

[17]

S. Geman and C.-R. Hwang, Diffusions for global optimization, SIAM J. Control Optim., 24 (1986), 1031-1043.  doi: 10.1137/0324060.

[18]

B. Gidas, Global optimization via the Langevin equation, In 24th IEEE Conference on Decision and Control, (1985), 774–778.

[19]

U. Grenander, Tutorial in Pattern Theory, Division of Applied Mathematics. Brown University, 1983.

[20]

X. Guo, J. Han, M. Tajrobehkar and W. Tang., Perturbed gradient descent with occupation time, arXiv: 2005.04507, 2020.

[21]

R. A. HolleyS. Kusuoka and D. W. Stroock, Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Funct. Anal., 83 (1989), 333-347.  doi: 10.1016/0022-1236(89)90023-2.

[22]

P. Jain and P. Kar., Non-convex optimization for machine learning, Found. Trends Mach. Learn., 10 (2017), 142-336. 

[23]

C. Jin, R. Ge, P. Netrapalli, S. Kakade and M. I. Jordan, How to escape saddle points efficiently, In ICML, (2017), 1724–1732.

[24]

S. KirkpatrickJ. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.  doi: 10.1126/science.220.4598.671.

[25]

C. KoulamasS. Antony and R. Jaen, A survey of simulated annealing applications to operations research problems, Omega, 22 (1994), 41-56.  doi: 10.1287/opre.42.6.1025.

[26]

H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7 (1940), 284-304. 

[27]

Y. MaY. ChenC. JinN. Flammarion and M. I. Jordan, Sampling can be faster than optimization, Proc. Natl. Acad. Sci. USA, 116 (2019), 20881-20885.  doi: 10.1073/pnas.1820003116.

[28]

Y.-A. MaN. S. ChatterjiX. ChengN. FlammarionP. L. Bartlett and M. I. Jordan, Is there an analog of Nesterov acceleration for gradient-based MCMC?, Bernoulli, 27 (2021), 1942-1992.  doi: 10.3150/20-bej1297.

[29]

D. Márquez, Convergence rates for annealing diffusion processes, Ann. Appl. Probab., 7 (1997), 1118-1139.  doi: 10.1214/aoap/1043862427.

[30]

G. Menz and A. Schlichting, Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape, Ann. Probab., 42 (2014), 1809-1884.  doi: 10.1214/14-AOP908.

[31]

G. Menz, A. Schlichting, W. Tang and T. Wu, Ergodicity of the infinite swapping algorithm at low temperature, arXiv: 1811.10174, 2018.

[32]

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab., 25 (1993), 518-548.  doi: 10.2307/1427522.

[33]

L. Miclo, Recuit simulé sur $\mathbb{R}^n$. Étude de l'évolution de l'énergie libre, Annales de l'Institut Henri Poincaré, 28 (1992), 235–266.

[34]

L. Miclo, Une étude des algorithmes de recuit simulé sous-admissibles, Ann. Fac. Sci. Toulouse Math., 4 (1995), 819-877. 

[35]

P. Monmarché, Hypocoercivity in metastable settings and kinetic simulated annealing, Probability Theory and Related Fields, (2018), 1–34. doi: 10.1007/s00440-018-0828-y.

[36] R. B. Myerson, Game Theory, Harvard University Press, 1991. 
[37]

I. Pavlyukevich, Lévy flights, non-local search and simulated annealing, J. Comput. Phys., 226 (2007), 1830-1844.  doi: 10.1016/j.jcp.2007.06.008.

[38]

M. Raginsky, A. Rakhlin and M. Telgarsky, Non-convex learning via stochastic gradient Langevin dynamics: a nonasymptotic analysis, In COLT, (2017), 1674–1703.

[39]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2, Itô Calculus, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., 1987.

[40]

G. Royer, An Initiation to Logarithmic Sobolev Inequalities, volume 14 of SMF/AMS Texts and Monographs, American Mathematical Society, 2007.

[41]

A. Schlichting, The Eyring-Kramers Formula for Poincaré and Logarithmic Sobolev Inequalities, PhD thesis, Universität Leipzig, 2012. Available at http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-97965.

[42]

Z. Shun and P. McCullagh, Laplace approximation of high-dimensional integrals, J. Roy. Statist. Soc. Ser. B, 57 (1995), 749-760. 

[43]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, volume 233 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, 1979.

[44]

W. Tang, Exponential ergodicity and convergence for generalized reflected Brownian motion, Queueing Syst., 92 (2019), 83-101.  doi: 10.1007/s11134-019-09610-5.

[45]

W. Tang, Y. Wu and X. Y. Zhou, Discrete simulated annealing: a convergence analysis via the Eyring–Kramers law, arXiv: 2102.02339, 2021.

[46]

A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer Series in Statistics, Springer, 2009. doi: 10.1007/b13794.

[47]

P. J. M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications, volume 37 of Mathematics and its Applications. D. Reidel Publishing Co., 1987. doi: 10.1007/978-94-015-7744-1.

[48]

H. WangT. Zariphopoulou and X. Y. Zhou, Reinforcement learning in continuous time and space: A stochastic control approach, Journal of Machine Learning Research, 21 (2020), 1-34. 

[49]

P. A. Zitt, Annealing diffusions in a potential function with a slow growth, Stochastic Process. Appl., 118 (2008), 76-119.  doi: 10.1016/j.spa.2007.04.002.

show all references

References:
[1]

D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der Mathematischen Wissenschaften, Springer, 2014. doi: 10.1007/978-3-319-00227-9.

[2]

Y. Bengio, J. Louradour, R. Collobert and J. Weston, Curriculum learning, In ICML, (2009), 41–48.

[3]

A. BovierM. EckhoffV. Gayrard and M. Klein, Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times, J. Eur. Math. Soc., 6 (2004), 399-424.  doi: 10.4171/JEMS/14.

[4]

A. BovierV. Gayrard and M. Klein, Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc., 7 (2005), 69-99.  doi: 10.4171/JEMS/22.

[5]

J. A. CarrilloR. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 179 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1.

[6]

V. Cerny, Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm, J. Optim. Theory Appl., 45 (1985), 41-51.  doi: 10.1007/BF00940812.

[7]

X. Chen, S. S. Du and X. T. Tong, On stationary-point hitting time and ergodicity of stochastic gradient Langevin dynamics, J. Mach. Learn. Res., 21 (2020), Paper No. 68.

[8]

A. S. Cherny and H.-J. Engelbert, Singular Stochastic Differential Equations, volume 1858 of Lecture Notes in Mathematics, Springer-Verlag, 2005. doi: 10.1007/b104187.

[9]

T.-S. ChiangC.-R. Hwang and S. J. Sheu, Diffusion for global optimization in Rn, SIAM J. Control Optim., 25 (1987), 737-753.  doi: 10.1137/0325042.

[10]

D. Delahaye, S. Chaimatanan and M. Mongeau, Simulated annealing: from basics to applications, In Handbook of Metaheuristics, volume 272 of Internat. Ser. Oper. Res. Management Sci., (2019), 1–35.

[11]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., 1986. doi: 10.1002/9780470316658.

[12]

H. Eyring, The activated complex in chemical reactions, J. Chem. Phys., 3 (1935), 107-115. 

[13]

H. FangM. Qian and G. Gong, An improved annealing method and its large-time behavior, Stochastic Process. Appl., 71 (1997), 55-74.  doi: 10.1016/S0304-4149(97)00069-0.

[14]

N. Fournier and C. Tardif, On the simulated annealing in $\mathbb{R}^d$, arXiv: 2003.06360, 2020. doi: 10.1016/j.jfa.2021.109086.

[15]

X. Gao, Z. Q. Xu and X. Y. Zhou, State-dependent temperature control for Langevin diffusions, arXiv: 2005.04507, 2020. doi: 10.1137/21M1429424.

[16]

R. Ge, F. Huang, C. Jin and Y. Yuan., Escaping from saddle points – online stochastic gradient for tensor decomposition, In COLT, (2015), 797–842.

[17]

S. Geman and C.-R. Hwang, Diffusions for global optimization, SIAM J. Control Optim., 24 (1986), 1031-1043.  doi: 10.1137/0324060.

[18]

B. Gidas, Global optimization via the Langevin equation, In 24th IEEE Conference on Decision and Control, (1985), 774–778.

[19]

U. Grenander, Tutorial in Pattern Theory, Division of Applied Mathematics. Brown University, 1983.

[20]

X. Guo, J. Han, M. Tajrobehkar and W. Tang., Perturbed gradient descent with occupation time, arXiv: 2005.04507, 2020.

[21]

R. A. HolleyS. Kusuoka and D. W. Stroock, Asymptotics of the spectral gap with applications to the theory of simulated annealing, J. Funct. Anal., 83 (1989), 333-347.  doi: 10.1016/0022-1236(89)90023-2.

[22]

P. Jain and P. Kar., Non-convex optimization for machine learning, Found. Trends Mach. Learn., 10 (2017), 142-336. 

[23]

C. Jin, R. Ge, P. Netrapalli, S. Kakade and M. I. Jordan, How to escape saddle points efficiently, In ICML, (2017), 1724–1732.

[24]

S. KirkpatrickJ. Gelatt and M. Vecchi, Optimization by simulated annealing, Science, 220 (1983), 671-680.  doi: 10.1126/science.220.4598.671.

[25]

C. KoulamasS. Antony and R. Jaen, A survey of simulated annealing applications to operations research problems, Omega, 22 (1994), 41-56.  doi: 10.1287/opre.42.6.1025.

[26]

H. A. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica, 7 (1940), 284-304. 

[27]

Y. MaY. ChenC. JinN. Flammarion and M. I. Jordan, Sampling can be faster than optimization, Proc. Natl. Acad. Sci. USA, 116 (2019), 20881-20885.  doi: 10.1073/pnas.1820003116.

[28]

Y.-A. MaN. S. ChatterjiX. ChengN. FlammarionP. L. Bartlett and M. I. Jordan, Is there an analog of Nesterov acceleration for gradient-based MCMC?, Bernoulli, 27 (2021), 1942-1992.  doi: 10.3150/20-bej1297.

[29]

D. Márquez, Convergence rates for annealing diffusion processes, Ann. Appl. Probab., 7 (1997), 1118-1139.  doi: 10.1214/aoap/1043862427.

[30]

G. Menz and A. Schlichting, Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape, Ann. Probab., 42 (2014), 1809-1884.  doi: 10.1214/14-AOP908.

[31]

G. Menz, A. Schlichting, W. Tang and T. Wu, Ergodicity of the infinite swapping algorithm at low temperature, arXiv: 1811.10174, 2018.

[32]

S. P. Meyn and R. L. Tweedie, Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes, Adv. in Appl. Probab., 25 (1993), 518-548.  doi: 10.2307/1427522.

[33]

L. Miclo, Recuit simulé sur $\mathbb{R}^n$. Étude de l'évolution de l'énergie libre, Annales de l'Institut Henri Poincaré, 28 (1992), 235–266.

[34]

L. Miclo, Une étude des algorithmes de recuit simulé sous-admissibles, Ann. Fac. Sci. Toulouse Math., 4 (1995), 819-877. 

[35]

P. Monmarché, Hypocoercivity in metastable settings and kinetic simulated annealing, Probability Theory and Related Fields, (2018), 1–34. doi: 10.1007/s00440-018-0828-y.

[36] R. B. Myerson, Game Theory, Harvard University Press, 1991. 
[37]

I. Pavlyukevich, Lévy flights, non-local search and simulated annealing, J. Comput. Phys., 226 (2007), 1830-1844.  doi: 10.1016/j.jcp.2007.06.008.

[38]

M. Raginsky, A. Rakhlin and M. Telgarsky, Non-convex learning via stochastic gradient Langevin dynamics: a nonasymptotic analysis, In COLT, (2017), 1674–1703.

[39]

L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2, Itô Calculus, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., 1987.

[40]

G. Royer, An Initiation to Logarithmic Sobolev Inequalities, volume 14 of SMF/AMS Texts and Monographs, American Mathematical Society, 2007.

[41]

A. Schlichting, The Eyring-Kramers Formula for Poincaré and Logarithmic Sobolev Inequalities, PhD thesis, Universität Leipzig, 2012. Available at http://nbn-resolving.de/urn:nbn:de:bsz:15-qucosa-97965.

[42]

Z. Shun and P. McCullagh, Laplace approximation of high-dimensional integrals, J. Roy. Statist. Soc. Ser. B, 57 (1995), 749-760. 

[43]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, volume 233 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, 1979.

[44]

W. Tang, Exponential ergodicity and convergence for generalized reflected Brownian motion, Queueing Syst., 92 (2019), 83-101.  doi: 10.1007/s11134-019-09610-5.

[45]

W. Tang, Y. Wu and X. Y. Zhou, Discrete simulated annealing: a convergence analysis via the Eyring–Kramers law, arXiv: 2102.02339, 2021.

[46]

A. B. Tsybakov, Introduction to Nonparametric Estimation, Springer Series in Statistics, Springer, 2009. doi: 10.1007/b13794.

[47]

P. J. M. van Laarhoven and E. H. L. Aarts, Simulated Annealing: Theory and Applications, volume 37 of Mathematics and its Applications. D. Reidel Publishing Co., 1987. doi: 10.1007/978-94-015-7744-1.

[48]

H. WangT. Zariphopoulou and X. Y. Zhou, Reinforcement learning in continuous time and space: A stochastic control approach, Journal of Machine Learning Research, 21 (2020), 1-34. 

[49]

P. A. Zitt, Annealing diffusions in a potential function with a slow growth, Stochastic Process. Appl., 118 (2008), 76-119.  doi: 10.1016/j.spa.2007.04.002.

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