Relative entropy minimization over Hilbert spaces via Robbins-Monro

Gideon Simpson; Daniel Watkins; Gideon Simpson; Daniel Watkins

doi:10.3934/math.2019.3.359

AIMS Mathematics

2019, Volume 4, Issue 3: 359-383. doi: 10.3934/math.2019.3.359

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Research article

Relative entropy minimization over Hilbert spaces via Robbins-Monro

Gideon Simpson ^{1
,
,},
Daniel Watkins ²

1.
Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA
2.
Department of Ocean, Earth, and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331, USA

Received: 25 February 2019 Accepted: 27 March 2019 Published: 15 April 2019
MSC : 65K10, 62L20, 60G15, 65C05

One way of getting insight into non-Gaussian measures is to first obtain good Gaussian approximations. These best fit Gaussians can then provide a sense of the mean and variance of the distribution of interest. They can also be used to accelerate sampling algorithms. This begs the question of how one should measure optimality, and how to then obtain this optimal approximation. Here, we consider the problem of minimizing the distance between a family of Gaussians and the target measure with respect to relative entropy, or Kullback-Leibler divergence. As we are interested in applications in the infinite dimensional setting, it is desirable to have convergent algorithms that are well posed on abstract Hilbert spaces. We examine this minimization problem by seeking roots of the first variation of relative entropy, taken with respect to the mean of the Gaussian, leaving the covariance fixed. We prove the convergence of Robbins-Monro type root finding algorithms in this context, highlighting the assumptions necessary for convergence to relative entropy minimizers. Numerical examples are included to illustrate the algorithms.
- Robbins-Monro,
- relative entropy,
- Hilbert space
Citation: Gideon Simpson, Daniel Watkins. Relative entropy minimization over Hilbert spaces via Robbins-Monro[J]. AIMS Mathematics, 2019, 4(3): 359-383. doi: 10.3934/math.2019.3.359

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Abstract

One way of getting insight into non-Gaussian measures is to first obtain good Gaussian approximations. These best fit Gaussians can then provide a sense of the mean and variance of the distribution of interest. They can also be used to accelerate sampling algorithms. This begs the question of how one should measure optimality, and how to then obtain this optimal approximation. Here, we consider the problem of minimizing the distance between a family of Gaussians and the target measure with respect to relative entropy, or Kullback-Leibler divergence. As we are interested in applications in the infinite dimensional setting, it is desirable to have convergent algorithms that are well posed on abstract Hilbert spaces. We examine this minimization problem by seeking roots of the first variation of relative entropy, taken with respect to the mean of the Gaussian, leaving the covariance fixed. We prove the convergence of Robbins-Monro type root finding algorithms in this context, highlighting the assumptions necessary for convergence to relative entropy minimizers. Numerical examples are included to illustrate the algorithms.

References

[1]	A. E. Albert and L. A. Gardner Jr., Stochastic approximation and nonlinear regression The MIT Press, 1967.
[2]	C. Andrieu and É. Moulines, On the ergodicity properties of some adaptive MCMC algorithms Ann. Appl. Probab. 16 (2006), 1462-1505. doi: 10.1214/105051606000000286
[3]	C. Andrieu and J. Thoms, A tutorial on adaptive MCMC Stat. Comput. 18 (2008), 343-373. doi: 10.1007/s11222-008-9110-y
[4]	D. M. Blei, A. Kucukelbir and J. D. McAuliffe, Variational Inference: A Review for Statisticians J. Am. Stat. Assoc. 112 (2017), 859-877. doi: 10.1080/01621459.2017.1285773
[5]	J. R. Blum, Approximation methods which converge with probability one Annals of Mathematical Statistics 25 (1954), 382-386. doi: 10.1214/aoms/1177728794
[6]	J. R. Blum, Multidimensional stochastic approximation methods Annals of Mathematical Statistics 25 (1954), 737-744. doi: 10.1214/aoms/1177728659
[7]	S. D. Chatterji, Martingale convergence and the Radon-Nikodym theorem in Banach spaces Math. Scand. 22 (1968), 21-41. doi: 10.7146/math.scand.a-10868
[8]	H.-F. Chen, Stochastic approximation and its applications 2002.
[9]	G. Da Prato, An Introduction to Infinite-Dimensional Analsysis Springer, 2006.
[10]	H. Haario, E. Saksman and J. Tamminen, An adaptive Metropolis algorithm Bernoulli 7 (2001), 223-242. doi: 10.2307/3318737
[11]	H. J. Kushner and G. Yin, Stochastic Approximation and Recursive Algorithms and Applications Springer, 2003.
[12]	J. Lelong, Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions Stat. Probabil. Lett. 78 (2008), 2632-2636. doi: 10.1016/j.spl.2008.02.034
[13]	Y. Lu, A. Stuart and H. Weber, Gaussian approximations for probability measures on r^d SIAM/ASA Journal on Uncertainty Quantification 5 (2017), 1136-1165. doi: 10.1137/16m1105384
[14]	Y. Lu, A. Stuart and H. Weber, Gaussian approximations for transition paths in brownian dynamics SIAM J. Math. Anal. 49 (2017), 3005-3047. doi: 10.1137/16m1071845
[15]	F. J. Pinski, G Simpson, A. M. Stuart, et al. Algorithms for Kullback-Leibler Approximation of Probability Measures in Infinite Dimensions SIAM J. Sci. Comput. 37 (2015), A2733-A2757. doi: 10.1137/14098171x
[16]	F. J. Pinski, G Simpson, A. M. Stuart, et al. Kullback-Leibler Approximation for Probability Measures on Infinite Dimensional Spaces SIAM J. Math. Anal. 47 (2015), 4091-4122. doi: 10.1137/140962802
[17]	H. Robbins and S. Monro, A stochastic approximation method The Annals of Mathematical Statistics (1950), 400-407.
[18]	G. O. Roberts and J. S. Rosenthal, Coupling and Ergodicity of Adaptive Markov Chain Monte Carlo Algorithms J. Appl. Probab. 44 (2007), 458-475. doi: 10.1017/s0021900200003090
[19]	D. Sanz-Alonso and A. M. Stuart, Gaussian approximations of small noise diffusions in kullback-leibler divergence Commun. Math. Sci. 15 (2017), 2087-2097. doi: 10.4310/cms.2017.v15.n7.a13
[20]	G. Yin and Y. M. Zhu, On H-valued Robbins-Monro processes J. Multivariate Anal. 34 (1990), 116-140. doi: 10.1016/0047-259x(90)90064-o

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