Research ArticleNo Access

Lagrangian and Hamiltonian dynamics for probabilities on the statistical bundle

Goffredo Chirco

Dipartimento di Fisica Ettore Pancini, Universitá di Napoli Federico II & INFN Sezione di Napoli, Napoli, Italy

E-mail Address: [email protected]

Corresponding author.

Search for more papers by this author

Luigi Malagò

Transylvanian Institute of Neuroscience, Cluj-Napoca, Romania

E-mail Address: [email protected]

Search for more papers by this author

, and

Giovanni Pistone

de Castro Statistics, Collegio Carlo Alberto, Torino, Italy

E-mail Address: [email protected]

Search for more papers by this author

https://doi.org/10.1142/S0219887822502140Cited by:7 (Source: Crossref)

Abstract

We provide an Information-Geometric formulation of accelerated natural gradient on the Riemannian manifold of probability distributions, which is an affine manifold endowed with a dually-flat connection. In a non-parametric formalism, we consider the full set of positive probability functions on a finite sample space, and we provide a specific expression for the tangent and cotangent spaces over the statistical manifold, in terms of a Hilbert bundle structure that we call the Statistical Bundle. In this setting, we compute velocities and accelerations of a one-dimensional statistical model using the canonical dual pair of parallel transports and define a coherent formalism for Lagrangian and Hamiltonian mechanics on the bundle. We show how our formalism provides a consistent framework for accelerated natural gradient dynamics on the probability simplex, paving the way for direct applications in optimization.

Keywords:

AMSC: 53B12, 62B11, 65K10

References

1. R. Abraham and J. E. Marsden, Foundations of Mechanics, Advanced Book Program, 2nd edn. (Benjamin/Cummings Publishing Company, Reading, Massachusetts, 1978), MR 515141. Google Scholar
2. P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds (Princeton University Press, 2008), With a foreword by Paul Van Dooren. Crossref, Google Scholar
3. K. Ahn and S. Sra, From Nesterov’s estimate sequence to Riemannian acceleration, Proc. Thirty Third Conf. Learning Theory, Proceedings of Machine Learning Research, Vol. 125, eds. J. Abernethy and S. Agarwal (PMLR, 2020), pp. 84–118. Google Scholar
4. J. Aitchison, The Statistical Analysis of Compositional Data, Monographs on Statistics and Applied Probability (Chapman & Hall, London, 1986), MR 865647. Crossref, Google Scholar
5. F. Alimisis, A. Orvieto, G. Bécigneul and A. Lucchi, A continuous-time perspective for modeling acceleration in Riemannian optimization, Proc. Twenty Third Int. Conf. Artificial Intelligence and Statistics (PMLR, 2019), pp. 1297–1307. Google Scholar
6. S.-i. Amari, Differential-Geometrical Methods in Statistics, Lecture Notes in Statistics, Vol. 28 (Springer-Verlag, 1985), MR 86m:62053. Crossref, Google Scholar
7. S.-i. Amari, Dual connections on the Hilbert bundles of statistical models, Geometrization of Statistical Theory (ULDM, Lancaster, 1987), pp. 123–151. Google Scholar
8. S.-i. Amari, Natural gradient works efficiently in learning, Neural Comput. 10(2) (1998) 251–276. Crossref, Web of Science, Google Scholar
9. S.-i. Amari and H. Nagaoka, Methods of Information Geometry, (American Mathematical Society, 2000), Translated from the 1993 Japanese original by D. Harada, MR 1 800 071. Google Scholar
10. V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics, Vol. 60 (Springer-Verlag, New York, 1989), Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second edition (1989), MR 1345386. Crossref, Google Scholar
11. H. Attouch, Z. Chbani, J. Peypouquet and P. Redont, Fast convergence of inertial dynamics and algorithms with asymptotic vanishing viscosity, Math. Program. 168(1) (2018) 123–175. Crossref, Web of Science, Google Scholar
12. N. Ay, J. Jost, H. V. Lê and L. Schwachhöfer, Information Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Vol. 64 (Springer, Cham, 2017), MR 3701408. Crossref, Google Scholar
13. H. Bateman, On dissipative systems and related variational principles, Phys. Rev. 38 (1931) 815–819. Crossref, Google Scholar
14. M. Betancourt, A conceptual introduction to Hamiltonian Monte Carlo, preprint (2017), arXiv:1701.02434. Google Scholar
15. L. Bottou, F. E. Curtis and J. Nocedal, Optimization methods for large-scale machine learning, SIAM Rev. 60(2) (2018) 223–311. Crossref, Web of Science, Google Scholar
16. G. Chirco, Rényi Relative Entropy from Homogeneous Kullback-Leibler Divergence Lagrangian, Geometric Science of Information (Cham), eds. F. Nielsen and F. Barbaresco (Springer International Publishing, 2021), pp. 744–751. Crossref, Google Scholar
17. G. Chirco, T. Josset and C. Rovelli, Statistical mechanics of reparametrization-invariant systems. It takes three to tango., Class. Quantum Grav. 33(4) (2016) 045005. Crossref, Web of Science, Google Scholar
18. F. M. Ciaglia, F. Di Cosmo, D. Felice, S. Mancini, G. Marmo and J. M. Pérez-Pardo, Hamilton-Jacobi approach to potential functions in information geometry, J. Math. Phys. 58(6) (2017) 063506. Crossref, Web of Science, Google Scholar
19. B. Efron and T. Hastie, Computer Age Statistical Inference: Algorithms, Evidence, and Data Science, Institute of Mathematical Statistics (IMS) Monographs, Vol. 5 (Cambridge University Press, New York, 2016), MR 3523956. Crossref, Google Scholar
20. D. Felice and N. Ay, Dynamical systems induced by canonical divergence in dually flat manifolds, preprint (2018), arXiv:1812.04461. Google Scholar
21. G. França, M. I. Jordan and R. Vidal, On dissipative symplectic integration with applications to gradient-based optimization, J. Stat. Mech. 2021 (2021) 043402. Crossref, Web of Science, Google Scholar
22. G. França, J. Sulam, D. P. Robinson and R. Vidal, Conformal symplectic and relativistic optimization, preprint (2019), arXiv:1903.04100. Google Scholar
23. P. Gibilisco and G. Pistone, Connections on non-parametric statistical manifolds by Orlicz space geometry, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1(2) (1998) 325–347, MR 1 628 177. Link, Web of Science, Google Scholar
24. D. M. Greenberger, A critique of the major approaches to damping in quantum theory, J. Math. Phys. 20(5) (1979) 762–770. Crossref, Web of Science, Google Scholar
25. L. Herrera, L. Núñez, A. Patiño and H. Rago, A variational principle and the classical and quantum mechanics of the damped harmonic oscillator, Am. J. Phys. 54 (1986) 273. Crossref, Web of Science, Google Scholar
26. R. E. Kass and P. W. Vos, Geometrical Foundations of Asymptotic Inference, Wiley Series in Probability and Statistics: Probability and Statistics (John Wiley & Sons, New York, 1997), MR 1461540 (99b:62032). Crossref, Google Scholar
27. W. P. A. Klingenberg, Riemannian Geometry, De Gruyter Studies in Mathematics, Vol. 1, 2nd edn. (Walter de Gruyter & Company, Berlin, 1995), MR 1330918. Crossref, Google Scholar
28. W. Krichene, A. Bayen and P. L. Bartlett, Accelerated Mirror Descent in Continuous and Discrete Time, Advances in Neural Information Processing Systems, Vol. 28, eds. C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama and R. Garnett (Curran Associates, 2015), pp. 2845–2853. Google Scholar
29. M. Kumon and S.-i. Amari, Differential geometry of testing hypothesis — a higher order asymptotic theory in multi-parameter curved exponential family, J. Fac. Eng. Univ. Tokyo Ser. B 39(3) (1988) 241–273, MR 1407894. Google Scholar
30. S. Lang, Differential and Riemannian Manifolds, Graduate Texts in Mathematics, Vol. 160, 3rd edn. (Springer-Verlag, 1995), MR 96d:53001. Crossref, Google Scholar
31. M. Leok and J. Zhang, Connecting information geometry and geometric mechanics, Entropy 19(10) (2017) 518. Crossref, Web of Science, Google Scholar
32. Y. Liu, F. Shang, J. Cheng, H. Cheng and L. Jiao, Accelerated first-order methods for geodesically convex optimization on Riemannian manifolds, Proc. 31st Int. Conf. Neural Information Processing Systems, NIPS’17 (Curran Associates, Red Hook, NY, 2017), p. 4875–4884. Crossref, Google Scholar
33. L. Malagò and G. Pistone, Combinatorial optimization with information geometry: Newton method, Entropy 16 (2014) 4260–4289. Crossref, Web of Science, Google Scholar
34. M. Michałek, B. Sturmfels, C. Uhler and P. Zwiernik, Exponential varieties, Proc. Lond. Math. Soc. (3) 112(1) (2016) 27–56, MR 3458144. Crossref, Web of Science, Google Scholar
35. A. Semenovich, A. S. Nemirovsky and D. B. Yudin, Problem Complexity and Method Efficiency in Optimization (Wiley, Chichester, 1983). Google Scholar
36. Y. Nesterov, A method for solving the convex programming problem with convergence rate $o (1 / k^{2})$ , Proc. USSR Acad. Sci. 269 (1983) 543–547. Web of Science, Google Scholar
37. Y. Nesterov, Smooth minimization of non-smooth functions, Math. Program. 103(1) (2005) 127–152. Crossref, Web of Science, Google Scholar
38. Y. Nesterov, Gradient methods for minimizing composite objective function, LIDAM Discussion Papers CORE 2007076 (Université catholique de Louvain, Center for Operations Research and Econometrics (CORE), 2007). Google Scholar
39. Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, 1st edn. (Springer Publishing Company, 2014). Google Scholar
40. G. Pistone, Examples of the application of nonparametric information geometry to statistical physics, Entropy 15(10) (2013) 4042–4065. Crossref, Web of Science, Google Scholar
41. G. Pistone, Nonparametric information geometry, Geometric Science of Information, Proc. First Int. Conf., GSI 2013, Lecture Notes in Computer Science, Vol. 8085, eds. F. Nielsen and F. Barbaresco (Springer, Heidelberg, 2013), pp. 5–36, MR 3126029. Crossref, Google Scholar
42. G. Pistone, Lagrangian function on the finite state space statistical bundle, Entropy 20(2) (2018) 139. Crossref, Web of Science, Google Scholar
43. G. Pistone, Information geometry of the probability simplex: A short course, Nonlinear Phenom. Complex Syst. 23(2) (2020) 221–242. Crossref, Web of Science, Google Scholar
44. G. Pistone and C. Sempi, An infinite-dimensional geometric structure on the space of all the probability measures equivalent to a given one, Ann. Statist. 23(5) (1995) 1543–1561, MR 97j:62006. Crossref, Web of Science, Google Scholar
45. A. Rényi, On measures of entropy and information, Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, Contributions to the Theory of Statistics, Vol. 1 (University of California Press, Berkeley, California, 1961), pp. 547–561. Google Scholar
46. H. Shima, The Geometry of Hessian Structures (World Scientific Publishing Company, Hackensack, NJ, 2007), MR 2293045. Link, Google Scholar
47. J.-M. Souriau, Structure des Systèmes Dynamiques (Dunod, 1970), Réimpression autorisées, $2^{e}$ tirage (Jacques Gabay, 2012). Google Scholar
48. W. Su, S. Boyd and E. J. Candès, A differential equation for modeling Nesterov’s accelerated gradient method: Theory and insights, J. Mach. Learn. Res. 17(153) (2016) 1–43. Google Scholar
49. A. Taghvaei and P. Mehta, Accelerated flow for probability distributions, Proc. Machine Learning Research (Long Beach, California, USA), Vol. 97, eds. K. Chaudhuri and R. Salakhutdinov (PMLR, 2019), pp. 6076–6085. Google Scholar
50. Y. Wang and W. Li, Accelerated information gradient flow, preprint (2019), arXiv:1909.02102. Google Scholar
51. A. Wibisono, A. C. Wilson and M. I. Jordan, A variational perspective on accelerated methods in optimization, Proc. Nat. Acad. Sci. 113(47) (2016) E7351–E7358. Crossref, Web of Science, Google Scholar
52. H. Zhang and S. Sra, Towards Riemannian accelerated gradient methods, preprint (2018), arXiv:1806.02812. Google Scholar