Adaptive importance sampling with scaled Langevin proposal adaptatioń

International audienceIn Bayesian inference, a statistical model is assumed between an unknown vector of parameters and a set of observations. The goal is to construct a posterior distribution of the unknowns conditioned to the given data. However, for most practical models, the posterior is not available in a closed form, mostly due to intractable integrals, and approximations must be performed via Monte Carlo (MC) methods [1]. Among the available classes of MC methods, importance sampling (IS) consists in simulating random samples from a proposal distribution and weighting them properly with the aim of building consistent estimators of the moments of the posterior. The performance of IS depends on the proposal distribution [2–4]. Adaptive IS (AIS) is an iterative version of IS where the proposals distributions are adapted based on its performance on previous iterations [5]. In the last decade, many AIS algorithms have been proposed in the literature [6–12]. There is a large diversity in the implementation of AIS methods. However, two main challenges are still to be tackled. First, although all the algorithms adapt the location parameter (i.e., the mean) of the proposal, only few adapt the scale parameter (i.e., the covariance). This is problematic for instance when the unknowns are in different orders of magnitude. Second, the use of the geometry of the target for adaptation rule has only been explored recently in the AIS literature [13, 14, 16]. Langevin-based schemes have also been considered for accelerating the convergence of MCMC algorithms in the context of large dimension problems [17–19]. The ULA update is then combined with an acceptance-reject step, leading to the so-called Metropolis adjusted Langevin algorithm (MALA). The performance of MALA can again be improved by adapting locally the drift term through the introduction of a scaling matrix in the ULA step [20, 21]. Several strategies have been investigated for the construction of such scaling matrix, relying on second-order information, Fisher metric [20], or majorization-minimization strategy [21]. In this work, we propose a new strategy for adapting efficiently the proposal in AIS algorithms. The proposal is adjusted using a scaled ULA step. The scaling matrix is related to the empirical covariance estimator obtained from the weighted samples of AIS, thus avoiding the computation of a costly Hessian matrix. Another novelty is the joint mean and covariance adaptation, with the advantage of fitting the proposal distributions locally, boosting the exploration and increasing the performance.References1. C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, 2004.2. V. Elvira, L. Martino, D. Luengo, and M. F. Bugallo, “Generalized multiple importance sampling,” Statistical Science, 2018, https://arxiv.org/abs/1511.03095.3. V. Elvira, L. Martino, D. Luengo, and M. F. Bugallo, “Efficient multiple importance sampling estimators,” Signal Processing Letters, vol. 22, no. 10, pp. 1757–1761, 2015.4. V. Elvira, L. Martino, D. Luengo, and M. F. Bugallo, “Heretical multiple importance sampling,” IEEE Signal Processing Letters, vol. 23, no. 10, pp. 1474–1478, 2016.5. M. F. Bugallo, V. Elvira, L. Martino, D. Luengo, J. M´ıguez, and P. M. Djuric, “Adaptive importance sampling: The past, the present, and the future,” IEEESignal Process. Mag., vol. 34, no. 4, pp. 60–79, 2017.6. O. Capp´e, R. Douc, A. Guillin, J. M. Marin, and C. P. Robert, “Adaptive IS in general mixture classes,” Stat. Comput., vol. 18, pp. 447–459, 2008.7. J. M. Cornuet, J. M. Marin, A. Mira, and C. P. Robert, “Adaptive multiple importance sampling,” Scand. Journal of Stat., vol. 39, no. 4, pp. 798–812, Dec. 2012.8. L. Martino, V. Elvira, D. Luengo, and J. Corander, “An adaptive population importance sampler,” IEEE International Conf. on Acoustics, Speech, and SignalProcessing (ICASSP), pp. 8088–8092, 2014.9. L. Martino, V. Elvira, D. Luengo, and J. Corander, “An adaptive population importance sampler: Learning from the uncertanity,” IEEE Transactions on SignalProcessing, vol. 63, no. 16, pp. 4422–4437, 2015.10. L. Martino, V. Elvira, D. Luengo, and J. Corander, “Layered adaptive importance sampling,” (to appear) Statistics and Computing, 2016.11. V. Elvira, L. Martino, D. Luengo, and M. F. Bugallo, “Improving Population Monte Carlo: Alternative weighting and resampling schemes,” Sig. Process., vol.131, no. 12, pp. 77–91, 2017.12. Y. El-Laham, V. Elvira, and M. F Bugallo, “Robust covariance adaptation in adaptive importance sampling,” IEEE Signal Processing Letters, 2018.13. I. Schuster, “Gradient importance sampling,” Tech. Rep., 2015.14. M. Fasiolo, F. E. de Melo, and S. Maskell, “Langevin incremental mixture importance sampling,” Stat. Comput., vol. 28, no. 3, pp. 549–561, 2018.15. G. O. Roberts and L. R. Tweedie, “Exponential convergence of Langevin distributions and their discrete approximations,” Bernoulli, vol. 2, no. 4, pp. 341–363,Dec. 1996.16. V. Elvira, L. Martino, L. Luengo, and J. Corander, “A gradient adaptive population importance sampler,” in Proc. IEEE Int. Conf. Acoust., Speech Signal Process.(ICASSP 2015), Brisbane, Australia, 19-24 April 2015, pp. 4075–4079.17. G. O. Roberts and O. Stramer, “Langevin diffusions and Metropolis-Hastings algorithms,” Methodol. Comput. Appl. Probab., vol. 4, no. 4, pp. 337–357, 2002.18. A. Durmus, E. Moulines, and M. Pereyra, “Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau,” SIAM J.Imaging Sci., vol. 11, no. 1, pp. 473–506, 2018.19. A. Schreck, G. Fort, S. Le Corff, and E. Moulines, “A shrinkage-thresholding Metropolis adjusted Langevin algorithm for Bayesian variable selection,” IEEE J.Sel. Top. Signal Process., vol. 10, no. 2, pp. 366–375, March 2016.20. M. Girolami and B. Calderhead, “Riemann manifold Langevin and Hamiltonian Monte Carlo methods,” J. R. Stat. Soc. Series B Stat. Methodol., vol. 73, no. 91,pp. 123–214, Mar. 2011.21. Y. Marnissi, E. Chouzenoux, A. Benazza-Benyahia, and J.C. Pesquet, “MajorizeMinimize adapted Metropolis-Hastings algorithm,” Tech. Rep., 2018