scholarly journals Stratification and Optimal Resampling for Sequential Monte Carlo

Biometrika ◽  
2021 ◽  
Author(s):  
Yichao Li ◽  
Wenshuo Wang ◽  
Ke Deng ◽  
Jun S Liu
Keyword(s):  
Monte Carlo ◽  
Sequential Monte Carlo ◽  
Optimal Transport ◽  
Wasserstein Distance ◽  
Hilbert Curve ◽  
Quasi Monte Carlo ◽  
Empirical Distributions ◽  
Monte Carlo Algorithms ◽  
Energy Distance ◽  
Optimal Resampling

Abstract Sequential Monte Carlo algorithms have been widely accepted as a powerful computational tool for making inference with dynamical systems. A key step in sequential Monte Carlo is resampling, which plays a role of steering the algorithm towards the future dynamics. Several strategies have been used in practice, including multinomial resampling, residual resampling, optimal resampling, stratified resampling, and optimal transport resampling. In the one-dimensional cases, we show that optimal transport resampling is equivalent to stratified resampling on the sorted particles, and they both minimize the resampling variance as well as the expected squared energy distance between the original and resampled empirical distributions. In general d-dimensional cases, if the particles are first sorted using the Hilbert curve, we show that the variance of stratified resampling is O(m-(1+2/d)), an improved rate compared to the previously known best rate O(m-(1+1/d)), where m is the number of resampled particles. We show this improved rate is optimal for ordered stratified resampling schemes, as conjectured in Gerber et al. (2019).We also present an almost sure bound on the Wasserstein distance between the original and Hilbert-curve-resampled empirical distributions. In light of these results, we show that, for dimension d > 1, the mean square error of sequential quasi-Monte Carlo with n particles can be O(n-1-4/{d(d+4)}) if Hilbert curve resampling is used and a specific low-discrepancy set is chosen. To our knowledge, this is the first known convergence rate lower than o(n-1).

scholarly journals A non-exponential extension of Sanov’s theorem via convex duality

2020 ◽  
Vol 52 (1) ◽  
pp. 61-101
Author(s):  
Daniel Lacker
Keyword(s):  
Optimal Transport ◽  
Large Deviation ◽  
Optimization Problems ◽  
Risk Measures ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Dual Pair ◽  
Wasserstein Distance ◽  
Empirical Measures ◽  
Empirical Distributions

AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

scholarly journals Sequential Monte Carlo Methods to Train Neural Network Models

2000 ◽  
Vol 12 (4) ◽  
pp. 955-993 ◽  
Author(s):  
J. F. G. de Freitas ◽  
M. Niranjan ◽  
A. H. Gee ◽  
A. Doucet
Keyword(s):  
Monte Carlo ◽  
Sequential Monte Carlo ◽  
Probability Distributions ◽  
Network Models ◽  
Computational Time ◽  
Optimization Strategy ◽  
Neural Network Models ◽  
Monte Carlo Techniques ◽  
Monte Carlo Algorithms ◽  
Importance Resampling

We discuss a novel strategy for training neural networks using sequential Monte Carlo algorithms and propose a new hybrid gradient descent/sampling importance resampling algorithm (HySIR). In terms of computational time and accuracy, the hybrid SIR is a clear improvement over conventional sequential Monte Carlo techniques. The new algorithm may be viewed as a global optimization strategy that allows us to learn the probability distributions of the network weights and outputs in a sequential framework. It is well suited to applications involving on-line, nonlinear, and nongaussian signal processing. We show how the new algorithm outperforms extended Kalman filter training on several problems. In particular, we address the problem of pricing option contracts, traded in financial markets. In this context, we are able to estimate the one-step-ahead probability density functions of the options prices.

The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets

2006 ◽  
Vol 105 (3) ◽  
pp. 413-455 ◽  
Author(s):  
Ligia L. Cristea ◽  
Josef Dick ◽  
Gunther Leobacher ◽  
Friedrich Pillichshammer
Keyword(s):  
Monte Carlo ◽  
Convergence Rate ◽  
Quasi Monte Carlo ◽  
Monte Carlo Algorithms ◽  
Digital Nets

scholarly journals Independent Resampling Sequential Monte Carlo Algorithms

2017 ◽  
Vol 65 (20) ◽  
pp. 5318-5333 ◽  
Author(s):  
Roland Lamberti ◽  
Yohan Petetin ◽  
Francois Desbouvries ◽  
Francois Septier
Keyword(s):  
Monte Carlo ◽  
Sequential Monte Carlo ◽  
Monte Carlo Algorithms

scholarly journals Transfer Learning for Wireless Fingerprinting Localization Based on Optimal Transport

Sensors ◽  
2020 ◽  
Vol 20 (23) ◽  
pp. 6994
Author(s):  
Siqi Bai ◽  
Yongjie Luo ◽  
Qun Wan
Keyword(s):  
Transfer Learning ◽  
Optimal Transport ◽  
Wasserstein Distance ◽  
Target Domain ◽  
The Public ◽  
Empirical Distributions ◽  
Feature Based ◽  
And Performance ◽  
The Cost ◽  
Radio Map

Wireless fingerprinting localization (FL) systems identify locations by building radio fingerprint maps, aiming to provide satisfactory location solutions for the complex environment. However, the radio map is easy to change, and the cost of building a new one is high. One research focus is to transfer knowledge from the old radio maps to a new one. Feature-based transfer learning methods help by mapping the source fingerprint and the target fingerprint to a common hidden domain, then minimize the maximum mean difference (MMD) distance between the empirical distributions in the latent domain. In this paper, the optimal transport (OT)-based transfer learning is adopted to directly map the fingerprint from the source domain to the target domain by minimizing the Wasserstein distance so that the data distribution of the two domains can be better matched and the positioning performance in the target domain is improved. Two channel-models are used to simulate the transfer scenarios, and the public measured data test further verifies that the transfer learning based on OT has better accuracy and performance when the radio map changes in FL, indicating the importance of the method in this field.

scholarly journals Application of ensemble transform data assimilation methods for parameter estimation in nonlinear problems

2020 ◽  
Author(s):  
Sangeetika Ruchi ◽  
Svetlana Dubinkina ◽  
Jana de Wiljes
Keyword(s):  
Monte Carlo ◽  
Sequential Monte Carlo ◽  
Optimal Transport ◽  
Hybrid Approach ◽  
Computational Cost ◽  
Nonlinear Problems ◽  
High Dimensional ◽  
Unknown Parameters ◽  
Sequential Monte Carlo Method ◽  
Ensemble Transform

Abstract. Identification of unknown parameters on the basis of partial and noisy data is a challenging task in particular in high dimensional and nonlinear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust, computationally cheap and often produce astonishingly accurate estimations despite the inherently wrong underlying assumptions. Yet there is a lot of room for improvement specifically regarding the description of the associated statistics. The tempered ensemble transform particle filter is an adaptive sequential Monte Carlo method, where resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity and the method is not as robust as the ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy inspired regularisation factor to the underlying optimal transport problem that allows to considerably reduce the high computational cost via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov Chain Monte Carlo methods results are computed as a benchmark.

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