Propagation of chaos: a review of models, methods and applications. II. Applications

Chaintron, Louis-Pierre; Diez, Antoine

doi:10.48550/arxiv.2106.14812

Cited by 5 publications

(5 citation statements)

References 208 publications

(449 reference statements)

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“…At the level of the nonlinear Markov process for the density on R N θ defined by the mean-field model, this corresponds to approximation by a linear Markov process for the density on R JN θ , where J is the number of particles; the concepts of exchangeability and propagation of chaos may be used to relate the two Markov processes. See [99,124,26] and the references therein.…”

Section: Gradient Flowsmentioning

confidence: 99%

Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance

Chen¹,

Huang²,

Huang³

et al. 2023

Preprint

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Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial distribution can be evolved to the desired minimizer (the target distribution) dynamically via gradient flows. Mean-field models, whose law is governed by the gradient flow in the space of probability measures, may also be identified; particle approximations of these mean-field models form the basis of algorithms. The gradient flow approach is also the basis of algorithms for variational inference, in which the optimization is performed over a parameterized family of probability distributions such as Gaussians, and the underlying gradient flow is restricted to the parameterized family.By choosing different energy functionals and metrics for the gradient flow, different algorithms with different convergence properties arise. In this paper, we concentrate on the Kullback-Leibler divergence as the energy functional after showing that, up to scaling, it has the unique property (among all f -divergences) that the gradient flows resulting from this choice of energy do not depend on the normalization constant of the target distribution. For the metrics, we focus on variants of the Fisher-Rao, Wasserstein, and Stein metrics; we introduce the affine invariance property for gradient flows, and their corresponding mean-field models, determine whether a given metric leads to affine invariance, and modify it to make it affine invariant if it does not.We study the resulting gradient flows in both the space of all probability density functions and in the subset of all Gaussian densities. The flow in the Gaussian space may be understood as a Gaussian approximation of the flow in the density space. We demonstrate that, under mild assumptions, the Gaussian approximation based on the metric and through moment closure coincide; the moment closure approach is more convenient for calculations. We establish connections between these approximate gradient flows, discuss their relation to natural gradient methods in parametric variational inference, and study their long-time convergence properties showing, for some classes of problems and metrics, the advantages of affine invariance. Furthermore, numerical experiments are included which demonstrate that affine invariant gradient flows have desirable convergence properties for a wide range of highly anisotropic target distributions.

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Section: Gradient Flowsmentioning

confidence: 99%

Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance

Chen¹,

Huang²,

Huang³

et al. 2023

Preprint

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show abstract

“…In the last part of the manuscript, we sketch the proof of the so-called propagation of chaos, 39,[46][47][48][49] relying on a martingale-based technique developed in Merle and Salez. 50 We emphasize that the proof presented here is a modification of Theorem 6 in Cao et al 18 We equip the space (R + ) with the Wasserstein distance with exponent 1, which is defined via…”

Section: Propagation Of Chaosmentioning

confidence: 99%

Explicit decay rate for the Gini index in the repeated averaging model

Cao

2022

Math Methods in App Sciences

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We investigate the repeated averaging model for money exchanges: two agents picked uniformly at random share half of their wealth to each other. It is intuitively convincing that a Dirac distribution of wealth (centered at the initial average wealth) will be the long time equilibrium for this dynamics. In other words, the Gini index should converge to zero. To better understand this dynamics, we investigate its limit as the number of agents goes to infinity by proving the so-called propagation of chaos, which links the stochastic agent-based dynamics to a (limiting) nonlinear partial differential equation (PDE). This deterministic description has a flavor of the classical Boltzmann equation arising from statistical mechanics of dilute gases. We prove its convergence towards its Dirac equilibrium distribution by showing that the associated Gini index of the wealth distribution converges to zero with an explicit rate. KEYWORDS agent-based model, econophysics, Gini index, propagation of chaos, repeated averaging MSC CLASSIFICATION 91B80, 91B70

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“…[3]). The McKean-Vlasov process was initially proposed by [4] to give a probabilistic interpretation for nonlinear Vlasov equations, and has found a wide range of applications (see [5,6] and references therein). Although the PoC property is widely applied in computational problems for McKean-Vlasov processes (see [7][8][9], etc.…”

Section: Introductionmentioning

confidence: 99%

Sequential propagation of chaos

Jiang

2023

Preprint

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A new class of particle systems with sequential interaction is proposed to approximate the McKean-Vlasov process that originally arises as the limit of the mean-field interacting particle system. The weighted empirical measure of this particle system is proved to converge to the law of the McKean-Vlasov process as the system grows. Based on the Wasserstein metric, quantitative propagation of chaos results are obtained for two cases: the finite time estimates under the monotonicity condition and the uniform in time estimates under the dissipation and the non-degenerate conditions. Numerical experiments are implemented to demonstrate the theoretical results.

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Propagation of chaos: a review of models, methods and applications. II. Applications

Cited by 5 publications

References 208 publications

Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance

Gradient Flows for Sampling: Mean-Field Models, Gaussian Approximations and Affine Invariance

Explicit decay rate for the Gini index in the repeated averaging model

Sequential propagation of chaos

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