Uniform in time weak propagation of chaos on the torus

Delarue, François; Tse, Alvin

doi:10.48550/arxiv.2104.14973

Cited by 7 publications

(10 citation statements)

References 54 publications

(139 reference statements)

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“…By ( 39) and ( 40), the first two terms are bounded by −c 1 N i f (r i t ) with c given in (17). By Lemma 19 the last term in (76) is bounded by…”

Section: Proof Of Sectionmentioning

confidence: 92%

“…, where f is defined by (37), M 1 by (18), c by (17) and C is a finite constant depending on γ ∞ , L and the second moment of μ0 and given in (77).…”

Section: Uniform In Time Propagation Of Chaosmentioning

confidence: 99%

“…Later on, assuming that the interaction potential is loosing strict convexity only in a finite number of points (e.g., W (x) = |x| 3 ), Cattiaux, Guillin and Malrieu [15] have shown uniform in time propagation of chaos with a rate getting worse with the degeneracy in convexity. In a very recent work, Delarue and Tse [17] prove uniform in time weak propagation of chaos (i.e., observable by observable) on the torus via Lions derivative methods. Remarkably, their results are not limited to the unique invariant measure case.…”

Section: Introductionmentioning

confidence: 99%

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Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

Durmus¹,

Eberle²,

Guillin³

et al. 2022

Preprint

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We develop a new approach to study the long time behaviour of solutions to nonlinear stochastic differential equations in the sense of McKean, as well as propagation of chaos for the corresponding mean-field particle system approximations. Our approach is based on a sticky coupling between two solutions to the equation. We show that the distance process between the two copies is dominated by a solution to a one-dimensional nonlinear stochastic differential equation with a sticky boundary at zero. This new class of equations is then analyzed carefully. In particular, we show that the dominating equation has a phase transition. In the regime where the Dirac measure at zero is the only invariant probability measure, we prove exponential convergence to equilibrium both for the one-dimensional equation, and for the original nonlinear SDE. Similarly, propagation of chaos is shown by a componentwise sticky coupling and comparison with a system of one dimensional nonlinear SDEs with sticky boundaries at zero. The approach applies to equations without confinement potential and to interaction terms that are not of gradient type.

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“…By ( 39) and ( 40), the first two terms are bounded by −c 1 N i f (r i t ) with c given in (17). By Lemma 19 the last term in (76) is bounded by…”

Section: Proof Of Sectionmentioning

confidence: 92%

“…, where f is defined by (37), M 1 by (18), c by (17) and C is a finite constant depending on γ ∞ , L and the second moment of μ0 and given in (77).…”

Section: Uniform In Time Propagation Of Chaosmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

Durmus¹,

Eberle²,

Guillin³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Here the interaction covariance function of the EnKF particle filter is not a bounded nor a Lipschitz function as is it is commonly assumed in traditional nonlinear and interacting diffusion theory. As a result, none of the stochastic tools developed in this field, including the rather elementary variational methodology for nonlinear diffusions developed in [9,10] nor the more recent powerful differential calculus developed in [31], can be used to analyse this class of continuous-time models equipped with a nonlinear quadratic-type interacting function. Here, the sample covariance matrices satisfy a rather sophisticated quadratic-type stochastic Riccati matrix diffusion equation, which requires one to develop new stochastic analysis tools [18].…”

Section: Some Comments and Comparisonsmentioning

confidence: 99%

A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters

Moral¹,

Horton²

2021

Preprint

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Despite the widespread usage of discrete generation Ensemble Kalman particle filtering methodology to solve nonlinear and high dimensional filtering and inverse problems, little is known about their mathematical foundations. As genetic-type particle filters (a.k.a. sequential Monte Carlo), this ensemble-type methodology can also be interpreted as mean-field particle approximations of the Kalman-Bucy filtering equation. In contrast with conventional mean-field type interacting particle methods equipped with a globally Lipschitz interacting drift-type function, Ensemble Kalman filters depend on a nonlinear and quadratic-type interaction function defined in terms of the sample covariance of the particles.Most of the literature in applied mathematics and computer science on these sophisticated interacting particle methods amounts to designing different classes of useable observer-type particle methods. These methods are based on a variety of inconsistent but judicious ensemble auxiliary transformations or include additional inflation/localisationtype algorithmic innovations, in order to avoid the inherent time-degeneracy of an insufficient particle ensemble size when solving a filtering problem with an unstable signal.To the best of our knowledge, the first and the only rigorous mathematical analysis of these sophisticated discrete generation particle filters is developed in the pioneering articles by Le Gland-Monbet-Tran and by Mandel-Cobb-Beezley, which were published in the early 2010s. Nevertheless, besides the fact that these studies prove the asymptotic consistency of the Ensemble Kalman filter, they provide exceedingly pessimistic meanerror estimates that grow exponentially fast with respect to the time horizon, even for linear Gaussian filtering problems with stable one dimensional signals.In the present article we develop a novel self-contained and complete stochastic perturbation analysis of the fluctuations, the stability, and the long-time performance of these discrete generation ensemble Kalman particle filters, including time-uniform and non-asymptotic mean-error estimates that apply to possibly unstable signals. To the best of our knowledge, these are the first results of this type in the literature on discrete generation particle filters, including the class of genetic-type particle filters and discrete generation ensemble Kalman filters. The stochastic Riccati difference equations considered in this work are also of interest in their own right, as a prototype of a new class of stochastic rational difference equation.

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“…For example, we refer a reader to [JŠS19][Thms. 8 and 9], in which such analysis has been carried out in the context of training recurrent neural networks, and to [DT21], where uniform in time weak particles approximation error has been studied. The approximations errors between given by (1.7) and , with = , are of (1/ ) + ( ) uniformly in time, and this can be seen as proxy for algorithmic complexity.…”

mentioning

confidence: 99%

Convergence of policy gradient for entropy regularized MDPs with neural network approximation in the mean-field regime

Kerimkulov¹,

Leahy²,

Šiška³

et al. 2022

Preprint

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We study the global convergence of policy gradient for infinite-horizon, continuous state and action space, entropy-regularized Markov decision processes (MDPs). We consider a softmax policy with (onehidden layer) neural network approximation in a mean-field regime. Additional entropic regularization in the associated mean-field probability measure is added, and the corresponding gradient flow is studied in the 2-Wasserstein metric. We show that the objective function is increasing along the gradient flow. Further, we prove that if the regularization in terms of the mean-field measure is sufficient, the gradient flow converges exponentially fast to the unique stationary solution, which is the unique maximizer of the regularized MDP objective. Lastly, we study the sensitivity of the value function along the gradient flow with respect to regularization parameters and the initial condition. Our results rely on the careful analysis of non-linear Fokker-Planck-Kolmogorov equation and extend the pioneering work of [MXSS20] and [AKLM20], which quantify the global convergence rate of policy gradient for entropy-regularized MDPs in the tabular setting.

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Uniform in time weak propagation of chaos on the torus

Cited by 7 publications

References 54 publications

Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement

A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters

Convergence of policy gradient for entropy regularized MDPs with neural network approximation in the mean-field regime

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