Mean-Field Langevin Dynamics: Exponential Convergence and Annealing

Chizat, Lénaïc

doi:10.48550/arxiv.2202.01009

Cited by 6 publications

(16 citation statements)

References 12 publications

(16 reference statements)

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“…Proof. We verify the assumptions of [9], noticing that their proof can be adapted without difficulty to our context of families of T probability measures with T ≥ 1. Assumption 1, about the stability and regularity of the first-variation V , is guaranteed Prop.…”

Section: Proof Of Theorem 33mentioning

confidence: 85%

“…• Under these assumptions, the convergence of µ s to µ * in relative entropy and in Wasserstein distance, also holds, with the same rate [27,9].…”

Section: Exponential Convergencementioning

confidence: 86%

“…Let us recall the statement of Theorem 3.3 and prove it, by an application of a convergence result proved independently in [27] and [9].…”

Section: Proof Of Theorem 33mentioning

confidence: 93%

“…It follows, by Holley and Strook perturbation criterion [17] that e −V (i) [µ]/τ satisfies a LSI with constant ρ ≥ D −2 e −κ . Then [9,Thm. 3.2] guarantees the exponential convergence with rate e −Cs with C = 2τ ρ.…”

Section: Proof Of Theorem 33mentioning

confidence: 99%

“…It is tackled in the original paper by discretizing the space and then applying convex optimization methods. The goal of the present paper is to show that the specific structure of this estimator makes it amenable to a newly introduced class of grid-free stochastic methods called Mean-Field Langevin (MFL) dynamics [18,24], a non-linear generalization of Langevin dynamics which enjoys quantitative global convergence guarantees [27,9]. Instantiated in our context, this method consists in a family of point clouds (one per snapshot) coupled via entropy regularized optimal transport, a.k.a.…”

Section: Introductionmentioning

confidence: 99%

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Trajectory Inference via Mean-field Langevin in Path Space

Zhang¹,

Chizat²,

Heitz³

et al. 2022

Preprint

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Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals. To solve this task, a min-entropy estimator relative to the Wiener measure in path space was introduced in Lavenant et al. [20], and shown to consistently recover the dynamics of a large class of drift-diffusion processes from the solution of an infinite dimensional convex optimization problem. In this paper, we introduce a grid-free algorithm to compute this estimator. Our method consists in a family of point clouds (one per snapshot) coupled via Schrödinger bridges which evolve with noisy gradient descent. We study the mean-field limit of the dynamics and prove its global convergence at an exponential rate to the desired estimator. Overall, this leads to an inference method with end-to-end theoretical guarantees that solves an interpretable model for trajectory inference. We also present how to adapt the method to deal with mass variations, a useful extension when dealing with single cell RNA-sequencing data where cells can branch and die.

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Section: Proof Of Theorem 33mentioning

confidence: 85%

“…• Under these assumptions, the convergence of µ s to µ * in relative entropy and in Wasserstein distance, also holds, with the same rate [27,9].…”

Section: Exponential Convergencementioning

confidence: 86%

“…Let us recall the statement of Theorem 3.3 and prove it, by an application of a convergence result proved independently in [27] and [9].…”

Section: Proof Of Theorem 33mentioning

confidence: 93%

Section: Proof Of Theorem 33mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Trajectory Inference via Mean-field Langevin in Path Space

Zhang¹,

Chizat²,

Heitz³

et al. 2022

Preprint

Self Cite

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Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

Lu,

Slepčev,

Wang

2023

Nonlinearity

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Motivated by the challenge of sampling Gibbs measures with nonconvex potentials, we study a continuum birth–death dynamics. We improve results in previous works (Liu et al 2023 Appl. Math. Optim. 87 48; Lu et al 2019 arXiv:1905.09863) and provide weaker hypotheses under which the probability density of the birth–death governed by Kullback–Leibler divergence or by χ 2 divergence converge exponentially fast to the Gibbs equilibrium measure, with a universal rate that is independent of the potential barrier. To build a practical numerical sampler based on the pure birth–death dynamics, we consider an interacting particle system, which is inspired by the gradient flow structure and the classical Fokker–Planck equation and relies on kernel-based approximations of the measure. Using the technique of Γ-convergence of gradient flows, we show that on the torus, smooth and bounded positive solutions of the kernelised dynamics converge on finite time intervals, to the pure birth–death dynamics as the kernel bandwidth shrinks to zero. Moreover we provide quantitative estimates on the bias of minimisers of the energy corresponding to the kernelised dynamics. Finally we prove the long-time asymptotic results on the convergence of the asymptotic states of the kernelised dynamics towards the Gibbs measure.

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High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation

Ba¹,

Erdogdu²,

Suzuki³

et al. 2022

Preprint

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We study the first gradient descent step on the first-layer parameters W in a two-layer neural network:, where W ∈ R d×N , a ∈ R N are randomly initialized, and the training objective is the empirical MSE loss: 1 n n i=1 (f (xi) − yi) 2 . In the proportional asymptotic limit where n, d, N → ∞ at the same rate, and an idealized student-teacher setting, we show that the first gradient update contains a rank-1 "spike", which results in an alignment between the first-layer weights and the linear component of the teacher model f * . To characterize the impact of this alignment, we compute the prediction risk of ridge regression on the conjugate kernel after one gradient step on W with learning rate η, when f * is a single-index model. We consider two scalings of the first step learning rate η. For small η, we establish a Gaussian equivalence property for the trained feature map, and prove that the learned kernel improves upon the initial random features model, but cannot defeat the best linear model on the input. Whereas for sufficiently large η, we prove that for certain f * , the same ridge estimator on trained features can go beyond this "linear regime" and outperform a wide range of random features and rotationally invariant kernels. Our results demonstrate that even one gradient step can lead to a considerable advantage over random features, and highlight the role of learning rate scaling in the initial phase of training.

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Mean-Field Langevin Dynamics: Exponential Convergence and Annealing

Cited by 6 publications

References 12 publications

Trajectory Inference via Mean-field Langevin in Path Space

Trajectory Inference via Mean-field Langevin in Path Space

Birth–death dynamics for sampling: global convergence, approximations and their asymptotics

High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation

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