动力学Langevin动力学离散化：一致最小化条件与收敛分析

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"动力学Langevin动力学离散化的一致最小化条件和收敛界" 本文研究了动力学Langevin动力学的离散化方案在总变差和V-范数下的收敛性。动力学Langevin动态系统在分子动力学和计算统计中广泛应用，用于近似采样目标分布。作者Alain Durmus、Aurélien Enfroy、Éric Moulines和Gabriel Stoltz在2021年9月1日更新的arXiv论文中探讨了这一主题。他们首先证明了一个重要的结果，即对于一个广泛的离散化方法类别，存在一个与步长无关的一致最小化条件。这个类别包括了常见的如Euler-Maruyama方法以及基于分割策略的方法。一致最小化条件是分析数值方法稳定性、效率和收敛性的一个关键工具，它保证了即使在改变步长时，离散化过程也能保持一定的性能。其次，作者提供了确保离散化方案在总变差和V-范数下收敛的温和条件。总变差是衡量概率分布变化的度量，而V-范数则常用于评估数值解的精度。这些条件对于实际应用中的算法设计和选择至关重要，因为它们提供了关于算法行为的理论保证。在动力学Langevin动力学的离散化中，通常面临的问题是如何平衡数值稳定性和计算效率。通过证明这种一致性条件和给出收敛界，研究者为解决这一问题提供了理论基础。这有助于开发更高效且可靠的数值算法，从而在计算成本有限的情况下更好地模拟复杂的物理或统计过程。此外，论文还可能涉及了数值方法的误差分析，以及如何通过调整步长和其他参数来优化算法性能。这些发现对理解动力学Langevin动力学的离散化过程的内在性质，以及如何在实践中有效地实现这些方法具有深远影响。这篇工作为动力学Langevin动力学的离散化提供了重要的理论框架，并为后续的算法开发和分析提供了坚实的基础。这对于计算机科学（特别是计算数学和统计计算领域）的研究人员和实践者来说，是一个有价值的贡献，可以帮助他们在处理复杂系统的模拟问题时做出更好的决策。

to evaluate or cannot even be access e d [2 2, 57]. In these contexts, the Langevin dynamics have been pri-

marily considered for either performing Bayesian inference [20, 1, 42] or optimizing a n objective function

[

47, 50, 7, 19]. In the ﬁrst case, b = ∇log π where π : R

→ R

is the a posteriori distribution of a

statistical model, which can generally be written as −log π =

k=1

, with N the number of obse rvations

and U

: R

→ R. It has been also proposed to use Langevin dynamics to ﬁnd an element of arg min

f for

some function f : R

→ R by setting b = −∇f and taking σ small. In particular, we are interested in the

situation where ∇f can only be estimated through estimators which can potentially be biased.

The use of stochastic approximation for b in these two settings can be formalized as follows. We suppose

that there exist a probability measure µ

on a measurable space (Y, Y), and a measura ble function H :

× Y → R

such that

∀x ∈ R

b(x) =

(y) µ

(dy) , sup

b − bk < +∞ .

In addition, we suppose that we can generate i.i.d. samples (Y

k+1

)

k∈N

from µ

. Then, a stochastic approx-

imation of the discretization of (

1) essentially consists in repla c ing the evaluation of b at each iteration k

by H

k+1

). For example, following [

13], the Euler Maruyama discretization (3) can be generalized as

k+1

= X

+ γV

, V

k+1

= (1 − κγ)V

+ γH

k+1

) +

√

γσZ

This numerical scheme ﬁts into the framework (2) upon taking W

= Y

for any k > 1,

= 1 − κγ , σ

= σ , D

= 0 , f

(x, v, z, w) = 0 , g

(x, v, z, w) = H

(w) .

The parameter δ is irrelevant. No te that A

1 and A2 hold under the condition that for any y ∈ Y, the

function x 7→ H

(y) is Lipschitz with a Lipschitz constant independent of y. The stochastic exponential

Euler and s plitting schemes presented in Sections

2.2.1 and 2.2.2 can also be adapted to take into acc ount

a stochastic approximation of b, for instance by relying on the methodology develop ed in [12].

2.3 Minorization, drift and convergence uniform in the timestep

To state our results, we introduce the Markov kernel R

associated with (2), i.e., for any (x, v) ∈ R

and

B ∈ B(R

((x, v), B) =

d+m



(x, v, (γσ

)

z, w)



ϕ(z) dz µ

(dw) ,

where ϕ(z) is the density of the d-dimensional standard normal distribution, and

(x, v, z, w) =



x + γv + γf



x, γ

v, γ

z, w



+ γ

z, τ

v + γg



x, γ

v, γ

z, w



+ z



. (13)

We present in the following our main results which a llow us to conclude to the uniform V -geometric

ergodicity for R

of the form kδ

(x,v)

− π

6 CV (x, v)ρ

γk

for any k ∈ N, γ ∈ (0, ¯γ] small enough,

and where C > 0 and ρ ∈ [0, 1) are independent of γ. The proof of this result standardly follows from

quantitative minorization and Lyapunov conditions.

Minorization conditio n uniform in the timestep. Our ﬁrst main result shows that R

⌈t

/γ⌉

satisﬁes

a minorization condition with a constant which depends only on the physical time t

> 0 (considered

suﬃciently small) and not on γ.

Theorem 1. Assume that A

1 and A2 hold. Then there exists

> 0 such that, for any t

∈





and

M > 0, there are ε

> 0 and ¯γ

> 0 for which, for any γ ∈ (0, ¯γ

] and (x, v), (x

′

, v

′

) ∈

, M),



⌊t

/γ⌋+1

((x, v), ·) − R

⌊t

/γ⌋+1

((x

′

, v

′

), ·)



6 2 (1 −ε

) . (14)

In words, Theorem

1 ensures that for any compa c t set K ⊂ R

, there exists t

, γ

> 0 for which, for

any γ ∈ (0, ¯γ

], K is 1-small for R

⌈t

/γ⌉

. The proof of this result can be read in Section

4. The main steps

are sketched out in the simple case of nondegenerate Langevin dynamics in Section

Lyapunov condition uniform in the timestep. To ensure the e xistence of a unique stationary distribu-

tion µ

for R

, and obtain a rate of conve rgence to stationarity, we need to consider additional assumptions.

We focus on conditions which allow to ensure the uniform V -geometric ergodicity of R

with a Lyapunov

function built upon a function U : R

→ R satisfying the following condition.

D 1. The function U : R

→ R is C

, U(x) > 0 for any x ∈ R

, and U(0 ) = 0, ∇U(0) = 0. In addition,

there exists L > 0 such that for any x, y ∈ R

, k∇U(x) − ∇U(y)k 6 L kx − yk.

The condition inf

d U > 0 is not restric tive since any function bounded from below can be shifted to

be nonnegative. The conditions U(0) = 0 and ∇U(0) = 0 could be relaxed but allow to simplify some

computations and are therefore considered for ease of presentation. When b comes from the gradient of

a potential (up to some p e rturbation) function, as discussed a round (18), then this potential is a natural

candidate for the function U. In this case, D

1 is necessary to ensure the stability of most of the schemes.

The L yapunov function W

γ,

: R

→ R we consider is parametrized by a consta nt  > 0, and is of the

following exponential form for any γ ∈ ( 0, ¯γ] and x, v ∈ R

γ,

(x, v) = exp





1 + W

(x, v)



, (15)

where W

: R

→ R

, given explicitly by (

73), is the s um of U(x) and a positive deﬁnite quadratic

form in (x, v). As in previous studies on discretization of Langevin dynamics such as [

44, Equation (8.3)],

the Lyapunov function W

γ,

depends on the stepsize γ through W

(in fact, W

converges as γ → 0 to a

Lyapunov function for the continuous dynamics (1), similarly to the family of L yapunov functions considere d

in [

44]; see [44, Equation (3.6)]).

By the estimates in Section 5 (see Le mma 22 and (90)), the design of W

ensures that there exist c

∈

and ¯γ

> 0 such that, for any γ ∈ (0, ¯γ

V 6 W

V , V (x, v) = kxk

+ kvk

+ U (x) .

Therefore, there exist 

, 

> 0 such that for any  > 0, γ ∈ (0, ¯γ

] and x, v ∈ R





(x, v)





γ,

(x, v) 6





(x, v)





, where



(x, v) = exp





1 + V (x, v)



. (16)

In particular, convergence bounds on kδ

(x,v)

− µ

γ,

for  > 0 (where µ

is the unique invariant

probability measure associated with R

) thus imply convergence bounds on kδ

(x,v)

− µ



In view of the minorization condition provided by Theorem

1 a nd using for example [21, Theorem 19.4.1],

it is suﬃcient to establish a Lyapunov condition for R

⌈t

/γ⌉

uniform in the stepsize γ > 0 in order to obta in

exp onential convergence bounds for R

. To this end, we ﬁrst establish a Lyapunov condition for R

, under

additional technical conditions on the family of functions f

, g

for γ ∈ (0, ¯γ], in re lation with the potential U

considered in D1. To state these conditions, we introduce the function F : R

×R

→ R

, such that

for any x, v, z ∈ R

and w ∈ R

F (x, v, z, w) =

k∇U(x)k

+ kvk

+ kzk

+ kwk

+ kxk .

Note that the position x appea rs through the two terms k∇U(x)k

and kxk (mind the fact that the latter

norm is not squared).

D 2. W = R

× R

, W = B(R

) ⊗ B(R

) and there exist α

, ζ

> 0, δ

∈ (0, 1] and C

> 0 for

which, for any γ ∈ (0, ¯γ], there is ϑ

∈ R with

sup

γ∈(0,¯γ]

|ϑ

| 6

ϑ ,

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