Deformations of Boltzmann Distributions Bálint Máté University of Geneva_2

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Deformations of Boltzmann Distributions

Bálint Máté

University of Geneva

balint.mate@unige.ch

François Fleuret

University of Geneva

francois.fleuret@unige.ch

Abstract

Consider a one-parameter family of Boltzmann distributions

pt(x) = 1

Zte−St(x)

This work studies the problem of sampling from

pt0

by ﬁrst sampling from

pt1

and then applying a transformation

Ψt0

so that the transformed samples follow

pt0

We derive an equation relating

and the corresponding family of unnormalized

log-likelihoods

. The utility of this idea is demonstrated on the

φ4

lattice ﬁeld

theory by extending its deﬁning action

to a family of actions

and ﬁnding a

such that normalizing ﬂows perform better at learning the Boltzmann distribution

pτthan at learning p0.

1 Introduction

Sampling from unnormalized densities has been studied by many due to its relevance for the sciences

[

–

]. The problem can be summarized as follows. Given an unnormalized log-density

S:Rn→R

can we efﬁciently generate samples from the probability density

p(x) = 1

Ze−S(x)

? In particular,

there are no samples given, all we have is the ability to evaluate

for any sample candidate. A

popular technique for attacking this problem is to use a normalizing ﬂow to parametrise a distribution

qθand optimize the parameters θto minimize the reverse KL divergence

KL[qθ, p] = Ex∼qθ(log qθ(x)−log p(x)) = Ex∼qθ(log qθ(x) + S(x)) + Z. (1)

As a motivating example for this paper, let

be the deﬁning action of the lattice

φ4

theory (See §4

for details), and consider the family of distributions

pβ(x)∝e−βS(x)

parametrized by

β∈R+

. In

terms of statistical physics

corresponds to the inverse temperature and controls how ordered the

given system is. As seen in Fig. 1 the performance of a normalizing ﬂow is sensitive to the parameter

. Continuous normalizing ﬂows converge faster at higher temperatures. Somewhat surprisingly,

the RealNVP architecture converges faster both at

β=.1

and

β= 10

than at

β= 1

. A possible

explanation is that both the smoother distribution (

β= 0.1

) and the more localized (

β= 10

) is easier

to learn than a combination of these characteristics at β= 1.

Motivated by this observation, this paper studies the following problem. Suppose we are given a

family of actions

St(x)

, parametrised by

, inducing a family of distributions,

pt(x)∝e−St(x)

We are interested in how these distributions are related for different values of

. From a practical

viewpoint, the goal is to sample from

pt2

by ﬁrst sampling from

pt1

and making the samples ﬂow

along some vector ﬁeld Vt. We will refer to this Vtas the deformation ﬁeld or transport ﬁeld.

The contributions of this paper can be summarized as:

•

In §3 we derive a PDE, the deformation equation, that translates between inﬁnitesimal

deformations of actions and the inﬁnitesimal deformations of the Boltzmann distributions

they induce.

•

In §4 we put the theory into practice in the case of a simple deformation of the lattice

φ4

theory and show that it improves the performance of normalizing ﬂows.

Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

arXiv:2210.13772v3 [hep-lat] 14 Nov 2022

Figure 1: Sensitivity of the training to the inverse temperature

on a

12 ×12

lattice. We trained

the continous normalizing ﬂow of Gerdes et al.

[7]

(left) and a RealNVP (right). In both cases we

used the action of the

φ4

theory with different values of

β∈ {0.1,1,10}

and with

and

values

same as in the work of Gerdes et al.

[7]

. The

-axis represents the number of training steps, while the

y-axis represents the performance metrics. Mean and standard deviation over 5 runs are shown.

Figure 2: Deformation of a two-dimensional standard Gaussian into a family of isotropic Gaussians

with different variances. Samples from a two-dimensional standard Gaussian (left). The deformation

ﬁeld (center). The samples in the left plot ﬂowing along the deformation ﬁeld from

follow a

Gaussian centered at the origin with covariance 1/t (right).

2 Background

Change of variables

Let

p0(z)

be a probability density on

and

Ψ : Rn→Rn

a diffeomorphism.

Pushing the the density forward along Ψinduces a new probability density pimplicitly deﬁned by

log p0(z) = log p(Ψz) + log |det JΨ(z)|.(2)

The term log |det JΨ(z)|measures how much the function Ψexpands volume locally at z.

Continous change of variables

Suppose

is a time-dependent vector ﬁeld. Let

Ψτ

denote the

diffeomorphism of following the trajectories of

from

. This family of diffeomorphisms

generates a one-parameter family of densities

pτ

. The amount of volume expansion a particle

experiences along this trajectory of

Ψτ

Rτ

0div Vt(Ψtz)dt

. The log-likelihoods are then related by

log p0(z) = log pτ(Ψτz) + Zτ

div Vt(Ψtz)dt. (3)

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摘要：

DeformationsofBoltzmannDistributionsBálintMátéUniversityofGenevabalint.mate@unige.chFrançoisFleuretUniversityofGenevafrancois.fleuret@unige.chAbstractConsideraone-parameterfamilyofBoltzmanndistributionspt(x)=1ZteSt(x).Thisworkstudiestheproblemofsamplingfrompt0byrstsamplingfrompt1andthenapplyingatra...

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