# Stochastic Langevin Monte Carlo for (weakly) log-concave posterior
distributions.

Marelys Crespo Navas1, Sébastien Gadat2,3, Xavier Gendre1
1 ISAE-SUPAERO, Université de Toulouse
2Toulouse School of Economics (CNRS UMR 5314), Université Toulouse I Capitole
3 Institut Universitaire de France

###### Abstract

In this paper, we investigate a continuous time version of the Stochastic
Langevin Monte Carlo method, introduced in [39], that incorporates a
stochastic sampling step inside the traditional over-damped Langevin
diffusion. This method is popular in machine learning for sampling posterior
distribution. We will pay specific attention in our work to the computational
cost in terms of $n$ (the number of observations that produces the posterior
distribution), and $d$ (the dimension of the ambient space where the parameter
of interest is living). We derive our analysis in the weakly convex framework,
which is parameterized with the help of the Kurdyka-Łojasiewicz (KL)
inequality, that permits to handle a vanishing curvature settings, which is
far less restrictive when compared to the simple strongly convex case. We
establish that the final horizon of simulation to obtain an $\varepsilon$
approximation (in terms of entropy) is of the order
$(d\log(n)^{2})^{(1+r)^{2}}[\log^{2}(\varepsilon^{-1})+n^{2}d^{2(1+r)}\log^{4(1+r)}(n)]$
with a Poissonian subsampling of parameter
$\left(n(d\log^{2}(n))^{1+r}\right)^{-1}$, where the parameter $r$ is involved
in the KL inequality and varies between $0$ (strongly convex case) and $1$
(limiting Laplace situation).

_Keywords:_ Langevin Monte Carlo sampling; Log concave models; Weak convexity.

_AMS classifications:_ Primary 6265C05; secondary ; 62C10; 65C30; 60H3520.

111 We are grateful to Patrick Cattiaux and Arnaud Guillin for helpful
discussions and references on functional inequalities and especially on weak
log Sobolev inequalities.

##  1 Markovian Stochastic Langevin Dynamics and main results

###  1.1 Introduction

##### Motivations

In the recent past years, a huge amount of methods have been developed in
machine learning to handle large scale massive datasets with a large number
$n$ of observations ($X_{1},\ldots,X_{n})$ embedded in a high dimensional
space $\mathbb{R}^{d}$. These methods generally involve either optimization of
a data-dependent function (for frequentist learning) or sampling a data-
dependent measure (for Bayesian learning with posterior distributions). In
both approaches, a bottleneck lies on the size of $n$ and $d$ that usually
generates numerical difficulties for the use of standard algorithms. We are
interested in this paper in the simulation of a posterior distribution
following a Bayesian point of view with a statistical model described by a
collection of densities $(p_{\theta})_{\theta\in\Theta}$ on $\mathcal{X}$,
where the parameter of interest $\theta^{\star}$ belongs to
$\Theta=\mathbb{R}^{d}$ and where the $(X_{i})_{1\leq i\leq n}$ are assumed to
be i.i.d. observations in $\mathcal{X}$ distributed according to
$p_{\theta^{\star}}$. A standard Bayesian approach consists in defining a
prior distribution $\pi_{0}$ on $\Theta$ and then sample the posterior
distribution denoted by $\mu_{n}$ (which will be denoted by
$\exp(-U_{\nu_{n}})$ below) using a numerical probabilistic approximation with
the help of an over-damped Langevin diffusion:

$\mathrm{d}\theta_{t}=-\nabla
U_{\nu_{n}}(t)\mathrm{d}t+\sqrt{2}\mathrm{d}B_{t}.$

In this work, we manage to deal with an adaptation of the Langevin Monte Carlo
(LMC) algorithm proposed in [39], that exploits some old ideas of stochastic
algorithms introduced in [36]: instead of using the previous equation, the
authors propose a modification of the diffusion that generates a noisy drift
in the LMC due to a sampling strategy among the set of observations
$(X_{i})_{1\leq i\leq n}$. Before we provide some details on the precise
objects and algorithm necessary to properly define this method, we first give
some literature insights related to it.

##### State of the art

Ergodicity and quantitative mixing properties of over-damped LMC and many
other sampling algorithms is a popular subject of research initiated in the
probabilistic works around, roughly speaking, two strategies. The first one
relies on pathwise considerations and dynamical properties of random dynamical
system and is built with some coupling argument and Lyapunov controls. We
refer to the seminal contributions [32, 27], that exploits the approach of the
Doeblin coupling and total variation (TV) bounds. Many extensions may be
derived from this Lyapunov approach and may lead to Wasserstein or
$\mathbb{L}^{2}$ upper bounds, we refer to [8] and the references therein of
the same authors for a description of the link between Lyapunov conditions and
ergodicity. The second strategy derives from spectral properties of Markov
operators and is related to famous functional inequalities (Poincaré and Log-
Sobolev among others). The general idea is to differentiate the distance along
the time-evolution and apply a Gronwall Lemma to obtain a quantitative
estimate of the long-time evolution of the semi-group. We refer to the seminal
contributions of [26, 2], and to [3] for an almost exhaustive survey of all
possible inequalities and consequences on the ergodicity of the Markov semi-
groups. Finally, let us emphasize that some strong links exist between the
spectral and the Lyapunov approaches, as pointed out by [9]. If functional
inequalities are then strongly related to mixing properties and especially
from a quantitative point of view, it is therefore necessary to develop a
machinery that is able to assess these inequalities carefully, especially with
a specific attention to our statistical setting of large $n$ and $d$ in the
completely non-trivial situation where the target measure is log-concave but
not strongly log-concave, which is a common feature of Bayesian posterior
distributions.

On the statistical side, the mixing properties of LMC has been largely
investigated during the past decade, strongly motivated by machine learning
methods such as Exponentially Weighted Aggregation introduced by [11], which
involves sampling a non log-concave and heavy tailed posterior distribution. A
first paper of Dalalyan [12] establishes the cost of LMC to obtain an
$\varepsilon$ TV bound in terms of $d$ and $\rho$ when the target measure is
$\rho$ strongly log-concave and proposes a penalized version of LMC to
circumvent the lack of strong log-concavity when the target distribution is
only log-concave. Since this pioneering paper, a huge impressive literature
expanded. Among others, we refer to [16] that gives a careful study of
discretized LMC, [14] for a kinetic version of LMC and [15] where the
penalized LMC in non strongly-concave situation is studied in depth. Among all
these papers, first, the lack of strong log-concavity is dealt with a
modification of the initial LMC using a surrogate and asymptotically vanishing
penalty. Second, these papers assume that a noiseless gradient of the log-
posterior is available at each iteration of the algorithm, which may not be
realistic, especially with large $n$.

Stochastic LMC (SLMC below) has attracted the interest of several works: [39]
introduced this method and described its efficiency from a numerical point of
view in the particular case of Bayesian learning, which is exactly our
framework. Some recent advances and related contributions may be also cited:
[13] studies a noisy version of LMC and derives some non-asymptotic upper
bounds (in terms of Wasserstein distance) of the sampling strategy in presence
of a possibly biased noise for strongly log-concave posterior distribution.
The recent contribution of [40] is also related to our work: the authors
develop a machinery for the study of SLMC essentially based on the Poincaré
inequality but the way the lower bound on the spectral gap involved in the LMC
is dealt with appears to be inappropriate. In particular, the diffusion
involved in (Stochastic)-LMC is used at a very low-temperature, proportional
to $1/n$, which generates some important troubles in the size of the spectral
gap in non strongly log-concave framework. In [35], the authors derives some
close bounds to our framework for optimization purpose, and the authors
identify the important dependency of the spectral gap denoted by $\lambda_{*}$
in their paper with the temperature level $1/\beta$ they introduced. They
obtain some very highly pessimistic bounds in some general situations (see
their discussion in [35][Section 4]), they conclude their discussion by the
urgent need to find some non-trivial situations where some better lower bounds
of $\lambda_{*}$ may be derived.

Indeed, the final remark of [35][Section 4]) is related to the well known
metastability phenomenon: at a low temperature, the mixing rates of a lot of
reversible and irreversible Markov semi-groups are strongly deteriorated by
the low temperature settings, which is implicitly induced by a Bayesian
posterior sampling problem with a large number $n$ of observations. In a
regime of variance noise of the order $O(\beta^{-1})$, the first study of
large deviation principle of invariant measures traces back to [18] where the
authors establish the asymptotic of the spectral gap of the over-damped
Langevin diffusion as $\exp(-I\beta)$ ( [18][Chapter 6]) where $I$ is an
explicit constant that depends on the potential of the Gibbs field. This
result has been extended in depth by [26], which leads to the first precise
analyses of the so-called simulated annealing method (see e.g. [24, 33]).
These works, and more recent contributions with irreversible dynamical systems
in a stochastic settings ([22, 19]) show that there is almost nothing to
expect in metastable situations in terms of asymptotic behaviour of the
spectral gap, and indirectly in terms of mixing rate. Hence, the only
situation that may lead to reasonable results is an intermediary situation
between the (almost) trivial strongly log-concave case and the metastable
multi-welled case. This is the purpose of the weakly log-concave situation
that is described by the family of Kurdyka-Łojasiewicz inequalities [28, 30]
used in optimization theory [5] that have shown to be efficient for stochastic
optimization [20] or for sampling [21]. We also refer to the recent
contributions [6] that derives some functional inequalities within an
intermediary framework in which the curvature $\rho$ is related to their
keystone function $\alpha$ that controls the constants involved in the
functional inequalities they are studying.

Taking together the statistical considerations and limitations, we are
motivated in this paper in the study of the continuous time Stochastic
Langevin Monte Carlo procedure. This process will be described precisely in
the next paragraph as well as the Kurdyka-Łojasiewicz setup parametrized by a
real value $r$, which varies between $0$ (strongly convex case) and $1$
(limiting Laplace asymptotic tail). We will show that the final horizon of
simulation to obtain an $\varepsilon$ approximation is of the order:

$(d\log(n)^{2})^{(1+r)^{2}}[\log^{2}(\varepsilon^{-1})+n^{2}d^{2(1+r)}\log^{4(1+r)}(n)]$

with a Poissonian subsampling of parameter $\frac{1}{n(d\log^{2}(n))^{1+r}}$.

The rest of the introduction consists in the definitions of the algorithm in
Subsection 1.2, the way we assess the quality of our result with an entropy
criterion in Subsection 1.3, as well as the quantitative weakly log-concave
assumption in Subsection 1.4. We finally state our main result in Subsection
1.5.

###  1.2 Continuous time evolution

Below, we briefly remind the continuous time SLMC algorithm for Bayesian
learning, for which a discretized form has been introduced in [39]. For this
purpose, we consider a statistical model that is built with the help of a
function $(x,\theta)\longmapsto p_{\theta}(x)$ where $\theta\in\mathbb{R}^{d}$
encodes the parameter of the statistical model and $x$ the observation in a
space denoted by $\mathcal{X}$. We then assume that we have $n$ i.i.d.
observations denoted by $(\mathbf{X}_{1},\ldots,\mathbf{X}_{n})$ distributed
according to $p_{\theta}$. Given a prior distribution $\pi_{0}$ on
$\mathbb{R}^{d}$, the posterior distribution $\mu_{n}$ is then defined as:

$\mu_{n}(\theta)\propto\pi_{0}(\theta)\times\prod_{i=1}^{n}p_{\theta}(\mathbf{X}_{i}).$

We introduce the log-parametrization that leads to the Gibbs form:

$U_{x}(\theta)=-[\log\pi_{0}(\theta)+n\log p_{\theta}(x)],$

and we then observe that:

$\mu_{n}(\theta)\propto\exp\left(-\frac{1}{n}\sum_{i=1}^{n}U_{\mathbf{X}_{i}}(\theta)\right)=\exp\left(-U_{\nu_{n}}(\theta)\right),$

where $\nu_{n}$ refers to the empirical distribution and $U_{\nu_{n}}$ the
average value of $U_{X}(\theta)$ when $X\sim\nu_{n}$:

$\nu_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}\delta_{\mathbf{X}_{i}}(x)\quad\text{and}\quad
U_{\nu_{n}}(\theta)=\mathbb{E}_{X\sim\nu_{n}}[U_{X}(\theta)].$

The standard Langevin Monte Carlo approach relies on the ergodic behaviour of
the stochastic differential equation:

$\mathrm{d}\theta_{t}=-\nabla
U_{\nu_{n}}(\theta_{t})\mathrm{d}t+\sqrt{2}\mathrm{d}B_{t},$ (1)

that possesses under some mild assumptions a unique invariant distribution
$\mu_{n}$.

The SLMC algorithm takes benefit of both sampling with a S.D.E. and
homogenization of the drift that may be written as an expectation on $X$ that
is sampled uniformly over the set of observations according to $\nu_{n}$. The
leading idea is to replace the expectation in $U_{\nu_{n}}$ that depends on
the overall set of observations $(\mathbf{X}_{1},\ldots,\mathbf{X}_{n})$ by a
single unique observation that is randomized uniformly all along the evolution
of the stochastic differential equation, and modified according to a Markov
exponential clock. That being said, we can write an explicit formal definition
of the algorithm as follows. We define $\left(\xi^{(n)}_{j}\right)_{j\geq 1}$
an infinite sequence of exponential random variables of mean $\alpha_{n}^{-1}$
that will be fixed later on.

We also consider a sequence $\left\\{V^{(n)}_{j}\right\\}_{j\geq 0}$ of i.i.d.
random variables uniformly distributed in $\\{1,2,\ldots,n\\}$. We then define
the process $(X_{t})_{t\geq 0}$ as a jump process that takes its values in
$\\{1,2,\ldots,n\\}$ such that:

$X_{t}=\left\\{\begin{array}[]{l c l}\mathbf{X}_{V^{(n)}_{1}},&\mbox{ if
}&0\leq t<\xi^{(n)}_{1},\\\ \\\ \mathbf{X}_{V^{(n)}_{j}},&\mbox{ if
}&\sum\limits_{k=1}^{j-1}\xi^{(n)}_{k}\leq
t<\sum\limits_{k=1}^{j}\xi^{(n)}_{k},\quad j>1.\end{array}\right.$ (2)

Informally, $(X_{t})_{t\geq 0}$ should be understood as follows: the process
takes the value of one observation uniformly chosen from the $n$ observations
$\mathbf{X}_{1},\ldots,\mathbf{X}_{n}$ during exponential times with intensity
$\alpha_{n}$. The stochastic Langevin over-damped diffusion we consider is
then given by the joint evolution $(\theta_{t},X_{t})_{t\geq 0}$ and that is
defined by:

$\mathrm{d}\theta_{t}=-\nabla_{\theta}U_{X_{t}}(\theta_{t})\mathrm{d}t+\sqrt{2}\mathrm{d}B_{t},\quad
t>0,$ (3)

where $(B_{t})_{t\geq 0}$ is a multivariate standard Brownian Motion.

Data: $(\mathbf{X}_{1},\ldots,\mathbf{X}_{n})$ i.i.d. observations, $n_{0}$
initial distribution, $\pi_{0}$ prior distribution

1 $t_{0}=0$

2 Generate $\theta_{0}$ according to $n_{0}$

3 for  _$k=0,1,\ldots$_ do

4 Pick $X_{k}$ uniformly in
$\left\\{\mathbf{X}_{1},\ldots,\mathbf{X}_{n}\right\\}$

5 Generate $\xi_{k}$ according to an Exponential distribution with mean
$\alpha_{n}^{-1}$

6 $t_{k+1}=t_{k}+\xi_{k}$

7
$\theta_{t_{k+1}}=\theta_{t_{k}}-\int_{t_{k}}^{t_{k+1}}\nabla_{\theta}U_{X_{k}}(\theta_{s})\mathrm{d}s+\sqrt{2}B_{\xi_{k}}$

8 end for

return $\lim\limits_{k\to\infty}\theta_{t_{k}}$

Algorithm 1 Stochastic Langevin over-damped

###  1.3 Entropic divergence

To assess the long-time behaviour of the SLMC, we introduce several notations
related to the pair $(\theta_{t},X_{t})_{t\geq 0}$. Below, we denote by
$\lambda_{d}$ the Lebesgue measure over $\mathbb{R}^{d}$. The semi-group
induced by $\mathcal{L}$ being elliptic on the $\theta$ coordinate, trivially
irreducible and finitely supported on the $x$ coordinate, makes the law of
$(\theta_{t},X_{t})$ absolutely continuous with respect to the measure
$\lambda_{d}\otimes\nu_{n}$ as soon as $t>0$.

We introduce the notation of $m_{t}$ to refer to the joint density of
$(\theta_{t},X_{t})$ at time $t$ with respect to $\lambda_{d}\otimes\nu_{n}$.
In the meantime, $n_{t}$ denotes the marginal distribution of $\theta_{t}$ and
$m_{t}(\cdot|\theta)$ the conditional distribution of $X_{t}$ given
$\theta_{t}=\theta$. That is:

$\mbox{Law}(\theta_{t},X_{t})=m_{t},\quad
n_{t}(\theta)=\sum\limits_{i=1}^{n}m_{t}(\theta,\mathbf{X}_{i}),\quad
m_{t}(x|\theta)=\frac{m_{t}(\theta,x)}{n_{t}(\theta)},$ (4)

for $\theta\in\mathbb{R}^{d}$ and
$x\in\\{\mathbf{X}_{1},\ldots,\mathbf{X}_{n}\\}$.

To show that the SLMC algorithm recovers the correct asymptotic behaviour,
i.e. that $n_{t}(\theta)\longrightarrow\mu_{n}$ when $t\longrightarrow\infty$,
we consider the relative entropy (or Kullback-Leibler divergence) of $n_{t}$
with respect to $\mu_{n}$ that is well defined thanks to the ellipticity, and
given by:

$J_{t}=Ent_{\mu_{n}}\left(\frac{n_{t}}{\mu_{n}}\right)=\int\limits_{\mathbb{R}^{d}}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}n_{t}(\theta).$
(5)

$J_{t}$ measures at any time $t>0$ a divergence between the instantaneous law
of the process at time $t$ and the (presumably) invariant distribution
$\mu_{n}$ of the process $(\theta_{t},X_{t})$. It would also be possible to
measure this difference between the two distributions in terms of the
$\mathbb{L}^{2}$ or the $\chi$-square distance and to produce a theoretical
analysis with the help of functional analysis but it would rely on stronger
assumptions on the function $U_{\nu_{n}}$.

In the meantime, we also introduce a weighted $\mathbb{L}^{2}$ distance
between the conditional distribution of $X_{t}$ given $\theta_{t}=\theta$ and
the measure $\nu_{n}$. This distance is denoted by $I_{t}$ and is defined as:

$I_{t}=\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left(\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right)^{2}\nu_{n}(\mathbf{X}_{i})\mathrm{d}n_{t}(\theta).$
(6)

This quantity measures the average closeness (w.r.t. $\theta$) of the
conditional law of $x$ given $\theta$ at time $t$ to $\nu_{n}$.

###  1.4 Main assumptions

##### Weak convexity

We will study the SLMC into a weakly convex framework, i.e. when $U_{\nu_{n}}$
is assumed to be convex but not necessarily strongly convex. SLMC has recently
received an important interest in the machine learning community and has been
studied essentially in its explicit Euler discretized form in various
situations where functional inequalities are involved. We refer to [38]
(uniform Log-Sobolev inequality), to [35] (uniform Poincaré inequality) where
the authors develop a Wasserstein-2 analysis of the algorithm, and to [40]
(uniform Poincaré inequality). In these works, the functional inequalities
play a crucial role to analyze the behaviour of SLMC and these inequalities
are assumed, which is an important hypothesis. Importantly, Poincaré or Log-
Sobolev inequalities are not so innocent since they generally require
convexity (see e.g. [4, 3]) to be reasonably dimension-dependent, and even
strong convexity to be dimension free. Otherwise, the constant involved in
these functional inequalities are exponentially degraded by the “temperature”
($n^{-1}(d\log^{2\beta}(n))^{-(1+r)}$ in our case) and the dimension ($d$ for
us) as indicated in [26].

In our work, we have chosen to parameterize this lack of strong convexity with
the help of the Kurdyka-Łojasiewicz inequality [28, 30], which is a standard
tool in optimization to describe the transition between convexity and strong
convexity and makes the bounds more explicit. This assumption allows to
observe how the entropy evolves according to the key exponent involved in the
KL inequality. In particular, it makes possible to understand the influence of
the lack of strong convexity that is more or less hidden in the uniform
Poincaré or Log-Sobolev inequalities that are assumed in the previous works.
We introduce a parametric form of the KL inequalities following [20].

For this purpose, for any $V$ twice differentiable function, we denote the
spectrum of the Hessian matrix of $V$ as $Sp(\nabla^{2}V(\theta))$.
Furthermore, if $V$ is convex, we denote:

$\underline{\lambda}_{\nabla^{2}V}(\theta)=\inf Sp(\nabla^{2}V(\theta)).$

Hypothesis $\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ We say that a
function $V:\mathbb{R}^{d}\to\mathbb{R}$ satisfies a
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$-condition if:

  * a)

$V$ is a $\mathcal{C}^{2}$-function.

  * b)

$V$ is a convex function and
$\mathrm{min}_{\theta\in\mathbb{R}^{d}}V(\theta)=V(\theta^{*})>0$.

  * c)

$\nabla V$ is $L$-Lipschitz.

  * d)

There exist some constants $0\leq r<1$ and $\mathfrak{c}>0$ such that:

$\mathfrak{c}V^{-r}(\theta)\leq\underline{\lambda}_{\nabla^{2}V(\theta)}\quad\forall\theta\in\mathbb{R}^{d}.$
(7)

Let us briefly comment this assumption.

  * •

In [21], a slightly different parametrization is used with the introduction of
another exponent $q$ related to $\overline{\lambda}_{\nabla^{2}V(\theta)}=\sup
Sp(\nabla^{2}V(\theta))$. The authors also assume the upper bound
$\overline{\lambda}_{\nabla^{2}V(\theta)}\leq\tilde{\mathfrak{c}}V^{-q}(\theta)$.
Here, we have chosen to simplify this assumption and use a rough upper bound
on the eigenvalues of the Hessian matrix given by the Lipschitz constant $L$,
i.e. in the last inequality we simply use $\tilde{\mathfrak{c}}=L$ and $q=0$.

  * •

We shall observe that if $V(\theta)=(1+\|\theta\|_{2}^{2})^{p}$ with
$p\in[1/2,1]$, then $V$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ with $r=\frac{1-p}{p}$ and
$\mathfrak{c}=2p(1-2(1-p))$, see Remark 7 of [21] for further details. In
particular, the larger $p$, the smaller $r$, which translates into a better
curvature of the potential function $V$.

  * •

When $r=q$, we recover a global standard KL inequality (see [20, 5]) and when
$r=1$ it corresponds to the limiting Laplace case.

  * •

The case $r=0$ is of course associated to the strongly convex situation where
the curvature of the function is uniformly lower bounded by $\mathfrak{c}$.

Hence, it is expected that the complexity of SLMC increases with the lack of
curvature, i.e. is an increasing function of $r$.

In section 4 we recall some important consequences of the KL inequality
obtained in Lemma 15 of [21]. In particular, the growth of any function that
satisfies $\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ is lower and upper
bounded by a positive power of the distance to its minimizer.

If inequality (7) holds for a constant $\mathfrak{c}$, then it holds for all
positive values less than $\mathfrak{c}$. For that reason, in section 5 we
assume $\mathfrak{c}\leq\left(\frac{8L}{(1+r)}\right)^{1+r}$.

##### Assumption on the prior $\pi_{0}$

We state below the important consequence of a “population”
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ assumption, but before, let us
state some mild assumptions on $\pi_{0}$.

Hypothesis $\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ $\pi_{0}$ is a log-
concave $\mathcal{C}^{2}$-function such that
$\mathrm{min}_{\theta\in\mathbb{R}^{d}}-\log\pi_{0}(\theta)>0$ and
$\theta\mapsto\nabla\log\pi_{0}(\theta)$ is $\ell_{0}$-Lipschitz.

Since the prior distribution is chosen by the user, our
$\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ hypothesis is not restrictive and
some typical examples satisfy these conditions, such as Gaussian, Weibull and
Gamma, both with shape parameter larger than 1, Gumbel, among others.

######  Proposition 1.1.

We assume $\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ and that there exist
$(\mathfrak{c},r)$ such that for any $x$: $\theta\longmapsto-\log
p_{\theta}(x)$ satisfies $\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then
$U_{\nu_{n}}$ satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$,
and in particular, for any $\mathbf{X}_{i}$, $U_{\mathbf{X}_{i}}$ satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$.

We introduce the notation $a\lesssim_{uc}b$ ($a\gtrsim_{uc}b$) which means
$a\leq cb$ ($a\geq cb$) where $c$ is a universal constant i.e. a positive
constant independent of $n$ and $d$.

We assume that the minimizers of the functions $U_{\mathbf{X}_{i}}$ are
contained in a ball of radius which depends of $n$ and $d$. Additionally, we
consider $\mathrm{min}_{\theta\in\mathbb{R}^{d}}U_{\mathbf{X}_{i}}$ to be at
most of order $d$.

Hypothesis $\mathbf{\mathcal{H}_{\mathrm{min}}}$ There exists $\beta\geq 0$
such that:

$\mathrm{max}_{i}\|\arg\mathrm{min}\,U_{\mathbf{X}_{i}}\|_{2}\lesssim_{uc}\sqrt{d}\log^{\beta}(n)\quad\mbox{
and }\quad\mathrm{max}_{i}\ \mathrm{min}_{\theta\in\mathbb{R}^{d}}\
U_{\mathbf{X}_{i}}(\theta)\lesssim_{uc}d.$

Assumption $\mathbf{\mathcal{H}_{\mathrm{min}}}$ is not restrictive. In
dimension $d=1$, it holds for many concentrated i.i.d. samples $(X_{i})_{1\leq
i\leq n}$ with a suitable sub-Gaussian like behaviour for which the Laplace
transform of $\mathrm{min}\ U_{X_{i}}$ is upper bounded as:

$\mathbb{E}[\exp(\lambda\mathrm{min}\
U_{X_{i}})]\leq\exp(\sigma^{2}\lambda^{k}),\quad\forall\lambda>0.$

The previous upper bound implies that, in this case, $\beta$ involved in
$\mathbf{\mathcal{H}_{\mathrm{min}}}$ is given by $\beta=\frac{k-1}{k}$. We
recover in particular the situation where $\beta=1/2$ when $k=2$. For larger
dimensions, the result may be extended using that $\|X\|_{2}^{2}\leq
d\,\mathrm{max}_{1\leq j\leq d}(X^{j})^{2}$, where $X^{j}$ is the $j$-th
component of $X$. We should keep in mind from this last discussion that even
if $\mathbf{\mathcal{H}_{\mathrm{min}}}$ is stated (and makes sense) for any
value of $\beta>0$, it holds in general for $\beta\leq 1$.

This $\mathbf{\mathcal{H}_{\mathrm{min}}}$ hypothesis together with
$\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ lead to an almost similar behaviour
of the minimizer and the minimum of $U_{\nu_{n}}$. Details appear in
Proposition 4.4.

###  1.5 Long-time entropy convergence

We introduce for any time $t\geq 0$ the density of $\mbox{Law}(\theta_{t})$
w.r.t. $\mu_{n}$, which is given by:

$f_{t}(\theta)=\frac{n_{t}(\theta)}{\mu_{n}(\theta)},$

and $n_{0}$ is chosen such that $\|f_{0}\|_{\infty}<+\infty$. The following
hypothesis guarantees this result which will be proved in Proposition 3.5.

Hypothesis $\mathbf{\mathcal{H}_{n_{0}}(L,\ell_{0})}$ A positive constant
$\sigma^{2}$ exists such that $n_{0}=\mathcal{N}(0,\sigma^{2}I_{d})$.
Moreover, there exist two universal constants $c_{1}$ and $c_{2}$ such that
$0<c_{1}\leq c_{2}<1$ and

$\frac{c_{1}}{nL+\ell_{0}}\leq\sigma^{2}\leq\frac{c_{2}}{nL+\ell_{0}}.$

Futhermore, in Proposition 3.5, as an immediate consequence of the boundedness
of $\|f_{0}\|_{\infty}$, we obtain that
$J_{0}\lesssim_{uc}nd^{1+r}\log^{2\beta(1+r)}(n)+d\log\left(\frac{d}{n}\right)$.

The next result assesses a mixing property in terms of decrease of the entropy
and therefore states the convergence of $n_{t}$ towards the correct measure
$\mu_{n}$.

######  Theorem 1.1.

Assume $\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$,
$\mathbf{\mathcal{H}_{\mathrm{min}}}$,
$\mathbf{\mathcal{H}_{n_{0}}(L,\ell_{0})}$ and that each $\theta\mapsto-\log
p_{\theta}(\mathbf{X}_{i})$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then

  * •

$U_{\nu_{n}}$ satisfies a Poincaré inequality of constant $C_{P}(\mu_{n})$,
indistinctly denoted as $C_{P}$.

  * •

Define $c_{n,d}:=n^{4}\left(d\log^{2\beta}(n)\right)^{1+r}$ and
$O_{n,d}:=\left(\frac{C_{1}d}{n}\right)^{\frac{dr}{2}}\exp\left(C_{2}n\left(d\log^{2\beta}(n)\right)^{1+r}\right)$,
where $C_{1}$ and $C_{2}$ are universal constants, then for any $t>0$:

$J_{t}\lesssim_{uc}\left(J_{0}+\frac{c_{n,d}}{\alpha_{n}}\left[1+\left(\frac{C_{P}}{\alpha_{n}}+\sqrt{C_{P}}\right)e^{\frac{\sqrt{C_{P}}}{\sqrt{\mathfrak{a}}}+\frac{C_{P}}{3\alpha_{n}}}\right]+O_{n,d}\right)(1+t)^{1/4}e^{-\frac{\sqrt{C_{p}}}{\sqrt{a}}(\sqrt{1+t}-1)}.$
(8)

  * •

For any $\varepsilon>0$, if
$\alpha_{n}=\frac{1}{n\left(d\log^{2\beta}(n)\right)^{1+r}}$, then:

$t\gtrsim_{uc}\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}}\left[\log^{2}(\varepsilon^{-1})+n^{2}\left(d\log^{2\beta}(n)\right)^{2(1+r)}+d^{2}\log^{2}d\right]\Longrightarrow
J_{t}\leq\varepsilon.$

If we denote $t_{\varepsilon}$ the smallest value such that
$J_{t_{\varepsilon}}\leq\varepsilon$, then the choice of
$\alpha_{n}=\frac{1}{n\left(d\log^{2\beta}(n)\right)^{1+r}}$ guarantees that
the mean number of jumps $\alpha_{n}t_{\varepsilon}$ of the process
$(X_{t})_{0\leq t\leq t_{\varepsilon}}$ is the minimum possible.

In order to proof the main result, we first present in Section 2 the classical
tools related to the Markov semi-group, which could be skipped by the
experienced reader in the subject. In Section 3 we prove the main result.
Sections 4 and 5 are reserved to the technical results of the
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ hypothesis and $U_{\nu_{n}}$,
and the Markov Dynamics respectively.

##  2 Markov tools

It is straightforward to verify that the joint evolution of
($\theta_{t},X_{t})_{t\geq 0}$ exists and is weakly unique (in law) with the
help of the Martingale Problem (MP below). For this purpose, we preliminary
define the operator $\mathcal{L}$ that acts on any function
$f\in\mathcal{C}^{2}(\mathbb{R}^{d}\times\mathcal{X})$ as:

$\mathcal{L}f(\theta,x)=\underbrace{-\langle\nabla_{\theta}U_{x}(\theta),\nabla_{\theta}f(\theta,x)\rangle+\Delta_{\theta}f(\theta,x)}_{:=\mathcal{L}_{1}f(\theta,x)}+\underbrace{\frac{\alpha_{n}}{n}\sum_{i=1}^{n}[f(\theta,\mathbf{X}_{i})-f(\theta,x)}_{\mathcal{L}_{2}f(\theta,x)}],$
(9)

for all $(\theta,x)\in\mathbb{R}^{d}\times\mathcal{X}$.

The operator $\mathcal{L}$ is divided into two terms, $\mathcal{L}_{1}$ acts
on the component $\theta$ and is associated to the diffusion part, while
$\mathcal{L}_{2}$ is the jump operator that acts on the $x$ component. Thanks
to the finiteness of the number of observations
$(\mathbf{X}_{1},\ldots,\mathbf{X}_{n})$, we can apply the results of Section
4 and 5 of chapter 4 of [17] and deduce the following result:

######  Proposition 2.1.

Assume that for any $x\in\mathcal{X}$, $U_{x}$ is
$\mathcal{C}^{2}(\mathbb{R}^{d})$ and $\nabla_{\theta}U_{x}$ is
$L_{x}$-Lipschitz, then for any initial distribution $\nu$ on
$\mathbb{R}^{d}\times\mathcal{X}$, the martingale problem $(\mathcal{L},\nu)$
is well-posed.

The associated (weakly) unique process $(\theta_{t},X_{t})_{t\geq 0}$ is a
Feller Markov process associated to the semi-group $\mathcal{L}$. In
particular, the $\theta$ component verifies the S.D.E. (3).

If we denote by $\mathcal{L}^{\star}$ the adjoint operator of $\mathcal{L}$ in
$\mathbb{L}^{2}(\mathbb{R}^{d})\times\nu_{n}$, the backward Kolmogorov
Equation yields:

$\partial_{t}m_{t}(\theta,x)=\mathcal{L}^{\star}m_{t}(\theta,x).$ (10)

Using the ellipticity of the semi-group $\mathcal{L}$ on the $\theta$
coordinate, we can use the result of [25] and deduce that for any $t>0$,
$n_{t}\in\mathcal{C}^{\infty}(\mathbb{R}^{d},\mathbb{R})$ and the
irreducibility yields $\forall t\geq 0,n_{t}>0$. We will prove in Proposition
3.5 some sufficient conditions that implies
$\|f_{0}\|_{\infty}=\|\frac{n_{0}(\theta)}{\mu_{n}(\theta)}\|_{\infty}<+\infty$
and an important and standard consequence of the maximum principle, is as
follows: if $\|f_{0}\|_{\infty}\leq M$, then

$\forall t\geq 0,\qquad\|f_{t}\|_{\infty}\leq M.$

We defer the details of this result to the Proposition 3.5 as they are not
central to our analysis and are rather technical.

Thanks to this master equation, it is possible to compute the derivative of
the semi-group on some time dependent function of $\theta$. For this purpose,
we introduce two keystone operators. The first one describes the infinitesimal
action on the $\theta$ coordinate under the average effect of $X_{t}$ at time
$t$ that applies $\forall f\in\mathcal{C}^{2}(\mathbb{R}^{d},\mathbb{R})$ as:

$\mathcal{G}_{t}f(\theta)=-\sum\limits_{i=1}^{n}\langle\nabla_{\theta}f(\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle
m_{t}(\mathbf{X}_{i}|\theta)+\Delta_{\theta}f(\theta).$ (11)

The second one is very close to the first one except that the average effect
of $X_{t}$ is replaced by the targeted ideal distribution $\nu_{n}$. It leads
to the definition $\forall f\in\mathcal{C}^{2}(\mathbb{R}^{d},\mathbb{R})$:

$\mathcal{G}f(\theta)=-\sum\limits_{i=1}^{n}\langle\nabla_{\theta}f(\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle\nu_{n}(\mathbf{X}_{i})+\Delta_{\theta}f(\theta)=-\langle\nabla_{\theta}f(\theta),\nabla_{\theta}U_{\nu_{n}}(\theta)\rangle+\Delta_{\theta}f(\theta).$
(12)

This derivative is given in the next result, whose proof is deferred to the
appendix.

######  Lemma 2.1.

Let be $h_{t}$ a twice differentiable function with uniformly bounded first
and second order derivatives on $\mathbb{R}^{d}$, then for $t>0$:

$\partial_{t}\left\\{\int_{\mathbb{R}^{d}}h_{t}(\theta)\mathrm{d}n_{t}(\theta)\right\\}=\int_{\mathbb{R}^{d}}\partial_{t}\\{h_{t}(\theta)\\}\mathrm{d}n_{t}(\theta)+\int_{\mathbb{R}^{d}}\mathcal{G}_{t}h_{t}(\theta)\mathrm{d}n_{t}(\theta),$
(13)

where $\mathcal{G}_{t}$ is the diffusion operator under the average effect of
$X_{t}$, defined in Equation (11).

##  3 Proof of the main results

###  3.1 Evolution of the entropy $J_{t}$

The entropy satisfies the following differential inequality.

######  Proposition 3.1.

Assume $\mathbf{\mathcal{H}_{\mathrm{min}}}$,
$\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ and for each $\mathbf{X}_{i}$,
$\theta\to-\log p_{\theta}(\mathbf{X}_{i})$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then a ”universal” constant
$C$ (independent from $n$ and $d$) exists such that $\forall t>0$:

$\partial_{t}\\{J_{t}\\}\leq-\int_{\mathbb{R}^{d}}\left\|\nabla_{\theta}\left(\sqrt{\frac{n_{t}(\theta)}{\mu_{n}(\theta)}}\right)\right\|_{2}^{2}\mathrm{d}\mu_{n}(\theta)+CI_{t}^{\frac{1}{3}}n^{\frac{11}{3}}\left(d\log^{2\beta}(n)\right)^{1+r}.$

###### Proof.

We shall use the standard preliminary estimate that may be derived from
Equation (3.14) of [29] for elliptic diffusions to apply Lemma 2.1 to
$f_{t}=\log(n_{t}\mu_{n}^{-1})$. From Equation (13), we have:

$\partial_{t}\\{J_{t}\\}=\int_{\mathbb{R}^{d}}\partial_{t}\left\\{\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\right\\}\mathrm{d}n_{t}(\theta)+\int_{\mathbb{R}^{d}}\mathcal{G}_{t}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}n_{t}(\theta),$

The first term vanishes since:

$\displaystyle\int_{\mathbb{R}^{d}}\partial_{t}\left\\{\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\right\\}\mathrm{d}n_{t}(\theta)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\frac{\partial_{t}\\{n_{t}(\theta)\\}}{n_{t}(\theta)}\mathrm{d}n_{t}(\theta)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\partial_{t}\left\\{n_{t}(\theta)\right\\}\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\partial_{t}\left(\int_{\mathbb{R}^{d}}\mathrm{d}n_{t}(\theta)\right)$
$\displaystyle=$ $\displaystyle 0.$

Then, the derivative is reduced to the second term, and we are led to:

$\displaystyle\partial_{t}\\{J_{t}\\}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\mathcal{G}_{t}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}n_{t}(\theta),$
(14) $\displaystyle=$
$\displaystyle\underbrace{\int_{\mathbb{R}^{d}}\mathcal{G}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}n_{t}(\theta)}_{J_{1,t}}+\underbrace{\int_{\mathbb{R}^{d}}\left(\mathcal{G}_{t}-\mathcal{G}\right)\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}n_{t}(\theta)}_{J_{2,t}}.$

We study the two terms $J_{1,t}$ and $J_{2,t}$ separately.

  * •

Study of $J_{1,t}$. Since $\mathcal{G}$ is a diffusion operator and $\mu_{n}$
is the invariant measure associated to $\mathcal{G}$, then we can use the
classical link between $J_{1,t}$ and the Dirichlet form (see [3]):

$\displaystyle\int_{\mathbb{R}^{d}}\mathcal{G}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}n_{t}(\theta)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\
\mathcal{G}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}\mu_{n}(\theta)$
(15) $\displaystyle=$
$\displaystyle-4\int_{\mathbb{R}^{d}}\left\|\nabla_{\theta}\left(\sqrt{\frac{n_{t}(\theta)}{\mu_{n}(\theta)}}\right)\right\|_{2}^{2}\mathrm{d}\mu_{n}(\theta).$

  * •

Study of $J_{2,t}$. We use the difference between $\mathcal{G}$ and
$\mathcal{G}_{t}$, for any twice differentiable function $f$:

$\displaystyle\left(\mathcal{G}_{t}-\mathcal{G}\right)f(\theta)$
$\displaystyle=$
$\displaystyle-\sum\limits_{i=1}^{n}\left\langle\nabla_{\theta}f(\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\rangle\left[m_{t}(\mathbf{X}_{i}|\theta)-\nu_{n}(\mathbf{X}_{i})\right]$
$\displaystyle=$
$\displaystyle-\sum\limits_{i=1}^{n}\langle\nabla_{\theta}f(\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]\nu_{n}(\mathbf{X}_{i}).$

Then, the term $J_{2,t}$ may be computed as:

$\displaystyle\left|J_{2,t}\right|$ $\displaystyle=$
$\displaystyle\left|\int_{\mathbb{R}^{d}}\left(\mathcal{G}_{t}-\mathcal{G}\right)\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\mathrm{d}n_{t}(\theta)\right|$
$\displaystyle=$
$\displaystyle\left|\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\langle\nabla_{\theta}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta)\right|.$

Using the Cauchy-Schwartz inequality with respect to the measure
$\nu_{n}(\mathbf{X}_{i})\times\mathrm{d}n_{t}(\theta)$ in the first line,
$2ab\leq a^{2}+b^{2}$ in the second line and $\nabla\log
f=2\nabla\log\sqrt{f}=2\frac{\nabla\sqrt{f}}{\sqrt{f}}$ in the third line, we
obtain that:

$\displaystyle\left|J_{2,t}\right|$ $\displaystyle\leq$
$\displaystyle\left(\int_{\mathbb{R}^{d}}\left\|\nabla_{\theta}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\right\|_{2}^{2}\mathrm{d}n_{t}(\theta)\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{2}\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]^{2}\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta)\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle\frac{3}{4}\int_{\mathbb{R}^{d}}\left\|\nabla_{\theta}\log\left(\frac{n_{t}(\theta)}{\mu_{n}(\theta)}\right)\right\|_{2}^{2}\mathrm{d}n_{t}(\theta)+\frac{1}{3}\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{2}\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]^{2}\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta)$ $\displaystyle\leq$ $\displaystyle
3\int_{\mathbb{R}^{d}}\left\|\nabla_{\theta}\left(\sqrt{\frac{n_{t}(\theta)}{\mu_{n}(\theta)}}\right)\right\|_{2}^{2}\mathrm{d}\mu_{n}(\theta)+\frac{1}{3}\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{2}\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]^{2}\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta).$

Using Equation (15) and the previous line yields:

$\displaystyle\partial_{t}\\{J_{t}\\}$ $\displaystyle\leq$
$\displaystyle-\int_{\mathbb{R}^{d}}\left\|\nabla_{\theta}\left(\sqrt{\frac{n_{t}(\theta)}{\mu_{n}(\theta)}}\right)\right\|_{2}^{2}\mathrm{d}\mu_{n}(\theta)+\frac{1}{3}\underbrace{\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{2}\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]^{2}\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta)}_{:=\Delta_{t}},$ (16)

We then focus on the second term of the right hand side. For this purpose, we
consider a non-negative function $g(t)$, which will be fixed later and we
split $\Delta_{t}$ into two terms as:

$\displaystyle\Delta_{t}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{2}\left(\mathbbm{1}_{\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\|_{2}\leq
g(t)}+\
\mathbbm{1}_{\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\|_{2}>g(t)}\right)\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]^{2}\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta)$ $\displaystyle\leq$ $\displaystyle
g^{2}(t)I_{t}+\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{2}\mathbbm{1}_{\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\|_{2}>g(t)}\left[\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right]^{2}\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta),$

where $I_{t}$ has been introduced in Equation (6) and measures the closeness
of $m_{t}(\mathbf{X}_{i}|\theta)$ to $\nu_{n}$. Finally, for the last term we
observe that $0\leq m_{t}(\mathbf{X}_{i}|\theta)\leq 1$ and
$\left|\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right|=n\left|m_{t}(\mathbf{X}_{i}|\theta)-\frac{1}{n}\right|\leq
n$, which implies that:

$\Delta_{t}\leq
g^{2}(t)I_{t}+n^{2}\underbrace{\frac{1}{n}\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{2}\mathbbm{1}_{\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\|_{2}>g(t)}\mathrm{d}n_{t}(\theta)}_{:=\tilde{\Delta}_{t}}.$
(17)

The Cauchy inequality leads to:

$\displaystyle\tilde{\Delta}_{t}$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{n}\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\right\|_{2}^{4}\mathrm{d}n_{t}(\theta)\right)^{\frac{1}{2}}\left(\frac{1}{n}\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathbbm{1}_{\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\|_{2}>g(t)}\mathrm{d}n_{t}(\theta)\right)^{\frac{1}{2}}$
(18) $\displaystyle=$
$\displaystyle\left(\frac{1}{n}\sum\limits_{i=1}^{n}\mathbb{E}\left[\left\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta_{t})\right\|_{2}^{4}\right]\right)^{\frac{1}{2}}\left(\frac{1}{n}\sum_{i=1}^{n}\mathbb{P}\left(\|\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta_{t})\|_{2}>g(t)\right)\right)^{\frac{1}{2}}.$

We then use Proposition 4.1 and obtain that:

$\displaystyle\tilde{\Delta}_{t}$ $\displaystyle\leq$
$\displaystyle\left(\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}\left[\left(2(nL+\ell_{0})U_{\mathbf{X}_{i}}^{2}(\theta_{t})\right)\right]\right)^{\frac{1}{2}}\left(\frac{1}{n}\sum\limits_{i=1}^{n}\mathbb{P}\left(2(nL+\ell_{0})U_{\mathbf{X}_{i}}(\theta_{t})>g^{2}(t)\right)\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
2(nL+\ell_{0})\left(n\mathbb{E}[U_{\nu_{n}}^{2}(\theta_{t})]\right)^{\frac{1}{2}}\left(\frac{1}{n}\sum_{i=1}^{n}\frac{2(nL+\ell_{0})}{g^{2}(t)}\mathbb{E}\left[U_{\mathbf{X}_{i}}(\theta_{t})\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq$
$\displaystyle[2(nL+\ell_{0})]^{\frac{3}{2}}n^{\frac{1}{2}}\frac{\mathbb{E}\left[U_{\nu_{n}}^{2}(\theta_{t})\right]^{\frac{1}{2}}\mathbb{E}\left[U_{\nu_{n}}(\theta_{t})\right]^{\frac{1}{2}}}{g(t)},$

where we used the Markov’s inequality and the relation
$\|.\|_{2}\leq\|.\|_{1}$ in $\mathbb{R}^{n}$. We apply Proposition 5.1 with
$\alpha=2$ and $\alpha=1$ and obtain that a constant $C>0$ exists (whose value
may change from line to line) such that:

$\displaystyle\tilde{\Delta}_{t}$ $\displaystyle\leq$ $\displaystyle
C\frac{n^{\frac{7}{2}}\left(d\log^{2\beta}(n)\right)^{\frac{3(1+r)}{2}}}{g(t)}.$

We use this last bound in (17) and we deduce that:

$\Delta_{t}\leq
g^{2}(t)I_{t}+C\frac{n^{\frac{11}{2}}\left(d\log^{2\beta}(n)\right)^{\frac{3(1+r)}{2}}}{g(t)}.$

Optimizing this last bound with respect to $g(t)$ leads to the upper bound:

$\Delta_{t}\leq
CI_{t}^{\frac{1}{3}}n^{\frac{11}{3}}\left(d\log^{2\beta}(n)\right)^{1+r},\quad\forall
t\geq 0.$

∎

###  3.2 Evolution of the weighted $\mathbb{L}^{2}$ distance $I_{t}$

The quantity $I_{t}$ involved in Proposition 3.1 measures how close to
$\nu_{n}$ the conditional distribution of $X_{t}|\theta_{t}$ is. To study
$I_{t}$, we first remark that it may be rewritten in a simpler way.

$\displaystyle I_{t}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left(\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right)^{2}\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left(\frac{m_{t}^{2}(\mathbf{X}_{i}|\theta)}{\nu_{n}^{2}(\mathbf{X}_{i})}-2\frac{m_{t}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}+1\right)\nu_{n}(\mathbf{X}_{i})\
\mathrm{d}n_{t}(\theta)$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left(\frac{m_{t}^{2}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-2m_{t}(\mathbf{X}_{i}|\theta)+\nu_{n}(\mathbf{X}_{i})\right)\mathrm{d}n_{t}(\theta)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\left(\sum\limits_{i=1}^{n}\frac{m_{t}^{2}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}-1\right)\mathrm{d}n_{t}(\theta)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\frac{m_{t}^{2}(\mathbf{X}_{i}|\theta)}{\nu_{n}(\mathbf{X}_{i})}\mathrm{d}n_{t}(\theta)-1.$

Using that
$m_{t}(\mathbf{X}_{i}|\theta)n_{t}(\theta)=m_{t}(\theta,\mathbf{X}_{i})$ and
$\nu_{n}(\mathbf{X}_{i})=\frac{1}{n}$ for $i=1,2,\ldots,n$, we obtain that:

$I_{t}=n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\frac{m_{t}^{2}(\theta,\mathbf{X}_{i})}{n_{t}(\theta)}\mathrm{d}\theta-1.$
(19)

The next proposition then assesses how fast $I_{t}$ decreases to $0$ as
$t\longrightarrow+\infty$.

######  Proposition 3.2.

For any $t\geq 0$:

$I_{t}\leq I_{0}e^{-2\alpha_{n}t}\leq(n-1)e^{-2\alpha_{n}t}.$ (20)

###### Proof.

Our starting point is Equation (19). We compute its derivative with respect to
$t$:

$\displaystyle\partial_{t}\\{I_{t}\\}$ $\displaystyle=$ $\displaystyle
2n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\frac{m_{t}(\theta,\mathbf{X}_{i})}{n_{t}(\theta)}\partial_{t}m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\frac{m_{t}^{2}(\theta,\mathbf{X}_{i})}{n_{t}^{2}(\theta)}\partial_{t}n_{t}(\theta)\mathrm{d}\theta$
$\displaystyle=$ $\displaystyle
2n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}(\mathbf{X}_{i}|\theta)\partial_{t}m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\partial_{t}n_{t}(\theta)\mathrm{d}\theta.$

Using the Kolmogorov backward equation in the first line and
$\mathcal{L}=\mathcal{L}_{1}+\mathcal{L}_{2}$ in the second one where
$\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ are defined in Equation (9), we have:

$\displaystyle\partial_{t}\\{I_{t}\\}$ $\displaystyle=$ $\displaystyle
2n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}m_{t}(\mathbf{X}_{i}|\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\partial_{t}n_{t}(\theta)\mathrm{d}\theta$
(21) $\displaystyle=$
$\displaystyle\underbrace{2n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{1}m_{t}(\mathbf{X}_{i}|\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta}_{:=I_{3,t}}+\underbrace{2n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{2}m_{t}(\mathbf{X}_{i}|\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta}_{:=I_{1,t}}$
$\displaystyle\underbrace{-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\partial_{t}n_{t}(\theta)\mathrm{d}\theta}_{:=I_{2,t}}.$

Then, $\partial_{t}\\{I_{t}\\}$ may be splitted into three terms that are
studied separately.

  * •

Study of $I_{1,t}$. We observe that:

$\mathcal{L}_{2}m_{t}(\mathbf{X}_{i}|\theta)=\frac{\alpha_{n}}{n}\sum\limits_{j=1}^{n}[m_{t}(\mathbf{X}_{j}|\theta)-m_{t}(\mathbf{X}_{i}|\theta)]=\frac{\alpha_{n}}{n}-\alpha_{n}\
m_{t}(\mathbf{X}_{i}|\theta).$ (22)

We then use this last equation in the definition of $I_{1}(t)$ and obtain
that:

$\displaystyle I_{1,t}$ $\displaystyle=$ $\displaystyle
2n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{2}m_{t}(\mathbf{X}_{i}|\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta$ (23) $\displaystyle=$
$\displaystyle
2\alpha_{n}\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta-2\alpha_{n}n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}(\mathbf{X}_{i}|\theta)m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta$
$\displaystyle=$ $\displaystyle
2\alpha_{n}-2\alpha_{n}n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\frac{m_{t}^{2}(\theta,\mathbf{X}_{i})}{n_{t}(\theta)}\mathrm{d}\theta$
$\displaystyle=$ $\displaystyle-2\alpha_{n}I_{t}.$

  * •

Study of $I_{2,t}$. Using the definition of $n_{t}$, we obtain that:

$\displaystyle I_{2,t}$ $\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\partial_{t}n_{t}(\theta)\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\partial_{t}\left(\sum\limits_{j=1}^{n}m_{t}(\theta,\mathbf{X}_{j})\right)\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\partial_{t}m_{t}(\theta,\mathbf{X}_{j})\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{j=1}^{n}\left(\sum\limits_{i=1}^{n}\mathcal{L}m_{t}^{2}(\mathbf{X}_{i}|\theta)\right)m_{t}(\theta,\mathbf{X}_{j})\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta).$

where we used the Kolmogorov backward equation in the fourth line and again
the definition of $n_{t}$ in the last line. Again, the decomposition
$\mathcal{L}=\mathcal{L}_{1}+\mathcal{L}_{2}$ yields:

$\displaystyle I_{2,t}$ $\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{1}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta)-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{2}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta).$

We repeat some similar computations as those developed in Equation (22) to
study the action of the jump component induced by $\mathcal{L}_{2}$ on
$m_{t}^{2}$. We obtain that:

$\mathcal{L}_{2}m_{t}^{2}(\mathbf{X}_{i}|\theta)=\frac{\alpha_{n}}{n}\sum\limits_{k=1}^{n}[m_{t}^{2}(\mathbf{X}_{k}|\theta)-m_{t}^{2}(\mathbf{X}_{i}|\theta)]=\frac{\alpha_{n}}{n}\sum\limits_{k=1}^{n}m_{t}^{2}(\mathbf{X}_{k}|\theta)-\alpha_{n}\
m_{t}^{2}(\mathbf{X}_{i}|\theta).$

We use this last equation and obtain that:

$\displaystyle I_{2,t}$ $\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{1}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta)-\alpha_{n}\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\sum\limits_{k=1}^{n}m_{t}^{2}(\mathbf{X}_{k}|\theta)\
\mathrm{d}n_{t}(\theta)$ (24)
$\displaystyle+\alpha_{n}n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta)$ $\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{1}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta)-\alpha_{n}n\int_{\mathbb{R}^{d}}\sum\limits_{k=1}^{n}m_{t}^{2}(\mathbf{X}_{k}|\theta)\
\mathrm{d}n_{t}(\theta)$
$\displaystyle+\alpha_{n}n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta)$ $\displaystyle=$
$\displaystyle-n\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{1}m_{t}^{2}(\mathbf{X}_{i}|\theta)\
\mathrm{d}n_{t}(\theta).$

  * •

Study of $I_{2,t}+I_{3,t}$. We observe that this sum involves only
$\mathcal{L}_{1}$ (see Equation (9). We first compute:

$\mathcal{L}_{1}m_{t}(\mathbf{X}_{i}|\theta)=-\langle\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta),\nabla_{\theta}m_{t}(\mathbf{X}_{i}|\theta)\rangle+\Delta_{\theta}m_{t}(\mathbf{X}_{i}|\theta),$

and similarly:

$\displaystyle\mathcal{L}_{1}m_{t}^{2}(\mathbf{X}_{i}|\theta)$
$\displaystyle=-\langle\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta),\nabla_{\theta}m_{t}^{2}(\mathbf{X}_{i}|\theta),\rangle+\Delta_{\theta}m_{t}^{2}(\mathbf{X}_{i}|\theta)$
$\displaystyle=-2m_{t}(\mathbf{X}_{i}|\theta)\langle\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta),\nabla_{\theta}m_{t}(\mathbf{X}_{i}|\theta)\rangle+2\|\nabla_{\theta}m_{t}(\mathbf{X}_{i}|\theta)\|_{2}^{2}+2m_{t}(\mathbf{X}_{i}|\theta)\Delta_{\theta}m_{t}(\mathbf{X}_{i}|\theta).$

Using these two equations into $I_{2,t}+I_{3,t}$ and
$m_{t}(\mathbf{X}_{i}|\theta)n_{t}(\theta)=m_{t}(\theta,\mathbf{X}_{i})$, we
get:

$\displaystyle\frac{I_{2,t}+I_{3,t}}{n}$
$\displaystyle=2\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\langle\nabla_{\theta}m_{t}(\mathbf{X}_{i}|\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta$
$\displaystyle-2\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\|\nabla_{\theta}m_{t}(\mathbf{X}_{i}|\theta)\|_{2}^{2}\
n_{t}(\theta)\mathrm{d}\theta-2\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\Delta_{\theta}m_{t}(\mathbf{X}_{i}|\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta$
$\displaystyle-2\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\langle\nabla_{\theta}m_{t}(\mathbf{X}_{i}|\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta+2\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\Delta_{\theta}m_{t}(\mathbf{X}_{i}|\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta$
$\displaystyle=-\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\|\nabla_{\theta}m_{t}(\mathbf{X}_{i}|\theta)\|_{2}^{2}\
\mathrm{d}n_{t}(\theta)\leq 0.$

Gathering this last inequality with (23) into Equation (21) yields:

$\partial_{t}\\{I_{t}\\}\leq-2\alpha_{n}I_{t}.$

We conclude with a direct application of the Gronwall lemma while observing
that $I_{0}\leq n-1$. ∎

###  3.3 Functional (weak) log-Sobolev inequalities

####  3.3.1 Related works on functional inequalities

A straightforward consequence of Proposition 3.1 and Proposition 3.2 is the
following differential inequality on the relative entropy $J_{t}$:

$\partial_{t}\\{J_{t}\\}\leq-\int_{\mathbb{R}^{d}}\left\|\nabla_{\theta}\left(\sqrt{\frac{n_{t}(\theta)}{\mu_{n}(\theta)}}\right)\right\|_{2}^{2}\mathrm{d}\mu_{n}(\theta)+c_{n,d}e^{-\frac{2\alpha_{n}}{3}t},$
(25)

where $c_{n,d}$ is defined as:

$c_{n,d}\lesssim_{uc}n^{4}\left(d\log^{2\beta}(n)\right)^{1+r}.$ (26)

At this stage, we should observe that a standard approach consists in finding
a functional inequality that relates the key Dirichlet form $\mathcal{E}(f)$
defined by:

$\mathcal{E}(f)=\int_{\mathbb{R}^{d}}\|\nabla_{\theta}f(\theta)\|_{2}^{2}\mathrm{d}\mu_{n}(\theta),$
(27)

to $Ent_{\mu_{n}}(f^{2})$, the entropy itself with respect to $\mu_{n}$. These
approaches rely on the initial works of [23] where Logarithmic Sobolev
Inequality (LSI for short) were introduced. The consequences of LSI to
exponential ergodicity has then been an extensive field of research and we
refer to [3] for an overview on this topic. A popular sufficient condition
that ensures LSI is the log strong-convexity of the targeted measure (see
among other [2]) and an impressive amount of literature has been focused on
the existing links between these functional inequalities, ergodicity of the
semi-group, transport inequalities and Lyapunov conditions. We refer to [8, 1]
(these two works are far from being exhaustive). The great interest of LSI has
then been observed in machine learning and statistics more recently as
testified by the recent works in Monte Carlo samplings of [31, 34]. A popular
way to extend LSI from the strongly convex situation to a more general case
relies on the “strong convexity outside a ball” hypothesis using the
perturbation argument of the seminal contributions of [26]. If this method
proves to be suitable for the study of the simulated annealing process in
[33], [26], it appears to be doubtful for the study of sampling problems with
convex potentials that satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ as this settings do not imply
an asymptotic strong convexity of $\theta\longmapsto U(\theta)$ for large
values of $\|\theta\|_{2}$. That being said, and maybe an even worst
consequence of such approach, is the unavoidable dependency on the dimension
for the LSI constant when using a perturbation approach, which leads to a
serious exponential degradation of the convergence rates with the dimension of
the ambient space.

To overcome these difficulties, we have chosen to use a slightly different
functional inequality that may be considered as an innocent modification of
LSI, but that indeed appears to be well suited to weakly log-concave setting
described through an $\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$
assumption. For this purpose, we shall use weak log-Sobolev inequalities (WLSI
for short below) that have been introduced in [37] and whose interest has been
extensively studied in many works to obtain exponentially sub-linear rates of
mixing, see among others for example [7]. To derive such inequalities, our
starting point will be the contribution of [10] that makes the link between
Lyapunov conditions and WLSI. Our approach based on
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ certainly shares some
similarities with the recent work of [6] where some functional inequalities
(Poincaré and Transport inequalities) are obtained within a framework of
variable curvature bound.

####  3.3.2 Weak log Sobolev inequalities

We briefly introduce the key theoretical ingredients, that are exhaustively
described in [3]. We introduce the following assumption, that will be suitable
for the setting of bounded functions.

######  Definition 3.1 (Weak Log-Sobolev Inequality ).

For any measurable space $(\Omega,\mathcal{F},\mu)$ and for any nice function
$f$, let us define:

$Ent_{\mu}(f^{2}):=\int_{\Omega}f^{2}\log(f^{2})\mathrm{d}\mu-\int_{\Omega}f^{2}\mathrm{d}\mu\log\left(\int_{\Omega}f^{2}\mathrm{d}\mu\right).$

The measure $\mu$ satisfies a WLSI if a non-increasing function
$\varphi_{WLS}:(0,+\infty)\mapsto\mathbb{R}_{+}$ exists such that for any
$f\in\mathcal{C}^{\mathbbm{1}}_{b}(\Omega)$:

$Ent_{\mu}(f^{2})\leq\varphi_{WLS}(s)\mathcal{E}(f)+s\,Osc^{2}(f),$ (28)

where $Osc(f):=\sup f-\inf f$.

Before establishing how to use this functional inequality, we first state the
important relationship between Poincaré Inequality and WLSI.

######  Proposition 3.3.

Assume that $\mu$ satisfies a Poincaré Inequality of constant $C_{P}$, i.e.
for any smooth integrable function $f$:

$C_{p}(\mu)Var_{\mu}(f)=C_{p}(\mu)\int_{\Omega}(f-\mu[f])^{2}\mathrm{d}\mu\leq\int_{\Omega}|\nabla
f|^{2}\mathrm{d}\mu,$

then if $\log
c=\frac{3}{14e^{2}}\left(\frac{1}{e}+\frac{1}{2}\right)+1+\log\left(\frac{14}{3}\right)$,
then $\mu$ satisfies a WLSI with:

$\varphi_{WLS}(s)=\left\\{\begin{array}[]{r l}0,&s>\frac{1}{e}+\frac{1}{2}\\\
\frac{32}{C_{P}}\log\left(\frac{c}{s}\right),&s\leq\frac{1}{e}+\frac{1}{2}\\\
\end{array}\right..$

For the sake of readability, we introduce a universal $\mathfrak{a}>0$ such
that:

$\varphi_{WLS}(s)=\left\\{\begin{array}[]{r l}0,&s>\frac{1}{e}+\frac{1}{2}\\\
\mathfrak{a}\frac{1+\log\left(\frac{1}{s}\right)}{C_{P}},&s\leq\frac{1}{e}+\frac{1}{2}\\\
\end{array}\right..$ (29)

###### Proof of Proposition 3.3.

The proof of how the Poincaré Inequality implies the WLSI in the bounded
setting described in Definition 28 is given for the sake of completeness.
Technical details are skipped and we refer to the references below. We use the
measure-capacity inequality (see [3], Section 8.3). We know that the Poincaré
Inequality implies a capacity inequality (Proposition 8.3.1 of [3]) with a
constant equal to $2C_{P}$. Then, we can apply Theorem 2.2 of [7] that induces
a WLSI which is based on the function $\varphi_{WLS}$ given in the statement
of the proposition. ∎

####  3.3.3 Weak log Sobolev inequalities under
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$

Of course, in the previous result, the only important dependency will be the
one induced by $C_{P}$, which will deserve an ad-hoc study under Assumption
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$. The numbers 32 and $\log(c)$
will be dealt with as “universal constants” in what follows.

The next proposition states two lower bounds on the Poincaré constant within
the $\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ framework. The first one
always holds, regardless the value of $(X_{1},\ldots,X_{n})$ that may be been
randomly sampled. The second one has to be considered with high probability,
with respect to the sampling process $(X_{1},\ldots,X_{n})$.

######  Proposition 3.4.

Assume
$\mathbf{\mathcal{H}_{\mathrm{min}}}$,$\mathbf{\mathcal{H}_{n_{0}}(L,\ell_{0})}$,
$\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ and for any $x$,
$\theta\mapsto-\log p_{\theta}(x)$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then:

  * $i)$

For any sample $(X_{1},\ldots,X_{n})$, it holds:

$C_{P}(\mu_{n})\gtrsim_{uc}\frac{1}{\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}}}$

  * $ii)$

Assume that $\theta\mapsto\mathbb{P}_{\theta}$ is injective and $\theta_{0}$
exists such that $(X_{1},\ldots,X_{n})\sim\mathbb{P}_{\theta_{0}}$. If locally
around $\theta_{0}$,
$\theta\mapsto|\theta-\theta_{0}|^{-\alpha}W_{1}(\mathbb{P}_{\theta},\mathbb{P}_{\theta_{0}})$
does not vanish, then:

$\mathbb{E}_{(X_{1},\ldots,X_{n})\sim\mathbb{P}_{\theta_{0}}}[C_{P}(\mu_{n})]\gtrsim_{uc}\left(\frac{n}{Ld\log
n}\right)^{\alpha}.$

We are finally led to upper bound the oscillations of the function involved in
the WLSI introduced in (28), i.e. we are looking for an upper bound of
$Osc^{2}\left(\sqrt{\frac{n_{t}}{\mu_{n}}}\right)$ for any time $t>0$. For
this purpose, we observe that the Markov semi-group induces that
$f_{t}=\frac{n_{t}}{\mu_{n}}=P_{t}f_{0}$ where $f_{0}=\frac{n_{0}}{\mu_{n}}$.
The next proposition implies the boundedness of $f_{t}$ over $\mathbb{R}^{d}$
when $n_{0}$ is chosen as a Gaussian distribution with a carefully tuned
covariance matrix.

######  Proposition 3.5.

Assume
$\mathbf{\mathcal{H}_{\mathrm{min}}}$,$\mathbf{\mathcal{H}_{n_{0}}(L,\ell_{0})}$,
$\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ and that, for any $x$,
$\theta\mapsto-\log p_{\theta}(x)$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then:

  * $i)$

Two positive constants $C_{1}$ and $C_{2}$ exist, which are independent from
$n$ and $d$ and such that:

$\|f_{0}\|_{\infty}\lesssim_{uc}\left(\frac{C_{1}d}{n}\right)^{\frac{dr}{2}}\exp\left(C_{2}nd^{1+r}\log^{2\beta(1+r)}(n)\right).$

  * $ii)$

As a consequence:

$Osc^{2}(\sqrt{f_{t}})\leq
Osc^{2}(\sqrt{f_{0}})\lesssim_{uc}\left(\frac{C_{1}d}{n}\right)^{\frac{dr}{2}}\exp\left(C_{2}nd^{1+r}\log^{2\beta(1+r)}(n)\right).$

  * $iii)$

Moreover, a straightforward consequence of $i)$ is:

$J_{0}=\int_{\mathbb{R}^{d}}\log\left(f_{0}(\theta)\right)\mathrm{d}n_{0}(\theta)\lesssim_{uc}nd^{1+r}\log^{2\beta(1+r)}(n)+d\log\left(\frac{d}{n}\right).$

###  3.4 Entropic convergence of the SLMC

The purpose of this paragraph is to prove the main result of the paper, i.e.
Theorem 1.1 that guarantees the convergence of the SLMC algorithm.

###### Proof of Theorem 1.1.

Our starting point is the semi-group inequality (25) associated with the
functional WLSI inequality (28). Using $c_{n,d}$ defined in (26), we obtain
for any $s>0$:

$\displaystyle\partial_{t}\\{J_{t}\\}$
$\displaystyle\leq-\mathcal{E}\left(\sqrt{\frac{n_{t}}{\mu_{n}}}\right)+c_{n,d}e^{-\frac{2\alpha_{n}}{3}t}$
$\displaystyle\leq-\frac{J_{t}}{\varphi_{WLS}(s)}+\frac{s}{\varphi_{WLS}(s)}Osc^{2}\left(\sqrt{\frac{n_{t}}{\mu_{n}}}\right)+c_{n,d}e^{-\frac{2\alpha_{n}}{3}t}$
$\displaystyle\leq-\frac{J_{t}}{\varphi_{WLS}(s)}+\frac{s\,O_{n,d}}{\varphi_{WLS}(s)}+c_{n,d}e^{-\frac{2\alpha_{n}}{3}t},$

where we applied Proposition 3.5 in the last line with
$O_{n,d}\lesssim_{uc}\left(\frac{C_{1}d}{n}\right)^{\frac{dr}{2}}\exp\left(C_{2}nd^{1+r}\log^{2\beta(1+r)}(n)\right)$
and $C_{1}$ and $C_{2}$ two universal constants. We then choose $s$ (that
depends on $t$) such that:

$s_{t}=e^{-A\sqrt{t+1}}\quad\text{with}\quad A>1\quad\text{that will be chosen
later on}.$

We observe that $s_{t}<e^{-1}+1/2$, so that Equation (29) of Proposition 3.3
yields:

$\varphi_{WLS}(s_{t})=\mathfrak{a}\frac{1+\log\left(\frac{1}{s_{t}}\right)}{C_{P}}=\mathfrak{a}\frac{1+A\sqrt{1+t}}{C_{P}}.$

We introduce
$\psi(t)=\exp\left(\frac{C_{P}}{\mathfrak{a}}\int_{0}^{t}\frac{\mathrm{d}u}{1+A\sqrt{1+u}}\right)$
and deduce that

$\psi(t)=\exp\left(\frac{C_{P}}{\mathfrak{a}}\frac{2A(\sqrt{1+t}-1)-2\log\left(\frac{1+A\sqrt{1+t}}{1+A}\right)}{A^{2}}\right)\leq\exp\left(\frac{2C_{P}}{\mathfrak{a}A}(\sqrt{1+t}-1)\right).$

We now apply the Gronwall Lemma:

$\displaystyle\partial_{t}\left\\{\psi(t)J_{t}\right\\}$
$\displaystyle=\left(\frac{C_{P}}{\mathfrak{a}(1+A\sqrt{1+t})}J_{t}+J^{\prime}_{t}\right)\psi(t)$
$\displaystyle\leq\left[\frac{C_{P}O_{n,d}}{\mathfrak{a}}\frac{e^{-A\sqrt{t+1}}}{1+A\sqrt{1+t}}+c_{n,d}e^{-\frac{2\alpha_{n}}{3}t}\right]\psi(t)$
$\displaystyle\leq\frac{C_{P}O_{n,d}}{\mathfrak{a}}e^{-(A-\frac{2C_{P}}{\mathfrak{a}A})\sqrt{1+t}}+c_{n,d}e^{\frac{2C_{P}}{\mathfrak{a}A}(\sqrt{1+t}-1)-\frac{2\alpha_{n}}{3}t}.$

We denote by $t_{0}$ the positive real value that solves the equation
$\frac{2C_{P}}{\mathfrak{a}A}\sqrt{1+t_{0}}=\frac{\alpha_{n}t_{0}}{3}.$ We
then observe that:

$\displaystyle\int_{0}^{t}e^{\frac{2C_{P}}{\mathfrak{a}A}(\sqrt{1+u}-1)-\frac{2\alpha_{n}}{3}u}\mathrm{d}u$
$\displaystyle\leq\int_{0}^{t_{0}}e^{\frac{2C_{P}}{\mathfrak{a}A}\sqrt{1+u}}\mathrm{d}u+\int_{t_{0}}^{+\infty}e^{-\frac{\alpha_{n}}{3}u}\mathrm{d}u$
$\displaystyle\leq
t_{0}e^{\frac{2C_{P}}{\mathfrak{a}A}\sqrt{1+t_{0}}}+\frac{3}{\alpha_{n}}=t_{0}e^{\frac{\alpha_{n}t_{0}}{3}}+\frac{3}{\alpha_{n}}.$

If $A$ is chosen such that $A>\frac{2C_{P}}{\mathfrak{a}A}$, we then deduce
that:

$\displaystyle J_{t}$
$\displaystyle\leq\left(J_{0}+c_{n,d}t_{0}e^{\frac{\alpha_{n}t_{0}}{3}}+\frac{3c_{n,d}}{\alpha_{n}}\right)\psi(t)^{-1}+\frac{C_{P}O_{n,d}}{\mathfrak{a}}\psi(t)^{-1}\int_{0}^{t}e^{-\left(A-\frac{2C_{P}}{\mathfrak{a}A}\right)\sqrt{1+u}}\mathrm{d}u$
$\displaystyle\leq\left(J_{0}+c_{n,d}t_{0}e^{\frac{\alpha_{n}t_{0}}{3}}+\frac{3c_{n,d}}{\alpha_{n}}\right)\psi(t)^{-1}+\frac{2C_{P}O_{n,d}}{\mathfrak{a}\left(A-\frac{2C_{P}}{\mathfrak{a}A}\right)^{2}}\psi(t)^{-1},$

where we used in the previous line the bound:

$\int_{0}^{t}e^{-b\sqrt{1+u}}\mathrm{d}u\leq\int_{0}^{+\infty}e^{-b\sqrt{1+u}}\mathrm{d}u\leq\frac{2}{b^{2}}.$

To obtain the lowest upper bound, we are led to choose $A$ such that
$\frac{2C_{P}}{\mathfrak{a}A}$ as large as possible and below $A$, which
naturally drives to the choice:

$\frac{2C_{P}}{\mathfrak{a}A}=\frac{A}{2}\Longrightarrow
A=\frac{2}{\sqrt{\mathfrak{a}}}\sqrt{C_{P}}.$

Using this value of $A$ in the previous bound, we observe that
$t_{0}\leq\frac{3\sqrt{C_{P}}}{\alpha_{n}\sqrt{\mathfrak{a}}}+\frac{C_{P}}{\alpha_{n}^{2}}$,
so that a constant $C$ exists such that:

$J_{t}\leq
C\left(J_{0}+\frac{c_{n,d}}{\alpha_{n}}\left[1+\left(\frac{C_{P}}{\alpha_{n}}+\sqrt{C_{P}}\right)e^{\frac{\sqrt{C_{P}}}{\sqrt{\mathfrak{a}}}+\frac{C_{P}}{3\alpha_{n}}}\right]+O_{n,d}\right)(1+t)^{1/4}e^{-\frac{\sqrt{C_{p}}}{\sqrt{a}}(\sqrt{1+t}-1)}.$
(30)

In Proposition 3.4 we obtained
$C_{P}\geq\frac{\kappa}{\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}}}$. If
instead of using the constant $C_{P}$, we use directly
$\frac{\kappa}{\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}}}$ with $\kappa<1$,
then all the previous computations remain the same only replacing $C_{P}$ by
its lower bound and:

$J_{t}\leq
C\left(J_{0}+\frac{c_{n,d}}{\alpha_{n}}e^{\frac{\sqrt{\kappa}\left(\frac{1}{\sqrt{a}}+\frac{1}{3\alpha_{n}}\right)}{\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}/2}}}+O_{n,d}\right)(1+t)^{1/4}e^{-\frac{\sqrt{\kappa}(\sqrt{1+t}-1)}{\sqrt{a}\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}/2}}}.$
(31)

Using the values of $O_{n,d}$, $c_{n,d}$ and the upper bound of $J_{0}$, we
finally observe that if
$\alpha_{n}=\frac{1}{n\left(d\log^{2\beta}(n)\right)^{1+r}}$, then:

$t\geq\aleph\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}}\left[\log^{2}(\varepsilon^{-1})+n^{2}\left(d\log^{2\beta}(n)\right)^{2(1+r)}+d^{2}\log^{2}d\right]\Longrightarrow
J_{t}\leq\varepsilon.$

∎

##  4 Technical results on KL and $U_{\nu_{n}}$

###  4.1 Growth properties under the Kurdyka-Łojasiewicz inequality

We remind here some important consequences of the KL inequality that implies
several relationships between the function and the norm of its gradient. The
proof of these inequalities may be found in Lemma 15 of [21] (a small mistake
appears and we correct the statement with a factor $2$ in our work).

######  Proposition 4.1.

Assume that a function $V$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then:

$\frac{2\mathfrak{c}}{1-r}\left[V^{1-r}(\theta)-\mathrm{min}(V)^{1-r}\right]\leq\|\nabla
V(\theta)\|^{2}_{2}\leq
2L\left[V(\theta)-\mathrm{min}(V)\right],\quad\forall\theta\in\mathbb{R}^{d}.$

It is furthermore possible to assess a minimal and maximal growth property of
any function that satisfies $\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$,
which is necessarily lower and upper bounded by a positive power of the
distance to its minimizer.

######  Proposition 4.2.

Assume that a function $V$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then,
$\forall\theta\in\mathbb{R}^{d}$:

$V^{1+r}(\theta)-\mathrm{min}(V)^{1+r}\geq\frac{(1+r)\mathfrak{c}}{2}\|\theta-\arg\mathrm{min}\,V\|_{2}^{2},$

and

$V(\theta)-\mathrm{min}(V)\leq\frac{L}{2}\|\theta-\arg\mathrm{min}\,V\|_{2}^{2}.$

A straightforward consequence of the first inequality is then

######  Proposition 4.3.

Assume that a function $V$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then,
$\forall\theta\in\mathbb{R}^{d}$:

$V(\theta)\geq
2^{-\frac{r}{1+r}}\left(\mathrm{min}(V)+\left(\frac{(1+r)\mathfrak{c}}{2}\right)^{\frac{1}{1+r}}\|\theta-\arg\mathrm{min}\,V\|_{2}^{\frac{2}{1+r}}\right).$

###  4.2 Properties of $U_{\nu_{n}}$

###### Proof of Proposition 1.1.

First, we observe that if each $\theta\mapsto\nabla\log
p_{\theta}(\mathbf{X}_{i})$ is $L$-Lipschitz and
$\theta\mapsto\nabla\log\pi_{0}$ is $\ell_{0}$-Lipschitz, then the triangle
inequality implies that

$\|\nabla U_{\nu_{n}}(\theta_{1})-\nabla
U_{\nu_{n}}(\theta_{2})\|_{2}\leq(nL+\ell_{0})\|\theta_{1}-\theta_{2}\|_{2}.$

Second, we consider the lower-bound property on the curvature and observe
that:

$\underline{\lambda}_{\nabla^{2}U_{\nu_{n}}(\theta)}=\inf_{e\in\mathbb{R}^{d}:|e|=1}e^{T}(\nabla^{2}U_{\nu_{n}})(\theta)e\geq\frac{1}{n}\sum_{i=1}^{n}\inf_{e\in\mathbb{R}^{d}:|e|=1}e^{T}(\nabla^{2}U_{\mathbf{X}_{i}})(\theta)e.$

The log concavity of the prior yields

$\underline{\lambda}_{\nabla^{2}U_{\nu_{n}}(\theta)}\geq\frac{1}{n}\sum_{i=1}^{n}\underline{\lambda}_{\nabla^{2}(-n\log
p_{\theta}(\mathbf{X}_{i}))}=\sum_{i=1}^{n}\underline{\lambda}_{\nabla^{2}(-\log
p_{\theta}(\mathbf{X}_{i}))}.$

Then, the $\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$ property applied to
each term of the sum above and
$\mathrm{min}_{\theta\in\mathbb{R}^{d}}-\log\pi_{0}(\theta)>0$ yields

$\underline{\lambda}_{\nabla^{2}U_{\nu_{n}}(\theta)}\geq\mathfrak{c}\sum_{i=1}^{n}[-\log
p_{\theta}(\mathbf{X}_{i})]^{-r}\geq\mathfrak{c}n^{r}\sum_{i=1}^{n}U_{\mathbf{X}_{i}}^{-r}(\theta)=\mathfrak{c}n^{1+r}\left(\frac{1}{n}\sum_{i=1}^{n}U_{\mathbf{X}_{i}}^{-r}(\theta)\right).$

From the Jensen inequality, we finally deduce that:

$\underline{\lambda}_{\nabla^{2}U_{\nu_{n}}(\theta)}\geq\mathfrak{c}n^{1+r}\left(\frac{1}{n}\sum_{i=1}^{n}U_{\mathbf{X}_{i}}^{-r}(\theta)\right)\geq\mathfrak{c}n^{1+r}U_{\nu_{n}}^{-r}(\theta).$

We conclude that $U_{\nu_{n}}$ satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$.
For $U_{\mathbf{X}_{i}}$, the proof is similar. ∎

######  Proposition 4.4.

We assume $\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$,
$\mathbf{\mathcal{H}_{\mathrm{min}}}$ and that for any $x$:
$\theta\longmapsto-\log p_{\theta}(x)$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then:

$\|\arg\mathrm{min}\,U_{\nu_{n}}\|_{2}\lesssim_{uc}d^{\frac{1+r}{2}}\log^{\beta(1+r)}(n)\quad\mbox{
and }\quad\mathrm{min}_{\theta\in\mathbb{R}^{d}}\
U_{\nu_{n}}(\theta)\lesssim_{uc}nd\log^{2\beta}(n).$

###### Proof.

Proposition 1.1 shows that $U_{\nu_{n}}$ satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$.
Therefore, we can apply Proposition 4.2 with $\theta=0$ and deduce that:

$\|\arg\mathrm{min}\,U_{\nu_{n}}\|_{2}^{2}\leq\frac{2}{(1+r)\mathfrak{c}n^{1+r}}\left(U_{\nu_{n}}^{1+r}(0)-\mathrm{min}\
U_{\nu_{n}}^{1+r}\right).$

To obtain an upper bound of $U_{\nu_{n}}(0)$ we first bound
$U_{\mathbf{X}_{i}}(0)$ using Proposition 4.2, for all $i$, as follows:

$U_{\mathbf{X}_{i}}(0)\leq\mathrm{min}\
U_{\mathbf{X}_{i}}+\frac{nL+\ell_{0}}{2}\|\arg\mathrm{min}\,U_{\mathbf{X}_{i}}\|_{2}^{2}\lesssim_{uc}d+nd\log^{2\beta}(n)\lesssim_{uc}nd\log^{2\beta}(n),$

then $U_{\nu_{n}}(0)\lesssim_{uc}nd\log^{2\beta}(n)$. We deduce that:

$\|\arg\mathrm{min}\,U_{\nu_{n}}\|_{2}^{2}\leq\frac{2}{(1+r)\mathfrak{c}n^{1+r}}U_{\nu_{n}}^{1+r}(0)\lesssim_{uc}d^{1+r}\log^{2\beta(1+r)}(n).$

The second part comes from $\mathrm{min}\ U_{\nu_{n}}\leq U_{\nu_{n}}(0)$. ∎

##  5 Smoothness and boundedness of the semi-group

###### Proof of Proposition 3.4.

$i)$. The proof relies on an argument set up with a ”fixed” sample
$(X_{1},\ldots,X_{n})$. Our starting point is Proposition 4.2 and the
consequences of the Kurdyka-Łojasiewicz inequality. Since
$\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$ and $\theta\mapsto-\log
p_{\theta}(\mathbf{X}_{i})$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$, then Proposition 1.1 shows
that $U_{\nu_{n}}$ satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$.
Therefore, we can apply Proposition 4.2 and deduce that:

$\|\theta-\arg\mathrm{min}\,U_{\nu_{n}}\|_{2}^{2}\leq\frac{2}{(1+r)\mathfrak{c}n^{1+r}}\left(U_{\nu_{n}}^{1+r}(\theta)-\mathrm{min}\,U_{\nu_{n}}^{1+r}\right)\leq\frac{2}{(1+r)\mathfrak{c}n^{1+r}}U_{\nu_{n}}^{1+r}(\theta).$

If $I_{d}$ refers to the identity map, we use the fact that for any
distribution $\mu$, we have $Var[\mu]\leq\mu[\|I_{d}-a\|_{2}^{2}]$ for any
$a\in\mathbb{R}^{d}$ so that a straightforward consequence with
$a=\arg\mathrm{min}\,U_{\nu_{n}}$ is then:

$Var(\mu_{n})\leq\int_{\mathbb{R}^{d}}\|\theta-\arg\mathrm{min}\,U_{\nu_{n}}\|_{2}^{2}\mathrm{d}\mu_{n}(\theta)\leq\frac{2}{(1+r)\mathfrak{c}n^{1+r}}\mu_{n}[U_{\nu_{n}}^{1+r}].$

We then use the ergodic behaviour of $(\theta_{t})_{t\geq 0}$ and observe that
there exists a constant $C$ independent from $n$ and $d$ such that:

$\displaystyle Var(\mu_{n})$
$\displaystyle\leq\frac{2}{(1+r)\mathfrak{c}n^{1+r}}\lim\sup_{t\geq
0}\mathbb{E}[U_{\nu_{n}}^{1+r}(\theta_{t})]$ $\displaystyle\leq
C\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}},$

where the last inequality comes from Proposition 5.1. We now use the Bobkov
bound on the Poincaré constant for log-concave distribution (see Theorem 1.2
of [4]) and deduce that a universal constant $K$ exists such that:

$C_{P}(\mu_{n})\geq\frac{1}{4K^{2}Var(\mu_{n})}.$

Using the upper bound of the variance, we deduce that a universal $\kappa>0$
exists such that:

$C_{P}(\mu_{n})\geq\frac{\kappa}{\left(d\log^{2\beta}(n)\right)^{(1+r)^{2}}}.$

$ii)$. For the second point, we consider a situation on average over the
samples and the result uses the concentration of the posterior distribution
around its mean. We know from Theorem 3 of [21] that a constant $c>0$ exists
such that:

$\mathbb{E}_{(X_{1},\ldots,X_{n})\sim\mathbb{P}_{\theta_{0}}}[\mathbb{V}ar(\mu_{n})]\leq
c\epsilon_{n,d}^{2},$

with $\epsilon_{n,d}=\left(\frac{Ld\log n}{n}\right)^{\alpha^{-1}}$. The
result follows using the Jensen inequality and the Bobkov bound. ∎

###### Proof of Proposition 3.5.

$i)$. We first establish the boundedness of $f_{0}$. From our assumptions, we
apply Proposition 1.1 and obtain that $U_{\nu_{n}}$ satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$.
If $\theta_{n}^{\star}=\arg\mathrm{min}\,U_{\nu_{n}}$, we then deduce from
Proposition 4.2 that:

$\displaystyle
f_{0}(\theta)=\frac{n_{0}(\theta)}{\mu_{n}(\theta)}=\frac{Z_{n}e^{-\frac{\|\theta\|_{2}^{2}}{2\sigma^{2}}+U_{\nu_{n}}(\theta)}}{(2\pi)^{d/2}\sigma^{d}}$
$\displaystyle\leq\frac{Z_{n}e^{-\frac{\|\theta\|_{2}^{2}}{2\sigma^{2}}+U_{\nu_{n}}(\theta_{n}^{\star})+\frac{(nL+\ell_{0})}{2}\|\theta-\theta_{n}^{\star}\|_{2}^{2}}}{(2\pi)^{d/2}\sigma^{d}}.$
(32)

We compute an upper bound of $Z_{n}$ and use the lower bound of $U_{\nu_{n}}$
induced by Proposition 4.3:

$\displaystyle Z_{n}$
$\displaystyle=\int_{\mathbb{R}^{d}}e^{-U_{\nu_{n}}(\theta)}\mathrm{d}\theta$
$\displaystyle\leq\int_{\mathbb{R}^{d}}e^{-2^{-\frac{r}{1+r}}\left[U_{\nu_{n}}(\theta_{n}^{\star})+n\left(\frac{(1+r)\mathfrak{c}}{2}\right)^{\frac{1}{1+r}}\|\theta-\theta_{n}^{\star}\|_{2}^{\frac{2}{1+r}}\right]}\mathrm{d}\theta$
$\displaystyle\leq
e^{-2^{-\frac{r}{1+r}}U_{\nu_{n}}(\theta_{n}^{\star})}\int_{\mathbb{R}^{d}}e^{-na_{r}\|\theta\|_{2}^{\frac{2}{1+r}}}\mathrm{d}\theta,$

with $a_{r}=\frac{\left((1+r)\mathfrak{c}\right)^{\frac{1}{1+r}}}{2}$. Using
the well known equality:

$\int_{\mathbb{R}^{d}}e^{-a|\theta|^{\ell}}\mathrm{d}\theta=\frac{d\pi^{d/2}\Gamma(d/\ell)}{\ell
a^{d/\ell}\Gamma(d/2+1)},\quad\forall a>0,\quad\forall\ell>0.$

we then deduce with $a=na_{r}$ and $\ell=\frac{2}{1+r}$ that:

$Z_{n}\leq
e^{-2^{-\frac{r}{1+r}}U_{\nu_{n}}(\theta_{n}^{\star})}\int_{\mathbb{R}^{d}}e^{-na_{r}\|\theta\|_{2}^{\frac{2}{1+r}}}\mathrm{d}\theta\leq\frac{d(1+r)}{2}\frac{\pi^{d/2}}{(na_{r})^{\frac{d(1+r)}{2}}}\frac{\Gamma\left(\frac{d(1+r)}{2}\right)}{\Gamma\left(\frac{d}{2}+1\right)}.$

From standard relationships on the Gamma function:

$Z_{n}\leq
2\left(\frac{2^{1+r}\pi}{\mathfrak{c}n^{1+r}}\right)^{\frac{d}{2}}d^{\frac{dr}{2}}.$
(33)

We gather Equations (32) and (33) and obtain that:

$\displaystyle f_{0}(\theta)\leq
2e^{U_{\nu_{n}}(\theta_{n}^{\star})}\left(\frac{2}{\mathfrak{c}\sigma^{2}n^{1+r}}\right)^{\frac{d}{2}}d^{\frac{dr}{2}}e^{-\frac{\|\theta\|_{2}^{2}}{2\sigma^{2}}+\frac{(nL+\ell_{0})}{2}\|\theta-\theta_{n}^{\star}\|_{2}^{2}}.$

For all $\sigma^{2}<\frac{1}{nL+\ell_{0}}$, a straightforward optimization on
$\theta$ yields :

$\|f_{0}\|_{\infty}\leq
2e^{U_{\nu_{n}}(\theta_{n}^{\star})}\left(\frac{2}{\mathfrak{c}\sigma^{2}n^{1+r}}\right)^{\frac{d}{2}}d^{\frac{dr}{2}}\exp\left(\frac{(nL+\ell_{0})}{2(1-\sigma^{2}(nL+\ell_{0}))}\|\theta_{n}^{\star}\|_{2}^{2}\right).$

Then, the choice
$\frac{c_{1}}{nL+\ell_{0}}\leq\sigma^{2}\leq\frac{c_{2}}{nL+\ell_{0}}$, where
$0<c_{1}\leq c_{2}<1$ in $\mathbf{\mathcal{H}_{n_{0}}(L,\ell_{0})}$ and the
bounds of $\|\theta_{n}^{\star}\|_{2}^{2}$ and
$U_{\nu_{n}}(\theta_{n}^{\star})$ in Proposition 4.4 lead to :

$\|f_{0}\|_{\infty}\leq
2\left(\frac{C_{1}d}{n}\right)^{\frac{dr}{2}}\exp\left(C_{2}nd^{1+r}\log^{2\beta(1+r)}(n)\right),$

where $C_{1}$ and $C_{2}$ are universal constants.

$ii)$. This result is an almost standard consequence of the maximum principle
for a Markov semi-group property with a Brownian diffusion. For any bounded
measurable $h>0$, we observe that $P_{t}h>0$ using the Markov property, and we
are led to define $g_{t}$ as the following function $g_{t}:=\sqrt{P_{t}h}$. We
then introduce $\overline{\theta}(t)$ and $\underline{\theta}(t)$ as:

$\overline{\theta}(t)=\arg\mathrm{max}\,g_{t}(\theta)\quad\mbox{ and
}\quad\underline{\theta}(t)=\arg\mathrm{min}\,g_{t}(\theta).$

The chain rule yields:

$\displaystyle\frac{d}{dt}Osc(g_{t})$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\left(g_{t}(\overline{\theta}(t))-g_{t}(\underline{\theta}(t))\right)$
(34) $\displaystyle=$
$\displaystyle\frac{dg_{t}}{dt}(\overline{\theta}(t))+\left\langle\nabla
g_{t}(\overline{\theta}(t)),\frac{d\overline{\theta}(t)}{dt}\right\rangle-\frac{dg_{t}}{dt}(\underline{\theta}(t))-\left\langle\nabla
g_{t}(\underline{\theta}(t)),\frac{d\underline{\theta}(t)}{dt}\right\rangle.$

We compute:

$\displaystyle\frac{dg_{t}}{dt}(\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{P_{t}h}}\frac{dP_{t}h}{dt}(\theta)$ (35)
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{P_{t}h}}\mathcal{G}_{t}P_{t}h(\theta)$
$\displaystyle=$
$\displaystyle\frac{1}{2\sqrt{P_{t}h(\theta)}}\left[-\sum\limits_{i=1}^{n}\langle\nabla_{\theta}P_{t}h(\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle
m_{t}(\mathbf{X}_{i}|\theta)+\Delta_{\theta}P_{t}h(\theta)\right].$

Now, we use that
$\overline{\theta}(t)=\arg\mathrm{max}\,g_{t}=\arg\mathrm{max}\,P_{t}h$, (a
similar argument holds for $\underline{\theta}(t)$):

$\nabla_{\theta}g_{t}(\overline{\theta}(t))=0,\quad\nabla_{\theta}P_{t}h(\overline{\theta}(t))=0\quad\text{and}\quad\Delta_{\theta}P_{t}h(\overline{\theta}(t))\leq
0.$

then:

$\displaystyle\frac{d}{dt}Osc(g_{t})$ $\displaystyle=$
$\displaystyle\frac{dg_{t}}{dt}(\overline{\theta}(t))-\frac{dg_{t}}{dt}(\underline{\theta}(t))$
$\displaystyle=$
$\displaystyle\frac{\Delta_{\theta}P_{t}h}{2\sqrt{P_{t}h}}(\overline{\theta}(t))-\frac{\Delta_{\theta}P_{t}h}{2\sqrt{P_{t}h}}(\underline{\theta}(t))$
$\displaystyle\leq$ $\displaystyle 0.$

We have therefore shown that $Osc(\sqrt{P_{t}h})$ is decreasing in $t\geq 0$,
which ends the proof. ∎

###### Proof of Lemma 2.1.

We proceed as in Proposition 3 of [33] to justify the use of the Lebesgue
dominated convergence theorem for the derivation of the integral involved in
our statement. We can then deduce that:

$\partial_{t}\left\\{\int_{\mathbb{R}^{d}}f_{t}(\theta)\mathrm{d}n_{t}(\theta)\right\\}=\int_{\mathbb{R}^{d}}\partial_{t}\\{f_{t}(\theta)\\}\mathrm{d}n_{t}(\theta)+\int_{\mathbb{R}^{d}}f_{t}(\theta)\partial_{t}\\{n_{t}(\theta)\\}\mathrm{d}\theta.$

We leave the first term unchanged and now focus on the second term:

$\displaystyle\int_{\mathbb{R}^{d}}f_{t}(\theta)\partial_{t}\\{n_{t}(\theta)\\}\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}f_{t}(\theta)\partial_{t}\left\\{\sum\limits_{i=1}^{n}m_{t}(\theta,\mathbf{X}_{i})\right\\}\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}f_{t}(\theta)\partial_{t}\\{m_{t}(\theta,\mathbf{X}_{i})\\}\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}f_{t}(\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta,$

where we used the definition of $n_{t}$ in the first step and Kolmogorov
backward equation (10) in the last one. Since the function $f_{t}(\theta)$
does not depend on $x$, we observe that $\mathcal{L}_{2}f_{t}(\theta)=0$ and
we only need to compute the remaining term $\mathcal{L}_{1}f_{t}(\theta)$:

$\displaystyle\int_{\mathbb{R}^{d}}f_{t}(\theta)\partial_{t}\\{n_{t}(\theta)\\}\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\mathcal{L}_{1}f_{t}(\theta)\
m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\left[-\langle\nabla_{\theta}f_{t}(\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle+\Delta_{\theta}f_{t}(\theta)\right]m_{t}(\theta,\mathbf{X}_{i})\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-\int_{\mathbb{R}^{d}}\sum\limits_{i=1}^{n}\langle\nabla_{\theta}f_{t}(\theta),\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle
m_{t}(\mathbf{X}_{i}|\theta)\mathrm{d}n_{t}(\theta)+\int_{\mathbb{R}^{d}}\Delta_{\theta}f_{t}(\theta)\mathrm{d}n_{t}(\theta)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\mathcal{G}_{t}f_{t}(\theta)\mathrm{d}n_{t}(\theta),$
(38)

where we used the fact that
$m_{t}(\theta,\mathbf{X}_{i})=m_{t}(\mathbf{X}_{i}|\theta)n_{t}(\theta)$. ∎

###  5.1 Moments upper bounds

######  Proposition 5.1.

Assume $\mathbf{\mathcal{H}_{n_{0}}(L,\ell_{0})}$,
$\mathbf{\mathcal{H}_{\pi_{0}}(\ell_{0})}$,
$\mathbf{\mathcal{H}_{\mathrm{min}}}$ and that for each $\mathbf{X}_{i}$,
$\theta\mapsto-\log p_{\theta}(\mathbf{X}_{i})$ satisfies
$\mathbf{\mathcal{H}^{r}_{KL}}(\mathfrak{c},L)$. Then:

  * $i)$

Three positive constants $C_{1}$, $C_{2}$ and $C_{3}$, independent from $n$
and $d$, exist such that for any $t>0$:

$\mathbb{E}\left[e^{\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}(\|\theta_{t}\|_{2}^{2}+1)^{\frac{1}{1+r}}}\right]\leq
C_{1}\left(d\log^{2\beta}(n)\right)^{\frac{r}{1+r}}e^{C_{2}nd\log^{2\beta}(n)}+C_{3}^{d}e^{\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}}.$

  * $ii)$

For any $t>0$ and for any $\alpha\geq 1$:

$\mathbb{E}[U_{\nu_{n}}^{\alpha}(\theta_{t})]\lesssim_{uc}n^{\alpha}\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}.$

###### Proof of $i)$.

We consider the function
$f(\theta)=\exp\left(\frac{a}{2}(\|\theta\|_{2}^{2}+1)^{\rho}\right)$ where
$0<\rho<1$, which is twice differentiable. The gradient of $f$ is computed as:

$\nabla f(\theta)=a\rho(\|\theta\|_{2}^{2}+1)^{\rho-1}f(\theta)\theta.$

The Laplace operator is given as:

$\Delta
f(\theta)=a\rho(\|\theta\|_{2}^{2}+1)^{\rho-2}f(\theta)\left[a\rho(\|\theta\|_{2}^{2}+1)^{\rho}\|\theta\|_{2}^{2}+(d+2\rho-2)\|\theta\|_{2}^{2}+d\right].$

We then deduce that for any $\theta\in\mathbb{R}^{d}$:

$\displaystyle\mathcal{G}_{t}f(\theta)$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\langle\nabla U_{\mathbf{X}_{i}},\nabla
f(\theta)\rangle m_{t}(\mathbf{X}_{i}|\theta)+\Delta f(\theta)$
$\displaystyle=$ $\displaystyle
a\rho(\|\theta\|_{2}^{2}+1)^{\rho-2}f(\theta)\Big{[}-(\|\theta\|_{2}^{2}+1)\sum\limits_{i=1}^{n}\langle\theta,\nabla_{\theta}U_{\mathbf{X}_{i}}(\theta)\rangle
m_{t}(\mathbf{X}_{i}|\theta)$
$\displaystyle+a\rho(\|\theta\|_{2}^{2}+1)^{\rho}\|\theta\|_{2}^{2}+\left(d+2\rho-2\right)\|\theta\|_{2}^{2}+d\Big{]}$
$\displaystyle\leq$ $\displaystyle
a\rho(\|\theta\|_{2}^{2}+1)^{\rho-2}f(\theta)\Big{[}-(\|\theta\|_{2}^{2}+1)\sum_{i=1}^{n}\left(U_{\mathbf{X}_{i}}(\theta)-U_{\mathbf{X}_{i}}(0)\right)m_{t}(\mathbf{X}_{i}|\theta)$
$\displaystyle+a\rho(\|\theta\|_{2}^{2}+1)^{\rho+1}+d\left(\|\theta\|_{2}^{2}+1\right)\Big{]}$
$\displaystyle\leq$ $\displaystyle
a\rho(\|\theta\|_{2}^{2}+1)^{\rho-1}f(\theta)\left[-\sum_{i=1}^{n}\left(U_{\mathbf{X}_{i}}(\theta)-U_{\mathbf{X}_{i}}(0)\right)m_{t}(\mathbf{X}_{i}|\theta)+a\rho(\|\theta\|_{2}^{2}+1)^{\rho}+d\right],$

where we used the convexity of $U_{x}$ for any position $x$.

Let us establish the bounds of $U_{\mathbf{X}_{i}}(\theta)$ and
$U_{\mathbf{X}_{i}}(0)$. We denote by
$\theta_{i}=\arg\mathrm{min}\,U_{\mathbf{X}_{i}}$ and from Hypothesis
$\mathbf{\mathcal{H}_{\mathrm{min}}}$, there exist two positive constants
$\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ independent on $n$ and $d$ such that:

$\mathrm{max}_{i}\,\|\theta_{i}\|_{2}^{2}\leq\mathcal{K}_{1}d\log^{2\beta}(n)\quad\mbox{
and
}\quad\mathrm{max}_{i}\,U_{\mathbf{X}_{i}}(\theta_{i})\leq\mathcal{K}_{2}d.$

We apply Proposition 4.2 to each non-negative function $U_{\mathbf{X}_{i}}$
that satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$,
then we obtain that:

$U_{\mathbf{X}_{i}}(\theta)\geq
n\left[\frac{(1+r)\mathfrak{c}}{2}\right]^{\frac{1}{1+r}}\|\theta-\theta_{i}\|_{2}^{\frac{2}{1+r}}.$

Since $\frac{2}{1+r}>1$, the Jensen inequality yields
$(u+v)^{\frac{2}{1+r}}\leq
2^{\frac{1-r}{1+r}}\left[u^{\frac{2}{1+r}}+v^{\frac{2}{1+r}}\right]$, for all
$(u,v)\in\mathbb{R}_{+}^{2}$ and we deduce that:

$\|\theta-\theta_{i}\|_{2}^{\frac{2}{1+r}}\geq
2^{\frac{r-1}{1+r}}\|\theta\|_{2}^{\frac{2}{1+r}}-\|\theta_{i}\|_{2}^{\frac{2}{1+r}}\geq
2^{\frac{r-1}{1+r}}\|\theta\|_{2}^{\frac{2}{1+r}}-\left(\mathcal{K}_{1}d\log^{2\beta}(n)\right)^{\frac{1}{1+r}}.$

Then we use this inequality to obtain a lower bound of $U_{\mathbf{X}_{i}}$:

$U_{\mathbf{X}_{i}}(\theta)\geq
2n\left[\frac{(1+r)\mathfrak{c}}{8}\right]^{\frac{1}{1+r}}\|\theta\|_{2}^{\frac{2}{1+r}}-n\left[\frac{(1+r)\mathfrak{c}}{2}\right]^{\frac{1}{1+r}}(\mathcal{K}_{1}d\log^{2\beta}(n))^{\frac{1}{1+r}}.$

Moreover an upper bound of $\mathrm{max}\,U_{\mathbf{X}_{i}}(0)$ comes from
Proposition 1.1 and 4.2 as follows:

$U_{\mathbf{X}_{i}}(0)\leq
U_{\mathbf{X}_{i}}(\theta_{i})+\frac{nL+\ell_{0}}{2}\|\theta_{i}\|_{2}^{2}\leq\mathcal{K}_{2}d+\frac{\mathcal{K}_{1}(nL+\ell_{0})d\log^{2\beta}(n)}{2}.$

Using the previous bounds and the fact that
$\sum_{i=1}^{n}m_{t}(\mathbf{X}_{i}|\theta)=1$, it yields:

$\displaystyle\sum_{i=1}^{n}\left(U_{\mathbf{X}_{i}}(\theta)-U_{\mathbf{X}_{i}}(0)\right)m_{t}(\mathbf{X}_{i}|\theta)$
$\displaystyle\geq$ $\displaystyle
2n\left[\frac{(1+r)\mathfrak{c}}{8}\right]^{\frac{1}{1+r}}\|\theta\|_{2}^{\frac{2}{1+r}}-n\left[\frac{(1+r)\mathfrak{c}}{2}\right]^{\frac{1}{1+r}}(\mathcal{K}_{1}d\log^{2\beta}(n))^{\frac{1}{1+r}}-\mathcal{K}_{2}d-\frac{\mathcal{K}_{1}(nL+\ell_{0})d\log^{2\beta}(n)}{2}$
$\displaystyle\geq$
$\displaystyle\frac{n\mathfrak{c}^{\frac{1}{1+r}}}{4}\|\theta\|_{2}^{\frac{2}{1+r}}-n\mathfrak{c}^{\frac{1}{1+r}}(\mathcal{K}_{1}d\log^{2\beta}(n))^{\frac{1}{1+r}}-\mathcal{K}_{2}d-\frac{\mathcal{K}_{1}(nL+\ell_{0})d\log^{2\beta}(n)}{2},$

where we used some uniform upper bounds when $r\in[0,1)$. We then choose
$\rho=\frac{1}{1+r}$ and we deduce that:

$\displaystyle\mathcal{G}_{t}f(\theta)$ $\displaystyle\leq$
$\displaystyle\frac{a}{1+r}(\|\theta\|_{2}^{2}+1)^{-\frac{r}{1+r}}f(\theta)\left[-\frac{n\mathfrak{c}^{\frac{1}{1+r}}}{4}\|\theta\|_{2}^{\frac{2}{1+r}}+n\mathfrak{c}^{\frac{1}{1+r}}(\mathcal{K}_{1}d\log^{2\beta}(n))^{\frac{1}{1+r}}+\mathcal{K}_{2}d\right.$
$\displaystyle\left.+\frac{\mathcal{K}_{1}(nL+\ell_{0})d\log^{2\beta}(n)}{2}+\frac{a}{(1+r)}(\|\theta\|_{2}^{2}+1)^{\frac{1}{1+r}}+d\right]$
$\displaystyle\leq$
$\displaystyle\frac{a}{1+r}(\|\theta\|_{2}^{2}+1)^{-\frac{r}{1+r}}f(\theta)\left[-\left(\frac{n\mathfrak{c}^{\frac{1}{1+r}}}{4}-\frac{a}{(1+r)}\right)\|\theta\|_{2}^{\frac{2}{1+r}}+n\mathfrak{c}^{\frac{1}{1+r}}(\mathcal{K}_{1}d\log^{2\beta}(n))^{\frac{1}{1+r}}\right.$
$\displaystyle\left.+(\mathcal{K}_{2}+1)d+\frac{\mathcal{K}_{1}(nL+\ell_{0})d\log^{2\beta}(n)}{2}+\frac{a}{(1+r)}\right],$

where we used
$(\|\theta\|_{2}^{2}+1)^{\frac{1}{1+r}}\leq\|\theta\|_{2}^{\frac{2}{1+r}}+1$
in the second line.

We now fix $a=\frac{n(1+r)\mathfrak{c}^{\frac{1}{1+r}}}{8}$ and deduce that:

$\displaystyle\frac{\mathcal{G}_{t}f(\theta)}{f(\theta)}$ $\displaystyle\leq$
$\displaystyle\frac{n^{2}\mathfrak{c}^{\frac{2}{1+r}}}{64}(\|\theta\|_{2}^{2}+1)^{-\frac{r}{1+r}}\left[-\|\theta\|_{2}^{\frac{2}{1+r}}+8(\mathcal{K}_{1}d\log^{2\beta}(n))^{\frac{1}{1+r}}+\right.$
(39)
$\displaystyle\left.+\frac{8(\mathcal{K}_{2}+1)d+4\mathcal{K}_{1}(nL+\ell_{0})d\log^{2\beta}(n)}{n\mathfrak{c}^{\frac{1}{1+r}}}+1\right].$

We then study two complementary situations and below, we denote by $K_{n,d}$
the radius of the key compact set involved by the previous Lyapunov
contraction:

$K_{n,d}^{\frac{2}{1+r}}=Cd\log^{2\beta}(n).$

$\bullet$ When $\|\theta\|_{2}$ is large enough ($\|\theta\|_{2}\geq
K_{n,d}$), we observe that a large enough $C>0$ independent from $n$ and $d$
exists such that:

$\|\theta\|_{2}^{\frac{2}{1+r}}\geq
Cd\log^{2\beta}(n)\Longrightarrow\frac{\mathcal{G}_{t}f(\theta)}{f(\theta)}\leq-\frac{n^{2}\left(d\log^{2\beta}(n)\right)^{\frac{1}{1+r}}\mathfrak{c}^{\frac{2}{1+r}}}{128}=-a_{n,d}.$
(40)

$\bullet$ When $\|\theta\|_{2}$ is upper bounded ($\|\theta\|_{2}\leq
K_{n,d}$), we use the upper bound stated in Equation (39) and obtain that a
universal $C_{1}$ (whose value may change from line to line) exists such that
:

$\displaystyle\|\theta\|_{2}^{\frac{2}{1+r}}\leq
Cd\log^{2\beta}(n)\Longrightarrow$ $\displaystyle\mathcal{G}_{t}f(\theta)\leq
C_{1}n^{2}f(\theta)\left[8(\mathcal{K}_{1}d\log^{2\beta}(n))^{\frac{1}{1+r}}+\frac{8(\mathcal{K}_{2}+1)d+4\mathcal{K}_{1}(nL+\ell_{0})d\log^{2\beta}(n)}{n\mathfrak{c}^{\frac{1}{1+r}}}+1\right]$
$\displaystyle\leq
C_{1}n^{2}d\log^{2\beta}(n)\exp\left(\frac{(C+1)c^{\frac{1}{1+r}}nd\log^{2\beta}(n)}{8}\right)\leq
b_{n,d}e^{\delta_{n,d}}.$ (41)

We then use Equations (40) and (41) as follows. We define the function
$\psi_{n,d}$ as $\psi_{n,d}(t)=\mathbb{E}[f(\theta_{t})]$ and use Lemma 2.1:

$\displaystyle\psi^{\prime}_{n,d}(t)$ $\displaystyle=$
$\displaystyle\mathbb{E}[\mathcal{G}_{t}f(\theta_{t})]$ $\displaystyle=$
$\displaystyle\mathbb{E}\left[\mathcal{G}_{t}f(\theta_{t})\left[\mathbbm{1}_{\|\theta_{t}\|_{2}\geq
K_{n,d}}+\mathbbm{1}_{\|\theta_{t}\|_{2}\leq K_{n,d}}\right]\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left[-a_{n,d}f(\theta_{t})\mathbbm{1}_{\|\theta_{t}\|_{2}\geq
K_{n,d}}+b_{n,d}e^{\delta_{n,d}}\mathbbm{1}_{\|\theta_{t}\|_{2}\leq
K_{n,d}}\right]$ $\displaystyle\leq$ $\displaystyle-
a_{n,d}\psi_{n,d}(t)+a_{n,d}\sup_{\|\theta\|_{2}\leq
K_{n,d}}f(\theta)+b_{n,d}e^{\delta_{n,d}}$ $\displaystyle\leq$ $\displaystyle-
a_{n,d}\psi_{n,d}(t)+(a_{n,d}+b_{n,d})e^{\delta_{n,d}}.$

We apply the Gronwall Lemma and obtain that:

$\forall
t>0\qquad\psi_{n,d}(t)\leq\left(1+\frac{b_{n,d}}{a_{n,d}}\right)e^{\delta_{n,d}}+\psi_{n,d}(0)e^{-a_{n,d}t}.$
(42)

Using that $n_{0}$ is a Gaussian distribution, which was fixed in
$\mathbf{\mathcal{H}_{n_{0}}(L,\ell_{0})}$ hypothesis, we find an upper bound
for
$\psi_{n,d}(0)=E[f(\theta_{0})]=\int_{\mathbb{R}^{d}}f(\theta)\mathrm{d}n_{0}(\theta)$
as follows :

$\displaystyle\psi_{n,d}(0)$ $\displaystyle=$
$\displaystyle\left(2\pi\sigma^{2}\right)^{-\frac{d}{2}}\int_{\mathbb{R}^{d}}e^{\frac{a}{2}\left(\|\theta\|_{2}^{2}+1\right)^{\frac{1}{1+r}}-\frac{\|\theta\|_{2}^{2}}{2\sigma^{2}}}\mathrm{d}\theta$
$\displaystyle\leq$
$\displaystyle\left(2\pi\sigma^{2}\right)^{-\frac{d}{2}}e^{\frac{a}{2}}\int_{\mathbb{R}^{d}}e^{-\frac{\|\theta\|_{2}^{2}}{2}\left(\frac{1}{\sigma^{2}}-a\right)}\mathrm{d}\theta,$

if $\sigma^{2}\leq\frac{1}{a}=\frac{8}{n(1+r)\mathfrak{c}^{\frac{1}{1+r}}}$
then the integral above is finite. Since
$c_{2}<1\leq\frac{8L}{(1+r)\mathfrak{c}^{\frac{1}{1+r}}}$, it guarantees
$\sigma^{2}<\frac{1}{a}$, then:

$\displaystyle\psi_{n,d}(0)$ $\displaystyle\leq$
$\displaystyle\left(1-a\sigma^{2}\right)^{-\frac{d}{2}}e^{\frac{a}{2}}$
$\displaystyle\leq$ $\displaystyle
C_{3}^{d}e^{\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}},$

where $C_{3}$ is a constant independent from $n$ and $d$.

Finally, using the value of $a_{n,d}$ and $b_{n,d}$ in (42), we deduce that:

$\mathbb{E}\left[e^{\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}(\|\theta_{t}\|_{2}^{2}+1)^{\frac{1}{1+r}}}\right]\leq
C_{1}\left(d\log^{2\beta}(n)\right)^{\frac{r}{1+r}}e^{C_{2}nd\log^{2\beta}(n)}+C_{3}^{d}e^{\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}},\quad\forall
t>0.$

where $C_{2}$ is another universal constant, which concludes the proof. ∎

###### Proof of $ii)$.

We consider $\alpha>1$ and below, $C>0$ refers to a “constant” independent
from $n$ and $d$, whose value may change from line to line. Our starting point
is the upper bound of the exponential moments obtained in $i)$. Proposition
1.1 shows that $U_{\nu_{n}}$ satisfies
$\mathbf{\mathcal{H}_{KL}^{r}}\left(\mathfrak{c}n^{1+r},nL+\ell_{0}\right)$,
then thanks to Proposition 4.2:

$\mathbb{E}[U_{\nu_{n}}^{\alpha}(\theta_{t})]\leq\mathbb{E}\left[\left(\mathrm{min}\,U_{\nu_{n}}+Cn\|\theta_{t}-\theta^{*}_{n}\|_{2}^{2}\right)^{\alpha}\right]\leq\mathbb{E}\left[\left(\mathrm{min}\,U_{\nu_{n}}+Cn\|\theta^{*}_{n}\|_{2}^{2}+Cn\|\theta_{t}\|_{2}^{2}\right)^{\alpha}\right],$

where $\theta^{*}_{n}=\arg\mathrm{min}\,U_{\nu_{n}}$.

By using Proposition 4.4 and the inequality derived from the Jensen inequality
$(a+b)^{\beta}\leq c_{\beta}(a^{\beta}+b^{\beta})$ for
$(a,b)\in\mathbb{R}_{+}^{2}$ and $\beta\geq 1$, we obtain that:

$\displaystyle\left(\mathrm{min}\,U_{\nu_{n}}+\right.$
$\displaystyle\left.Cn\|\theta^{*}_{n}\|_{2}^{2}+Cn\|\theta_{t}\|_{2}^{2}\right)^{\alpha}$
$\displaystyle\leq
C\left[nd\log^{2\beta}(n)+nd^{1+r}\log^{2\beta(1+r)}(n)+n\|\theta_{t}\|_{2}^{2}\right]^{\alpha}$
$\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+\|\theta_{t}\|_{2}^{2\alpha}\right]$
$\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+k^{-\alpha(1+r)}\log^{\alpha(1+r)}\left(e^{k\|\theta_{t}\|_{2}^{\frac{2}{1+r}}}\right)\right]$
$\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+k^{-\alpha(1+r)}\log^{\alpha(1+r)}\left(e^{\alpha(1+r)-1+k\|\theta_{t}\|_{2}^{\frac{2}{1+r}}}\right)\right].$

The Jensen inequality and the concavity of $x\mapsto\log^{p}(x)$ on
$[e^{p-1},+\infty[$ when $p\geq 1$ yield

$\displaystyle\mathbb{E}[$ $\displaystyle U_{\nu_{n}}^{\alpha}(\theta_{t})]$
$\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+k^{-\alpha(1+r)}\mathbb{E}\left[\log^{\alpha(1+r)}\left(e^{\alpha(1+r)-1+k\|\theta_{t}\|_{2}^{\frac{2}{1+r}}}\right)\right]\right]$
$\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+k^{-\alpha(1+r)}\log^{\alpha(1+r)}\left[\mathbb{E}\left(e^{\alpha(1+r)-1+k\|\theta_{t}\|_{2}^{\frac{2}{1+r}}}\right)\right]\right]$
$\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+k^{-\alpha(1+r)}\left[\alpha(1+r)-1+\log\mathbb{E}\left(e^{k\|\theta_{t}\|_{2}^{\frac{2}{1+r}}}\right)\right]^{\alpha(1+r)}\right]$
$\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+k^{-\alpha(1+r)}\left[\alpha(1+r)-1+\log\mathbb{E}\left(e^{k(\|\theta_{t}\|_{2}^{2}+1)^{\frac{1}{1+r}}}\right)\right]^{\alpha(1+r)}\right],$

where we used in the last inequality that
$\|\theta\|_{2}^{2}\leq\|\theta\|_{2}^{2}+1$.

We then apply $i)$ in Proposition 5.1, we choose
$k=\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}$ and obtain that:

$\displaystyle\mathbb{E}[U_{\nu_{n}}^{\alpha}(\theta_{t})]$ $\displaystyle\leq
Cn^{\alpha}\left[\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}+\frac{1}{n^{\alpha(1+r)}}\left[1+\log\mathbb{E}\left(e^{\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}(\|\theta_{t}\|_{2}^{2}+1)^{\frac{1}{1+r}}}\right)\right]^{\alpha(1+r)}\right]$
$\displaystyle\leq
C\left[n^{\alpha}\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)}\right.$
$\displaystyle\left.+\frac{1}{n^{\alpha
r}}\left[1+\log\left[C_{1}\left(d\log^{2\beta}(n)\right)^{\frac{r}{1+r}}e^{C_{2}nd\log^{2\beta}(n)}+C_{3}^{d}e^{\frac{(1+r)n\mathfrak{c}^{\frac{1}{1+r}}}{16}}\right]\right]^{\alpha(1+r)}\right]$
$\displaystyle\leq Cn^{\alpha}\left(d\log^{2\beta}(n)\right)^{\alpha(1+r)},$

where we used in the previous lines simple algebra and
$\log(a+b)\leq\log(2)+\log(a)+\log(b)$ when $a\geq 1$ and $b\geq 1$. This
concludes the proof. ∎

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