Universality of the Langevin diffusion as scaling limit of a family of Metropolis-Hastings processes

1. Introduction

The Metropolis-Hastings (MH) algorithm and the Langevin diffusion are among the most popular algorithms in the area of Markov chain Monte Carlo (MCMC), see for instance the survey

Roberts and Rosenthal [6] and the references therein. Under a Gaussian proposal with vanishing variance and Gibbs target distribution, Gelfand and Mitter [5] proves that the MH process converges weakly to the Langevin diffusion, thus highlighting the asymptotic connection between this two classes of Markov processes.

With the above classical result in mind, the aim of this paper is to investigate the scaling limit of an entirely different dynamics that we call the second MH process, introduced recently in Choi [2], Choi and Huang [3]. We first motivate our study of the second MH process by offering a geometric perspective: both the classical MH and the second MH minimize certain $L^{1}$ distance, extending the results by Billera and Diaconis [1]. Perhaps surprisingly, for fixed dimension, we shall prove in our main result Theorem 3.1 below that, upon scaling in time and in space, both the classical MH and the second MH converge to an universal Langevin diffusion. On a microscopic level, both the classical MH and the second MH exhibit different Markovian dynamics, yet however on a macroscopic level or on a large time-scale, both processes and their convex combinations converge to an universal rescaled Langevin diffusion. We note that in our paper the dimension is kept fixed, while for the case of optimal scaling of MCMC (see for example Roberts et al. [7]), the weak convergence result therein is obtained by taking the dimension going to infinity.

The rest of this paper is organized as follows. In Section 2, we recall the classical and the second MH process and fix our notations. The geometric interpretation of these processes are proved in Section 2.2. In Section 3, the weak convergence result is proved.

2. Preliminaries

2.1. Metropolis-Hastings generators: $M_{1}$ and $M_{2}$

In this section, we recall the construction of continuous-time Metropolis-Hastings (MH) Markov processes on a general state space $X$ . There are two inputs to the MH algorithm, namely the target distribution and the proposal chain. We refer readers to Roberts and Rosenthal [6] and the references therein for further pointers on this subject. We denote by $μ$ to be our target distribution and $Q$ to be the generator of the proposal Markov jump process. We assume that both $μ$ and $Q (x, \cdot)$ are absolutely continuous with respect to a common sigma-finite reference measure $ν$ on $X$ , and with a slight abuse of notations we still denote their densities by $μ$ and $Q (x, \cdot)$ respectively. Recall that $Q$ is the generator of a Markov jump process in the sense of [4, Chapter $4$ Section $2$ ] if and only if

sup x \in X \int_{y; y \neq x} Q (x, y) ν (d y) < \infty .

With these notations, we can now define the first MH generator as a transformation from $Q$ and $μ$ :

Definition 2.1 (The first MH generator).

Given a target distribution $μ$ on general state space $X$ and a proposal continuous-time Markov jump process with generator $Q$ , the first MH Markov process is a $μ$ -reversible Markov jump process with generator given by $M_{1} = M_{1} (Q, μ)$ , where for bounded $f$

	$M_{1} f (x)$	$= \int_{y; y \neq x} (f (y) - f (x)) M_{1} (x, y) ν (d y),$
	$M_{1} (x, y)$	$:= min {Q (x, y), \frac{μ (y)}{μ (x)} Q (y, x)}, x \neq y .$

Note that

sup x \in X \int_{y; y \neq x} M_{1} (x, y) ν (d y) ⩽ sup x \in X \int_{y; y \neq x} Q (x, y) ν (d y) < \infty .

In view of the earlier work by the author Choi [2], Choi and Huang [3], we would like to study the so-called second MH generator that replaces $min$ by $max$ in Definition 2.1. More precisely, we define it as follows.

Definition 2.2 (The second MH generator).

Given a target distribution $μ$ on general state space $X$ and a proposal continuous-time Markov jump process with generator $Q$ , define

M_{2} (x, y) := max {Q (x, y), \frac{μ (y)}{μ (x)} Q (y, x)}, x \neq y .

(2.1)

sup x \in X \int_{y; y \neq x} M_{2} (x, y) ν (d y) < \infty,

then the second MH Markov process is a $μ$ -reversible Markov jump process with generator given by $M_{2} = M_{2} (Q, μ)$ , where for bounded $f$

M_{2} f (x)

= \int_{y; y \neq x} (f (y) - f (x)) M_{2} (x, y) ν (d y) .

Comparing Definition 2.1 and 2.2, we see that in the former $M_{1}$ is always a generator of Markov jump process, while in the latter additional conditions on $μ$ and $Q$ are required so as to ensure (2.1). In our main results Section 3, we will consider the special case when $Q (x, \cdot)$

is a normal distribution with mean

x

and variance

ϵ

, and

μ

is the Gibbs distribution. Under the usual regularity conditions on

μ

as in Gelfand and Mitter [5], we will see that

M_{2}

as defined is a valid generator of a Markov jump process, see Proposition 3.1 below.

2.2. Geometric interpretation of $M_{1}$ and $M_{2}$

In order to motivate the definition of $M_{1}$ and $M_{2}$ as natural transformations from $Q$ and $μ$ , in this section we offer a geometric interpretation for both $M_{1}$ and $M_{2}$ , extending the results by Billera and Diaconis [1], Choi and Huang [3]. In our result Theorem 2.1 below, we prove that both $M_{1}$ and $M_{2}$ , as well as their convex combinations, minimize certain $L^{1}$ distance between $Q$ and the set of $μ$ -reversible generator of Markov jump processes on $X$ . As a result, in this sense they are natural transformations that maps a given generator $Q$ of Markov jump process to the set of $μ$ -reversible generators of Markov jump process.

We first introduce a few notations and define a metric to quantify the distance between two generators of Markov jump processes. We write $R (μ)$ to be the set of conservative $μ$ -reversible generators of Markov jump processes and $S (X)$ to be the set of generators of Markov jump processes on $X$ . For any $Q_{1}, Q_{2} \in S (X)$ , similar to [1, Section $4$ ] we define a metric $d_{μ}$ on $S (X)$ to be

d_{μ} (Q_{1}, Q_{2}) := \int_{X \times X ∖ Δ} μ (x) | Q_{1} (x, y) - Q_{2} (x, y) | ν (d x) ν (d y),

where $Δ := {(x, x); x \in X}$ is the set of diagonal in $X \times X$ . The distance between $Q$ and $R (μ)$ is then defined to be

(2.2)

d_{μ} (Q, R (μ)) := inf R \in R (μ) d_{μ} (Q, R) .

With the above notations in mind, we are now ready to state our result in this section:

Theorem 2.1.

Suppose that $Q$ and $μ$ are such that (2.1) is satisfied and $M_{2}$ is a generator of Markov jump process. The convex combinations $α M_{1} + (1 - α) M_{2}$ for $α \in [0, 1]$ minimize the distance $d_{μ}$ between $Q$ and $R (μ)$ . That is,

d_{μ} (Q, R (μ)) = d_{μ} (Q, α M_{1} + (1 - α) M_{2}) .

[Proof. ]The proof is inspired by the proof of Theorem $1$ in Billera and Diaconis [1] and Theorem $3.1$ in Choi and Huang [3]. We first define two helpful half spaces:

	$H^{<} = H^{<} (Q, μ)$	$:= {(x, y); μ (x) Q (x, y) < μ (y) Q (y, x)},$
	$H^{>} = H^{>} (Q, μ)$	$:= {(x, y); μ (x) Q (x, y) > μ (y) Q (y, x)} .$

We now show that for $R \in R (μ)$ , $d_{μ} (Q, R) ⩾ d_{μ} (Q, M_{2})$ . First, we note that

d_{μ} (Q, R) ⩾ \int_{(x, y) \in H^{<}} [μ (x) | Q (x, y) - R (x, y) | + μ (y) | Q (y, x) - R (y, x) |] ν (d x) ν (d y) .

As $R$ is $μ$ -reversible, setting $R (x, y) = Q (x, y) + ϵ_{x y}$ gives $R (y, x) = \frac{μ (x)}{μ (y)} (Q (x, y) + ϵ_{x y})$ . Plugging these expressions back yields

	$d_{μ} (Q, N)$	$⩾ \int_{(x, y) \in H^{<}} [μ (x) \| ϵ_{x y} \| + μ (y) ∣ ∣ ∣ Q (y, x) - \frac{μ (x)}{μ (y)} (Q (x, y) + ϵ_{x y}) ∣ ∣ ∣] ν (d x) ν (d y)$
		$= \int_{(x, y) \in H^{<}} [μ (x) \| ϵ_{x y} \| + ∣ ∣ μ (y) Q (y, x) - μ (x) Q (x, y) - μ (x) ϵ_{x y} ∣ ∣] ν (d x) ν (d y)$
		$⩾ \int_{(x, y) \in H^{<}} \| μ (y) Q (y, x) - μ (x) Q (x, y) \| ν (d x) ν (d y) = d_{μ} (Q, M_{2}),$

where we use the reverse triangle inequality $| a - b | ⩾ | a | - | b |$ in the second inequality. Similarly, we can show $d_{μ} (Q, N) ⩾ d_{μ} (Q, M_{1})$ via substituting $H^{<}$ by $H^{>}$ . To see that $d_{μ} (Q, M_{1}) = d_{μ} (Q, M_{2})$ , we have

	$d_{μ} (Q, M_{2})$	$= \int_{(x, y) \in H^{<}} \| μ (y) Q (y, x) - μ (x) Q (x, y) \| ν (d x) ν (d y)$
		$= \int_{(y, x) \in H^{>}} \| μ (y) Q (y, x) - μ (x) Q (x, y) \| ν (d x) ν (d y) = d_{μ} (Q, M_{1}) .$

As for convex combinations of $M_{1}$ and $M_{2}$ , we see that

	$d_{μ} (Q, α M_{1} + (1 - α) M_{2})$	$= (1 - α) \int_{(x, y) \in H^{<}} \| μ (y) Q (y, x) - μ (x) Q (x, y) \| ν (d x) ν (d y)$
		$+ α \int_{(x, y) \in H^{>}} \| μ (y) Q (y, x) - μ (x) Q (x, y) \| ν (d x) ν (d y)$
		$= (1 - α) d_{μ} (Q, M_{2}) + α d_{μ} (Q, M_{1}) = d_{μ} (Q, M_{1}) .$

3. Main results: universality of Langevin diffusion as scaling limit of random walk $M_{1}$ and $M_{2}$

In this section, we specialize into the case of $X = R^{d^{*}}$ with $d^{*} \in N$ , and we take the reference measure $ν$ to be the Lebesgue measure. Let $U : R^{d^{*}} \to R$ be a function satisfying the following regularity assumption:

Assumption 3.1.

$U$ is continuously differentiable, and its gradient $\nabla U$ is bounded and Lipschitz continuous.

Note that the same assumption on $U$ is imposed in Gelfand and Mitter [5] to obtain their weak convergence result that we will briefly recall later in this section. The target distribution $μ$ is the Gibbs distribution at temperature $T > 0$ with density given by

(3.1)

μ (x) = \frac{e^{- U (x) / T}}{\int e^{- U (x) / T} d x}, x \in R^{d^{*}} .

Writing $ϕ_{ϵ}$ to be the density of one-dimensional normal distribution with mean $0$ and variance $ϵ > 0$ , for the proposal Markov jump process, we take $Q_{ϵ} (x, y)$ to be

(3.2)

Q_{ϵ} (x, y) = \frac{1}{d^{*}} d^{*} \sum i = 1 ϕ_{ϵ} (y_{i} - x_{i}) \prod j \neq i δ (y_{j} - x_{j}),

where $δ$ is the Dirac delta function. In words, we pick one of the $d^{*}$ coordinates uniformly at random, say $i$ , and propose a new state at $i$ according to a normal distribution centered at $x_{i}$ and variance $ϵ$ while keeping other coordinates unchanged. Note that $Q_{ϵ} (x, y) = Q_{ϵ} (y, x)$ . If we write $x_{+} := max {x, 0}$ , we define $s_{M_{1}}$ and $s_{M_{2}}$ to be respectively

	$s_{M_{1}} (x, y)$	$:= e^{- (U (y) - U (x))_{+} / T}, s_{M_{1}} (i, x, y_{i}) := s_{M_{1}} ((x_{1}, \dots, x_{d^{}}), (x_{1}, \dots, x_{i - 1}, y_{i}, x_{i + 1}, \dots, x_{d^{}})),$
	$s_{M_{2}} (x, y)$	$:= e^{(U (x) - U (y))_{+} / T}, s_{M_{2}} (i, x, y_{i}) := s_{M_{2}} ((x_{1}, \dots, x_{d^{}}), (x_{1}, \dots, x_{i - 1}, y_{i}, x_{i + 1}, \dots, x_{d^{}})) .$

With the above notations, we can define $M_{1}$ and $M_{2}$ in this setting:

Proposition 3.1 ( $M_{1}$ and $M_{2}$ under Gibbs $μ$ and Gaussian proposal $Q_{ϵ}$ ).

Suppose that $U$ satisfies Assumption 3.1, $μ$ is the Gibbs distribution (3.1) and $Q_{ϵ}$ is the Gaussian proposal (3.2). Then both $M_{1}^{ϵ} = M_{1} (Q_{ϵ}, μ)$ and $M_{2}^{ϵ} = M_{2} (Q_{ϵ}, μ)$ are generators of Markov jump process. Furthermore, for $x \neq y$ ,

	$M_{1}^{ϵ} (x, y)$	$= \frac{1}{d^{}} d^{} \sum i = 1 s_{M_{1}} (i, x, y_{i}) ϕ_{ϵ} (y_{i} - x_{i}) \prod j \neq i δ (y_{j} - x_{j}),$
	$M_{2}^{ϵ} (x, y)$	$= \frac{1}{d^{}} d^{} \sum i = 1 s_{M_{2}} (i, x, y_{i}) ϕ_{ϵ} (y_{i} - x_{i}) \prod j \neq i δ (y_{j} - x_{j}) .$

We write $X^{M_{1}^{ϵ}} = (X^{M_{1}^{ϵ}} (t))_{t ⩾ 0}$ and $X^{M_{2}^{ϵ}} = (X^{M_{2}^{ϵ}} (t))_{t ⩾ 0}$ to be the Markov jump process with generator $M_{1}^{ϵ}$ and $M_{2}^{ϵ}$ respectively.

[Proof. ]We first prove the two formulae for $M_{1}^{ϵ} (x, y)$ and $M_{2}^{ϵ} (x, y)$ . As $Q_{ϵ} (x, y) = Q_{ϵ} (y, x)$ , we have

	$M_{1}^{ϵ} (x, y)$	$= Q_{ϵ} (x, y) min {\frac{μ (y)}{μ (x)}, 1} = Q_{ϵ} (x, y) s_{M_{1}} (x, y) = \frac{1}{d^{}} d^{} \sum i = 1 s_{M_{1}} (i, x, y_{i}) ϕ_{ϵ} (y_{i} - x_{i}) \prod j \neq i δ (y_{j} - x_{j}),$
	$M_{2}^{ϵ} (x, y)$	$= Q_{ϵ} (x, y) max {\frac{μ (y)}{μ (x)}, 1} = Q_{ϵ} (x, y) s_{M_{2}} (x, y) = \frac{1}{d^{}} d^{} \sum i = 1 s_{M_{2}} (i, x, y_{i}) ϕ_{ϵ} (y_{i} - x_{i}) \prod j \neq i δ (y_{j} - x_{j}) .$

Next, we show that $M_{2}^{ϵ}$ is a valid generator of Markov jump process. Let $M$ be an upper bound on

sup x \in R^{d^{*}} | \partial_{x_{i}} U (x) | ⩽ M .

for all $1 ⩽ i ⩽ d^{*}$ . By mean value theorem on $U$ , we have

s_{M_{2}} (i, x, y_{i}) ⩽ e^{M | y_{i} - x_{i} | / T} .

By writing $Z$

to be a normal random variable with mean

0

and variance

ϵ

, this leads to

\int_{y; x \neq y} M_{2}^{ϵ} (x, y) d y ⩽ E (e^{M | Z | / T}) < \infty,

where $E (e^{M | Z | / T})$

is the moment generating function of the half-normal distribution

| Z |

, which is independent of

x

In our main result of this paper, we are primarily interested in the scaling limit of $X^{M_{1}^{ϵ}}$ and $X^{M_{2}^{ϵ}}$ upon scaling in time as $ϵ \to 0$ . The scaling in space is embedded in the proposal $Q_{ϵ}$ . Let $D^{d^{*}} [0, \infty)$ be the space of $R^{d^{*}}$ -valued functions on $[0, \infty)$ that are right continuous with left limit, equipped with the Skorohod topology. We denote the weak convergence of processes in the Skorohod topology by $\Rightarrow$ .

Theorem 3.1 (Universality of the Langevin diffusion as scaling limit of $M_{1}^{ϵ}$ and $M_{2}^{ϵ}$ ).

Suppose that $U$ satisfies Assumption 3.1, and we let $X^{M_{1}^{ϵ}} = (X^{M_{1}^{ϵ}} (t))_{t ⩾ 0}$ and $X^{M_{2}^{ϵ}} = (X^{M_{2}^{ϵ}} (t))_{t ⩾ 0}$ to be the Markov jump process with generator $M_{1}^{ϵ}$ and $M_{2}^{ϵ}$ respectively, both with initial distribution $X^{M_{1}^{ϵ}} (t) = X^{M_{2}^{ϵ}} (t) = X_{0}$ independent of $ϵ$ . Let $X = (X (t))_{t ⩾ 0}$ be the following time-rescaled Langevin diffusion with $X (0) = X_{0}$ and stochastic differential equation given by

(3.3)

d X (t) = - \frac{\nabla U (X (t))}{2 T d^{*}} d t + \frac{1}{\sqrt{d^{*}}} d W (t),

where $(W (t))_{t ⩾ 0}$ is the standard $d^{*}$ -dimensional Brownian motion. Then we have

X^{M_{1}^{ϵ}} (\frac{\cdot}{ϵ}) \Rightarrow X (\cdot), X^{M_{2}^{ϵ}} (\frac{\cdot}{ϵ}) \Rightarrow X (\cdot)

weakly in $D^{d^{*}} [0, \infty)$ as $ϵ \to 0$ .

Remark 3.1.

The weak convergence of $X^{M_{1}^{ϵ}}$ is first proved by Gelfand and Mitter [5]. We shall only prove the case of $X^{M_{2}^{ϵ}}$ in Section 4.1.

Remark 3.2.

Denote by $(˜ X (t))_{t ⩾ 0}$ the Langevin diffusion with initial condition $˜ X (0) = X_{0}$ and stochastic differential equation

d ˜ X (t) = - \nabla U (˜ X (t)) d t + \sqrt{2 T} d W (t) .

Denote the clock process by $τ (t) := t / (2 T d^{*})$ , then $˜ X (τ (\cdot))$ has the same law as $X (\cdot)$ satisfying (3.3). In other words, (3.3) is the Langevin diffusion running at time scale $τ (\cdot)$ .

It is perhaps surprising that both $M_{1}^{ϵ}$ and $M_{2}^{ϵ}$ share the same scaling limit, given that they have entirely different dynamics. In fact, any Markov jump process whose generator is a convex combination of $M_{1}^{ϵ}$ and $M_{2}^{ϵ}$ converges to the same Langevin diffusion:

Corollary 3.1.

Suppose that $U$ satisfies Assumption 3.1, and we let $Y^{ϵ} = (Y^{ϵ} (t))_{t ⩾ 0}$ be the Markov jump process with generator $α M_{1}^{ϵ} + (1 - α) M_{2}^{ϵ}$ , $α \in [0, 1]$ and initial distribution $Y^{ϵ} (t) = X_{0}$ independent of $ϵ$ . Let $X = (X (t))_{t ⩾ 0}$ be the following time-rescaled Langevin diffusion with $X (0) = X_{0}$ and stochastic differential equation given by

d X (t) = - \frac{\nabla U (X (t))}{2 T d^{*}} d t + \frac{1}{\sqrt{d^{*}}} d W (t),

where $(W (t))_{t ⩾ 0}$ is the standard $d^{*}$ -dimensional Brownian motion. Then we have

Y^{ϵ} (\frac{\cdot}{ϵ}) \Rightarrow X (\cdot)

weakly in $D^{d^{*}} [0, \infty)$ as $ϵ \to 0$ .

In view of Theorem 2.1 and Corollary 3.1, we see that on one hand the convex combination of $α M_{1}^{ϵ} + (1 - α) M_{2}^{ϵ}$ may have different dynamics for different $α$ , yet interestingly they all minimize the distance $d_{μ}$ and converge weakly to the same Langevin diffusion.

The rest of the paper is devoted to the proof of Theorem 3.1 and Corollary 3.1 in Section 4.1 and 4.2 respectively.

4. Proofs of main results

4.1. Proof of Theorem 3.1

For notational convenience, we replace $U (\cdot)$ by $U (\cdot) / T$ . In view of Remark 3.1, we only prove the weak convergence of $X^{M_{2}^{ϵ}}$ . We let $G$ to be the generator of $X$ described by (3.3), where for $f \in C^{2} (R^{d^{*}})$ ,

G f (x) = \frac{1}{d^{*}} d^{*} \sum i = 1 \partial_{x_{i}} U (x) \partial_{x_{i}} f (x) + \frac{1}{2 d^{*}} d^{*} \sum i = 1 \partial_{x_{i}}^{2} f (x) .

Note that since the drift $\nabla U$ is Lipschitz continuous, by [4, Chapter $8$ Theorem $2.5$ ], the space of infinitely differentiable functions with compact support $C_{c}^{\infty} (R^{d^{*}})$ forms a core of $G$ . Thus to prove the desired weak convergence, by [4, Chapter $1$ Theorem $6.1$ ] it suffices to prove the uniform convergence of the generator in $C_{c}^{\infty} (R^{d^{*}})$ , that is, for $f \in C_{c}^{\infty} (R^{d^{*}})$ as $ϵ \to 0$ we would like to prove that

(4.1)

sup x \in R^{d^{*}} | (1 / ϵ) M_{2}^{ϵ} f (x) - G f (x) | \to 0.

Define for $x, y \in R^{d^{*}}$ , $⟨ x, y ⟩ := \sum_{i = 1}^{d^{*}} x_{i} y_{i}, ∥ x ∥^{2} := ⟨ x, x ⟩,$

	${^s}_{M_{2}} (x, y)$	$:= e^{⟨ \nabla U (x), x - y ⟩_{+}}, {^s}_{M_{2}} (i, x, y_{i}) := {^s}_{M_{2}} ((x_{1}, \dots, x_{d^{}}), (x_{1}, \dots, x_{i - 1}, y_{i}, x_{i + 1}, \dots, x_{d^{}})),$
	$g (x, y)$

We now present three lemmas that will aid our proof, and their proofs are deferred to Section 4.1.1, 4.1.2 and 4.1.3 respectively. The first auxiliary lemma bounds the distance between $s_{M_{2}}$ and ${^s}_{M_{2}}$ :

Lemma 4.1.

There exists positive constants $M$ and $c_{1}$ that only depend on $U$ and $T$ such that

| s_{M_{2}} (x, y) - {^s}_{M_{2}} (x, y) | ⩽ c_{1} e^{M \sum_{i = 1}^{d^{*}} | y_{i} - x_{i} |} ∥ y - x ∥^{2} .

Consequently, we have

| s_{M_{2}} (i, x, y_{i}) - {^s}_{M_{2}} (i, x, y_{i}) | ⩽ c_{1} e^{M | y_{i} - x_{i} |} | y_{i} - x_{i} |^{2} .

Our next lemma controls the upper bound on Lemma 4.1.

Lemma 4.2.

Recall that $Z$ follows a normal distribution with mean $0$ , variance $ϵ$

and probability density function

ϕ_{ϵ}

. Then for

t \in R

and

ϵ \to 0

we have

	$E (e^{t \| Z \|} \| Z \|^{3})$	$= O (ϵ^{3 / 2}),$
	$E (e^{t \| Z \|} \| Z \|^{4})$	$= O (ϵ^{2}) .$

With Lemma 4.1 and 4.2

, we prove the following estimates on the drift and volatility terms of

M_{2}^{ϵ}

ϵ \to 0

Lemma 4.3.

For $1 ⩽ i ⩽ d^{*}$ , as $ϵ \to 0$ ,

(4.2)	$(1 / ϵ) \int (y_{i} - x_{i}) M_{2}^{ϵ} (x, y) d y$	$= - \partial_{x_{i}} U (x) / (2 d^{*}) + O (ϵ^{1 / 2}),$
(4.3)	$(1 / ϵ) \int (y_{i} - x_{i})^{2} M_{2}^{ϵ} (x, y) d y$	$= 1 / d^{*} + O (ϵ^{1 / 2}),$
(4.4)	$(1 / ϵ) \int (y_{i} - x_{i})^{3} M_{2}^{ϵ} (x, y) d y$	$= O (ϵ^{1 / 2}),$

where the convergence are all uniform in $x \in R^{d^{*}}$ .

We proceed to complete the proof of Theorem 3.1. By Taylor expansion on $f \in C_{c}^{\infty} (R^{d^{*}})$ , there exists $z = α x + (1 - α) y$ such that

	$(1 / ϵ) M_{2}^{ϵ} f (x)$	$= (1 / ϵ) \int (f (y) - f (x)) M_{2}^{ϵ} (x, y) d y$
		$= (1 / ϵ) \int (d^{} \sum i = 1 \partial_{x_{i}} f (x) (y_{i} - x_{i}) + \frac{1}{2} d^{} \sum i, j = 1 \partial_{x_{i}} \partial_{x_{j}} f (x) (y_{i} - x_{i}) (y_{j} - x_{j})) M_{2}^{ϵ} (x, y) d y$

		$= (1 / ϵ) \int (d^{} \sum i = 1 \partial_{x_{i}} f (x) (y_{i} - x_{i}) + \frac{1}{2} d^{} \sum i = 1 \partial_{x_{i}}^{2} f (x) (y_{i} - x_{i})^{2}) M_{2}^{ϵ} (x, y) d y$
		$+ (1 / ϵ) \int (\frac{1}{6} d^{*} \sum i = 1 \partial_{x_{i}}^{3} f (z) (y_{i} - x_{i})^{3}) M_{2}^{ϵ} (x, y) d y$
		$= \frac{1}{d^{}} d^{} \sum i = 1 \partial_{x_{i}} U (x) \partial_{x_{i}} f (x) + \frac{1}{2 d^{}} d^{} \sum i = 1 \partial_{x_{i}}^{2} f (x) + O (ϵ^{1 / 2})$
		$= G f (x) + O (ϵ^{1 / 2}),$

where the fourth equality follows from Lemma 4.3 and the fact that $f$ has compact support. Note that the convergence is uniform in $x$ .

4.1.1. Proof of Lemma 4.1

First, by the Taylor expansion on $U$ and the fact that $U$ is Lipschitz continuous by Assumption 3.1, there exists constant $c_{1}$ such that

| g (x, y) | ⩽ c_{1} ∥ y - x ∥^{2} .

We would like to show the following inequality holds, by considering the possible signs of $U (x) - U (y)$ and $⟨ \nabla U (x), x - y ⟩$ :

(4.5)

1 - e^{⟨ \nabla U (x), x - y ⟩_{+} - (U (x) - U (y))_{+}} ⩽ 1 - e^{- | g (x, y) |} ⩽ | g (x, y) | ⩽ c_{1} ∥ y - x ∥^{2} .

Case 1: $U (x) - U (y) > 0$ , $⟨ \nabla U (x), x - y ⟩ > 0$
In this case, since $- (U (y) - U (x) + ⟨ \nabla U (x), x - y ⟩) ⩽ | g (x, y) |$ , upon rearranging we obtain the leftmost inequality of (4.5).
Case 2: $U (x) - U (y) ⩽ 0$ , $⟨ \nabla U (x), x - y ⟩ ⩽ 0$
The leftmost inequality of (4.5) holds trivially.
Case 3: $U (x) - U (y) > 0$ , $⟨ \nabla U (x), x - y ⟩ ⩽ 0$
Case 4: $U (x) - U (y) ⩽ 0$ , $⟨ \nabla U (x), x - y ⟩ > 0$

$1 - e^{⟨ \nabla U (x), x - y ⟩_{+} - (U (x) - U (y))_{+}} = 1 - e^{⟨ \nabla U (x), x - y ⟩} ⩽ 0 ⩽ 1 - e^{- | g (x, y) |} .$

Similarly, we would like to show the following inequality holds, by considering the possible signs of $U (x) - U (y)$ and $⟨ \nabla U (x), x - y ⟩$ :

(4.6)

1 - e^{- ⟨ \nabla U (x), x - y ⟩_{+} + (U (x) - U (y))_{+}} ⩽ 1 - e^{- | g (x, y) |} ⩽ | g (x, y) | ⩽ c_{1} ∥ y - x ∥^{2} .

Case 1: $U (x) - U (y) > 0$ , $⟨ \nabla U (x), x - y ⟩ > 0$

$1 - e^{- ⟨ \nabla U (x), x - y ⟩_{+} + (U (x) - U (y))_{+}} = 1 - e^{g (x, y)} ⩽ 1 - e^{- | g (x, y) |} .$
Case 2: $U (x) - U (y) ⩽ 0$ , $⟨ \nabla U (x), x - y ⟩ ⩽ 0$
The leftmost inequality of (4.6) holds trivially.
Case 3: $U (x) - U (y) > 0$ , $⟨ \nabla U (x), x - y ⟩ ⩽ 0$
Case 4: $U (x) - U (y) ⩽ 0$ , $⟨ \nabla U (x), x - y ⟩ > 0$

$1 - e^{- ⟨ \nabla U (x), x - y ⟩_{+} + (U (x) - U (y))_{+}} = 1 - e^{- ⟨ \nabla U (x), x - y ⟩} ⩽ 1 - e^{U (x) - U (y) - ⟨ \nabla U (x), x - y ⟩} = 1 - e^{g (x, y)} = 1 - e^{- | g (x, y) |} .$

As a result, collecting both (4.5) and (4.6) we have

	$s_{M_{2}} (x, y) - {^s}_{M_{2}} (x, y)$	$= e^{(U (x) - U (y))_{+}} (1 - e^{⟨ \nabla U (x), x - y ⟩_{+} - (U (x) - U (y))_{+}}) ⩽ c_{1} e^{M \sum_{i = 1}^{d^{*}} \| y_{i} - x_{i} \|} ∥ y - x ∥^{2},$
	${^s}_{M_{2}} (x, y) - s_{M_{2}} (x, y)$	$= e^{⟨ \nabla U (x), x - y ⟩_{+}} (1 - e^{- ⟨ \nabla U (x), x - y ⟩_{+} + (U (x) - U (y))_{+}}) ⩽ c_{1} e^{M \sum_{i = 1}^{d^{*}} \| y_{i} - x_{i} \|} ∥ y - x ∥^{2},$

where the inequalities in the two equations above follow from mean value theorem and the fact that $\nabla U$ is bounded by Assumption 3.1.

4.1.2. Proof of Lemma 4.2

Let $h (t) := E (e^{t | Z |})$ and we write $Φ (\cdot)$

to be the cumulative distribution function of standard normal. We also denote

ϕ_{ϵ}^{(i)}

to be the

i

-th derivative of

ϕ_{ϵ}

. By brute force differentiation and integration we note that

	$h (t)$	$= 2 e^{\frac{ϵ t^{2}}{2}} (1 - Φ (- \sqrt{ϵ} t)),$
	$\partial_{t} h (t)$	$= E (e^{t \| Z \|} \| Z \|) = 2 \sqrt{ϵ} e^{\frac{ϵ t^{2}}{} 2} ϕ$

	$d_{μ} (Q, N)$	$⩾ \int_{(x, y) \in H^{<}} [μ (x) \| ϵ_{x y} \| + μ (y) ∣ ∣ ∣ Q (y, x) - \frac{μ (x)}{μ (y)} (Q (x, y) + ϵ_{x y}) ∣ ∣ ∣] ν (d x) ν (d y)$
		$= \int_{(x, y) \in H^{<}} [μ (x) \| ϵ_{x y} \| + ∣ ∣ μ (y) Q (y, x) - μ (x) Q (x, y) - μ (x) ϵ_{x y} ∣ ∣] ν (d x) ν (d y)$
		$⩾ \int_{(x, y) \in H^{<}} \| μ (y) Q (y, x) - μ (x) Q (x, y) \| ν (d x) ν (d y) = d_{μ} (Q, M_{2}),$

Universality of the Langevin diffusion as scaling limit of a family of Metropolis-Hastings processes

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