SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Sifan Liu
sfliu@stanford.edu
Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful algorithms for sampling from complex high-dimensional distributions. To sample from a distribution with density  $\pi(\theta) \propto \exp(-U(\theta))$ , LMC iteratively generates the next sample by taking a step in the gradient direction  $\nabla U$  with added Gaussian perturbations. Expectations w.r.t. the target distribution  $\pi$  are estimated by averaging over LMC samples. In ordinary Monte Carlo, it is well known that the estimation error can be substantially reduced by replacing independent random samples by quasi-random samples like low-discrepancy sequences. In this work, we show that the estimation error of LMC can also be reduced by using quasi-random samples. Specifically, we propose to use completely uniformly distributed (CUD) sequences with certain low-discrepancy property to generate the Gaussian perturbations. Under smoothness and convexity conditions, we prove that LMC with a low-discrepancy CUD sequence achieves smaller error than standard LMC. The theoretical analysis is supported by compelling numerical experiments, which demonstrate the effectiveness of our approach.
Sampling from probability distributions is a crucial task in both statistics and machine learning. However, when the target distribution does not permit exact sampling, researchers often rely on Markov chain Monte Carlo (MCMC) methods. These techniques simulate a Markov chain that converges to the target distribution as its stationary distribution. Recently, MCMC samplers based on discretizing the continuous-time Langevin diffusion have become popular, due to its ease of implementation and ability to handle stochastic gradients (Welling and Teh, 2011).
The primary focus of this work is on the quality of samples generated by Langevin Monte Carlo (LMC) algorithms in terms of estimating the expectation  $\mathbb{E}_{\theta \sim \pi}[f(\theta)]$  for some integrand  $f$  by sample averages. In the context of Bayesian inference, the target distribution  $\pi$  is typically the posterior distribution, and computing the posterior expectation, posterior variance, or confidence intervals are of great interest. In the context of post-selection inference, the target distribution  $\pi$  is the probability distribution conditioned on the selection event, and computing the selection-adjusted p-value is the main task. LMC has been widely used in this problem as well (Markovic and Taylor, 2016; Shi et al., 2022). In all these situations, the accuracy of the sample average estimator is critical and affects the downstream data analysis.
In traditional Monte Carlo sampling, it is well known that using quasi-Monte Carlo (QMC) samples, instead of independent and identically distributed (i.i.d.) random samples, can lead to significant error reduction. So it is natural to ask whether we can apply QMC techniques to improve Langevin Monte Carlo sampling as well. In this work, we introduce the Langevin quasi-Monte Carlo (LQMC) algorithm, which replaces the i.i.d. random inputs in the LMC algorithm with quasi-random numbers.
Figure 1: Scatter plots of 251 points generated from Mersenne Twister 19937 (left) and 251 points generated from a linear congruential generator (LCG) of period 251. Points from an entire period of a pseudo-random number generator (right) fill the unit square more evenly than the same number of points from a PRNG with a larger period (left).
These quasi-random numbers are carefully designed to sample from the target distribution more evenly and more balanced, leading to improved estimation accuracy.
Not all quasi-Monte Carlo point sets are suitable for simulating Markov chains. Suppose the Markov chain is driven by a sequence of uniform random vectors in the unit cube. A sufficient condition for the sequence is known as completely uniformly distributed (CUD). In our implementation of the driving sequence, we use an entire period of a pseudo-random number generator (PRNG). While modern computer simulations often use PRNGs with a large period, such as Mersenne Twister with a period of  $2^{19937} - 1$ , our approach runs through the entire period of a PRNG with a relatively small period in the LMC algorithm. The advantage of using an entire period of a PRNG is that the points are more evenly distributed, which is more desirable for numerical integration. We illustrate the balancing property of an entire PRNG in Figure 1.
The main contributions of this paper are threefold. First, we propose a novel technique of using quasi-random numbers in Langevin-type algorithms, which can be applied to a wide range of such algorithms by substituting i.i.d. random numbers with a sequence of quasi-random numbers. The quasi-random numbers are constructed similarly as usual PRNGs, therefore no extra computational complexity is required. Second, we evaluate the performance of the proposed LQMC algorithm in a variety of numerical experiments, demonstrating that it can significantly reduce the mean squared error (MSE) of traditional LMC by a factor ranging from 2 to 500, depending on the problem. Finally, we provide theoretical analysis showing that LQMC can reduce the Monte Carlo part of the error from  $O(n^{-1/2})$  to  $O(n^{-1+\delta})$  for any  $\delta > 0$  in situations where the Markov chain is strongly contracting and the integrand function  $f$  is sufficiently regular. This error reduction is consistent with the usual improvement achieved by using quasi-Monte Carlo in place of plain Monte Carlo.
The rest of the paper is organized as follows. In Section 2, we provide some background on LMC and QMC, followed by a review of related work. Section 3 describes the LQMC algorithm and its implementation details. In Section 4, we present theoretical guarantees for the proposed method. Finally, in Section 5, we provide empirical results to evaluate the performance of LQMC and compare it with the standard LMC algorithm.
This section provides some background on Langevin Monte Carlo and quasi-Monte Carlo.
2.1 Langevin Monte Carlo
Suppose we want to sample from the target distribution  $\pi (\theta)\propto \exp (-U(\theta))$  where  $\theta \in \mathbb{R}^d$  and  $U$  is known as the potential function. LMC algorithms are based on Euler-Maruyama discretization of the Langevin diffusion  $\theta (t)$ , which satisfies the stochastic differential equation
$$
\mathrm {d} \theta (t) = - \nabla U (\theta (t)) \mathrm {d} t + \sqrt {2} \mathrm {d} W _ {t}, \tag {1}
$$
where  $\{W_t\}_{t \geq 0}$  is a  $d$ -dimensional standard Brownian motion. Under mild technical conditions, the Langevin diffusion  $\theta(t)$  has  $\pi$  as its unique invariant distribution (Roberts and Tweedie, 1996). With a discretization step size  $h$ , LMC updates the sample  $\theta_k$  by
$$
\theta_ {k + 1} \leftarrow \theta_ {k} - h \nabla U \left(\theta_ {k}\right) + \sqrt {2 h} \xi_ {k + 1} \tag {2}
$$
where  $\xi_k \stackrel{iid}{\sim} \mathcal{N}(0, I_d)$ .
In many applications, we are interested in computing the expectation  $\mu \coloneqq \mathbb{E}_{\theta \sim \pi}[f(\theta)]$  over  $\pi$  for some  $\pi$ -integrable function  $f$ . The LMC estimator of  $\mu$  is the sample average
$$
\hat {\mu} _ {n} = \frac {1}{n} \sum_ {k = 1} ^ {n} f (\theta_ {k}),
$$
where  $n$  is the number of iterations.
Teh et al. (2016) provide an asymptotic bias-variance decomposition of the MSE of the weighted average  $\frac{\sum_{k=1}^{n} h_k f(\theta_k)}{\sum_{k=1}^{n} h_k}$  and show that the optimal step size scales as  $h_k \asymp k^{-1/3}$ , leading to an MSE of order  $O(n^{-2/3})$ . Here  $h_k$  is the step size used at the  $k$ -th iteration. Vollmer et al. (2016) generalize this result to the non-asymptotic setting with a constant step size  $h$ . They show that the MSE is of order  $O(h^2 + \frac{1}{nh})$ , where  $h^2$  corresponds to the squared bias and  $\frac{1}{nh}$  corresponds to the variance.
2.2 Quasi-Monte Carlo
QMC is an alternative to Monte Carlo for numerical integration and is well-known for having much higher accuracy than Monte Carlo. QMC is primarily designed to numerically evaluate the integral  $\mu = \int_{[0,1]^d}f(\mathbf{u})\mathrm{d}\mathbf{u}$ . It estimates  $\mu$  by taking points  $\mathbf{u}_i\in [0,1]^d$  and let the estimator be
$$
\hat {\mu} = \frac {1}{n} \sum_ {i = 1} ^ {n} f (\mathbf {u} _ {i}).
$$
Unlike Monte Carlo which takes  $\mathbf{u}_i$  to be identically independently distributed (i.i.d.), QMC constructs the point set  $\{\mathbf{u}_i\}_{i=1}^n$  that aims to minimize the star discrepancy
$$
D _ {n} ^ {*} = D _ {n} ^ {*} \left(\mathbf {u} _ {1}, \dots , \mathbf {u} _ {n}\right) = \sup  _ {\mathbf {a} \in [ 0, 1 ] ^ {d}} \left| \frac {1}{n} \sum_ {i = 1} ^ {n} 1 \left\{\mathbf {u} _ {i} \in [ \mathbf {0}, \mathbf {a}) \right\} - \prod_ {j = 1} ^ {d} a _ {j} \right|. \tag {3}
$$
The star discrepancy measures the uniformity of the point sets by comparing the fraction of points inside  $[0, a)$  and the volume  $\prod_{j=1}^{d} a_j$ , taking supreme over all the rectangles inside  $[0, 1]^d$  anchored at  $0$ . QMC can generate points with  $D_n^* = O(n^{-1} (\log n)^{d-1})$ , thus QMC is also known as low-discrepancy sequence. Commonly used QMC points include Sobol' sequence (Sobol', 1967), Niederreiter's sequence (Niederreiter, 1987), Halton's sequence, and lattice rules. For a comprehensive survey, we refer to the monograph Dick and Pillichshammer (2010). If the integrand  $f$  has bounded variation in the sense of Hardy and Krause  $\| f \|_{\mathrm{HK}}$ , then the Koksma-Hlawka inequality (see e.g. Dick and Pillichshammer (2010)) bounds the integration error by
$$
\left| \hat {\mu} - \mu \right| \leq D _ {n} ^ {*} \cdot \| f \| _ {\mathrm {H K}} \leq O \left(n ^ {- 1} (\log n) ^ {d - 1}\right). \tag {4}
$$
While the Koksma-Hlawka inequality shows that QMC is asymptotically better than usual Monte Carlo, it doesn't provide a practical way to estimate the error. Moreover, integrands might have infinite Hardy-Krause variation.
One can apply randomization techniques to QMC to address both problems. Common randomization techniques include random shifts (Cranley and Patterson, 1976) and scrambling (Owen, 1995). For RQMC samples  $\mathbf{u}_1,\ldots ,\mathbf{u}_n$ , each  $\mathbf{u}_i\sim \mathrm{Unif}([0,1]^d)$  individually but they have the low-discrepancy property collectively with probability 1. One can estimate the error by multiple independent random replicates. For sufficiently smooth  $f$ , the scrambled Sobol' sequence has variance  $O(n^{-3}(\log n)^{d - 1})$  (Owen, 1997a,b).
2.3 Related work
The first attempt to apply quasi-random numbers to simulate stochastic differential equations was made by Hofmann and Mathé (1997). They showed that if a numerical scheme is weakly convergent with i.i.d. samples, then using completely uniformly distributed (CUD) sequences also leads to consistent estimation. They also demonstrated that certain low-discrepancy sequences are not suitable for simulating SDEs. There have also been some efforts to apply QMC to MCMC. Owen and Tribble (2005) proposed to apply CUD sequences to a Metropolis algorithm and showed that the method is consistent in problems with finite state spaces. Chen et al. (2011) generalized the consistency result to continuous state spaces under the assumption that the Markov chain is a contraction. More recently, Dick et al. (2016); Dick and Rudolf (2014) proved that there exists constructions of the driving sequence  $\{\mathbf{u}_k\}_{k\geq 1}$  such that the discrepancy between the empirical distribution of MCMC samples and the target distribution is bounded by  $O(n^{-1 / 2}(\log n)^{1 / 2})$ , the same rate achieved by random inputs. Another line of applying QMC to Markov chains is known as array-RQMC proposed by L'Ecuyer et al. (2008). Array-RQMC runs in parallel multiple Markov chains, and each iteration involves a complicated reordering of the states so that the low-discrepancy among the chains is maintained. Empirically, it achieves significantly smaller estimation error than usual MCMC, but theoretical guarantees remain a challenging open problem.
There has been a growing interest in using QMC techniques in various machine learning tasks, such as variational inference (Buchholz et al., 2018; Liu and Owen, 2021), policy learning and evaluation (Arnold et al., 2022), reinforcement learning with evolution strategies (Choromanski et al., 2019; Rowland et al., 2018), compression of large datasets (Dick and Feischl, 2021), example selection in stochastic gradient descent (SGD) (Lu et al., 2021), and deep learning for solving partial differential equations (Longo et al., 2021).
Numerous efforts have been devoted to improving LMC and stochastic gradient Langevin dynamics (SGLD). To overcome the instability of Euler-Maruyama discretization, various numerical schemes have been proposed, including higher-order integrators (Chen et al., 2015), underdamped LMC (Cheng et al., 2018), and stochastic Runge-Kutta diffusion (Li et al., 2019). For SGLD, variance reduction techniques such as SAGA and SVGR (Dubey et al., 2016) and control variates (Baker et al., 2019) have been proposed. LMC also provides a useful perspective for optimization, as demonstrated by the analyses in Chen et al. (2016); Dalalyan (2017); Raginsky et al. (2017); Xu et al. (2018); Erdogdu et al. (2018). Our contribution is orthogonal to all the aforementioned work, as our algorithm only modifies the random numbers used in the algorithm. Therefore, our method can be combined with other algorithms without interference.
3 QMC for LMC
In the LMC algorithm, we can think of the Markov chain as being driven by a sequence of uniform variables  $\mathbf{u}_k$  in the unit cube  $[0,1]^d$ . For instance, the Gaussian perturbation can be represented as  $\xi_k = \Phi^{-1}(\mathbf{u}_k)$ , where  $\Phi^{-1}$  denotes the inverse Gaussian CDF applied element-wise to  $\mathbf{u}_k$ . If a stochastic gradient is employed, the randomness associated with the stochastic gradient can also be expressed as uniform variables. Therefore, we can write the transition of the Markov chain as  $\theta_{k + 1} = \psi (\theta_k,\mathbf{u}_{k + 1})$ . In typical computer experiments,  $\mathbf{u}_k$  are not really i.i.d. but are deterministic pseudo-random numbers. In this section, we will describe an alternative method of generating the pseudo-random numbers  $\mathbf{u}_k$ , which are carefully constructed and can lead to more accurate sample averages.
The idea here is to use point sets that are more evenly distributed such as QMC points, which can lead to significant improvement in the usual Monte Carlo estimation. However, caution is required when using QMC points to simulate an SDE like (1). This is because the correlation between successive QMC samples may introduce undesired behavior in the Markov chain, as demonstrated in (Tribble, 2007, Section 3.2). To avoid the dependence among successive values, we require that the blocks of points  $(v_{i}, v_{i+1}, \ldots, v_{i+d-1})$  for any lag  $d$  are uniformly distributed. This notion of uniformity is formally known as completely uniformly distributed (CUD, Korobov (1948)), which we define next.
We say an infinite sequence  $\{\mathbf{u}_i\}_{i=1}^{\infty} \subseteq [0,1]^d$  is uniformly distributed on  $[0,1]^d$  if the star discrepancy  $D^{*}(\{\mathbf{u}_i\}_{i=1}^{n})$  goes to 0 as  $n \to \infty$ , where the star discrepancy is defined in Equation 3.
Definition 3.1 (Completely uniformly distributed sequence (CUD)). An infinite sequence  $\{v_{i}\}_{i = 0}^{\infty}\subset [0,1]$  is called completely uniformly distributed, if for all positive integer  $d$ , the sequence  $\{(v_{k},\ldots ,v_{k + d - 1})\}_{k = 0}^{\infty}\subseteq \mathbb{R}^{d}$  is uniformly distributed on  $[0,1]^d$ . A triangular array  $\mathbf{v}_n = (v_{n,1},\dots,v_{n,N_n})$  is called array-CUD, if for all positive integer  $d$ ,  $D^{*}((v_{n,1},\dots,v_{n,d}),(v_{n,2},\dots,v_{n,d + 1}),\dots,(v_{n,N_n - d + 1},\dots,v_{n,N_n}))\to 0$  as  $n\to \infty$ ,  $N_{n}\to \infty$ .
In other words, the subsequent  $d$ -tuples in a CUD sequence are uniformly distributed in the  $d$ -dimensional unit cube for any positive dimension  $d$ . Now we are ready to present the main algorithm.
3.1 LQMC algorithm
Let  $\{v_{i}\}_{i = 0}^{\infty}$  be a CUD sequence. Let  $\mathbf{u}_k = (v_{kd},\dots ,v_{(k + 1)d - 1})\in \mathbb{R}^d$  be the  $k$ -th non-overlapping  $d$ -tuple from the sequence  $(k\geq 0)$ . A CUD sequence is often constructed deterministically. They can further be randomized using the Cranley-Patterson (i.e. random shift) rotation (Cranley and Patterson, 1976)
$$
\mathbf {u} _ {k} \leftarrow \mathbf {u} _ {k} + \Delta \mod 1,
$$
where  $\Delta \sim \mathrm{Unif}([0,1]^d)$ . The Cranley-Patterson rotation randomly shifts each dimension of  $\mathbf{u}_k$  by a uniform random number separately. Then each  $\mathbf{u}_k$  is uniformly distributed on  $[0,1]^d$ . If we apply the inverse Gaussian CDF to each coordinate of  $\mathbf{u}_k$ , then  $\Phi^{-1}(\mathbf{u}_k) \sim \mathcal{N}(0, I_d)$ . In the Langevin-type algorithms, we will let  $\xi_k = \Phi^{-1}(\mathbf{u}_k)$  and use  $\xi_k$  as the Gaussian perturbation in the  $k$ -th iteration. Specifically, each iteration takes the form
$$
\theta_ {k + 1} = \theta_ {k} - h \nabla U (\theta_ {k}) + \sqrt {2 h} \cdot \Phi^ {- 1} (\mathbf {u} _ {k + 1}), \quad k \geq 0.
$$
Thus the transition map is  $\psi (\theta ,\mathbf{u}) = \theta -h\nabla U(\theta) + \sqrt{2h}\Phi^{-1}(\mathbf{u})$  . In practice, we can only run finite many iterations. In the following, we will describe how to construct a finite CUD sequence and feed it into the LMC algorithm.
3.2 Construction of CUD sequences
A finite CUD (array-CUD) sequence is often implemented by using an entire period of a pseudo random number generator with a small period (Tribble, 2007). There exist other constructions of CUD sequences. For further details, interested readers can refer to Levin (1999). We propose to use the linear-feedback shift register (LFSR) provided in Chen (2011), because it has demonstrated good performance and the computational effort required is comparable to other commonly used PRNGs.
The binary Galois LFSR (Tausworth the generator, Tausworth (1965)) of order  $m$  updates the states  $b_{i} \in \{0,1\}$  recursively by
$$
b _ {i} = \sum_ {j = 0} ^ {m - 1} a _ {j} b _ {i - m + j} \mod 2, \quad i \geq m
$$
with initial states  $b_{0}, b_{1}, \ldots, b_{m-1}$  pre-specified. The  $m$ -tuple  $(b_{i}, b_{i+1}, \ldots, b_{i+m-1}) \in \mathrm{GF}(2)^{m}$  can only take  $2^{m}$  different values. If there is an  $m$ -tuple that is all zero, then all  $b_{i}$ 's in this sequence must be zero. So the period of the sequence  $\{b_{i}\}_{i \geq 0}$  is at most  $n = 2^{m} - 1$ . Moreover, the period is exactly equal to  $2^{m} - 1$  if and only if the characteristic polynomial
$$
x ^ {m} + a _ {m - 1} x ^ {m - 1} + \dots + a _ {1} x + a _ {0}
$$
is a primitive polynomial over  $\mathrm{GP}(2)$  (Niederreiter, 1992, Lemma 9.1). Given the states  $\{b_i\}_{i\geq 0}$  and an offset  $s > 0$  such that  $\gcd (s,2^m -1) = 1$ ,  $v_{i}$  is computed with
$$
v _ {i} = \sum_ {j = 0} ^ {m - 1} b _ {s i + j} 2 ^ {- j - 1}, \quad i = 0, 1, \dots , 2 ^ {m} - 2.
$$
That is, for each  $i$ , we take the  $m$ -tuple  $(b_{si+j})_{0 \leq j < m}$  and interpret it as the binary expansion of  $v_i$ . For the next step, we jump  $s$  bits ahead in the sequence  $\{b_i\}_{i \geq 0}$  and use the  $m$ -tuple starting from  $b_{s(i+1)}$ . Chen (2011) provided a table of the LFSR generators for  $10 \leq m \leq 32$ . They searched the offsets so that the LFSR has good equi-distributed properties. Our experiments use the LFSR generators listed there.
Given the sequence  $\{v_{i}\}_{i = 0}^{n - 1}$  of length  $n$ , we repeat it  $d$  times and arrange  $v_{i}$ 's in the following  $n\times d$  matrix
$$
\left( \begin{array}{c c c c} v _ {0} & v _ {1} & \dots & v _ {d - 1} \\ v _ {d} & v _ {d + 1} & \dots & v _ {2 d - 1} \\ \vdots & \vdots & \ddots & \vdots \\ v _ {(n - 1) d} & v _ {(n - 1) d + 1} & \dots & v _ {n d - 1} \end{array} \right). \tag {5}
$$
We run the LMC algorithm  $n = 2^m - 1$  iterations. The  $k$ -th uniform vector  $\mathbf{u}_k$  is the  $k$ -th row of the above matrix. The procedure is summarized in Algorithm 1.
Algorithm 1 Langevin quasi-Monte Carlo (LQMC)
Input: Number of iterations  $n = 2^m - 1$  such that  $\gcd(2^m - 1, d) = 1$ , step size  $h$ , initial value  $\theta_0$ . Generate an LFSR sequence  $\{v_i\}_{i \geq 0}$  of period  $2^m - 1$ .
Let  $\mathbf{u}_k = (v_{(k - 1)d},\dots ,v_{kd - 1})\in [0,1]^d$  , for  $1\leq k\leq n$
Apply Cranley-Patterson rotation (random shift) to  $\mathbf{u}_k$ 's.
for  $k\gets 1,\dots ,n$  do
$$
\theta_ {k} \leftarrow \theta_ {k - 1} - h \nabla U (\theta_ {k - 1}) + \sqrt {2 h} \Phi^ {- 1} (\mathbf {u} _ {k})
$$
Output:  $\theta_{1},\ldots ,\theta_{n}$
If  $\gcd(n, d) = 1$ , then each column of the matrix (5) contains no repeated values. This means that among the  $n = 2^m - 1$  iterations of the LQMC algorithm, each dimension uses one value in each sub-interval  $\left(\frac{k}{2^m}, \frac{k + 1}{2^m}\right]$  at most once ( $0 \leq k \leq 2^m - 1$ ). This perfect one-dimensional stratification is one of the reasons why CUD may achieve smaller estimation error than pseudo-random numbers. If  $\gcd(n, d) > 1$ , then we take  $d'$  to be the smallest integer greater than  $d$  and co-prime with  $n$ . We then create the matrix in (5) similarly but with  $d'$  columns. In the LQMC algorithm, we take  $\mathbf{u}_k$  to be the  $k$ -th row of the matrix but only use the first  $d$  coordinates.
Algorithm 1 may seem to be restricted by having a fixed number of iterations,  $n = 2^m - 1$ . However, in practice, the LQMC algorithm can be started with an initial value of  $m$ . If the chain does not converge after  $2^m - 1$  iterations, one can continue the chain with another freshly generated LFSR, possibly with a larger period. This allows for flexibility in adjusting the number of iterations based on the convergence of the chain. Additionally, if a burn-in period is required, one can first run the algorithm with an LFSR of a small period to serve as the burn-in stage and then continue with a larger LFSR. Furthermore, running multiple chains with independent random shifts is embarrassingly parallel. We present the algorithm in the form of the basic LMC algorithm with accurate gradient and constant learning rate. However, as we noted previously, other Langevin-type algorithms can also utilize the CUD sequence directly by substituting the pseudo-random numbers with the LFSR sequence.
Here we study the estimation error  $|n^{-1}\sum_{k = 1}^{n}f(\theta_k) - \pi (f)|$  of LQMC for some test function  $f$  that is 1-Lipschitz and bounded. As the first attempt to prove the convergence rate of using QMC in LMC, we impose the relatively strong conditions of smoothness and convexity.
Assumption 1. The potential function  $U$  is  $L$ -smooth
$$
\| \nabla U (\theta) - \nabla U \left(\theta^ {\prime}\right) \| _ {2} \leq L \| \theta - \theta^ {\prime} \| _ {2}, \quad \forall \theta , \theta^ {\prime},
$$
and  $M$ -strongly convex
$$
U \left(\theta^ {\prime}\right) \geq U (\theta) + \nabla U (\theta) ^ {\intercal} \left(\theta^ {\prime} - \theta\right) + \frac {M}{2} \| \theta^ {\prime} - \theta \| _ {2} ^ {2}, \quad \forall \theta , \theta^ {\prime}.
$$
We will also assume a constant step size  $h$ . While LMC with vanishing step sizes converges weakly to the target distribution, in practice a constant step size is often used (Vollmer et al., 2016; Brosse et al., 2018). With a constant step size, we can derive a non-asymptotic error bound for LQMC.
Assumption 1 implies that if the step size  $h \leq \frac{2}{L + M}$ , then the transition map  $\psi$  is a strong contraction with parameter  $\rho = 1 - hM$ , i.e.
$$
\left\| \psi (\theta , \mathbf {u}) - \psi \left(\theta^ {\prime}, \mathbf {u}\right) \right\| _ {2} = \left\| \theta - \theta^ {\prime} - h \left(\nabla U (\theta) - \nabla U \left(\theta^ {\prime}\right)\right) \right\| _ {2} \leq \rho \| \theta - \theta^ {\prime} \| _ {2}. \tag {6}
$$