Original: R ANDOM C OORDINATE L ANGEVIN M ONTE C ARLO
Revised: R ANDOM C OORDINATE L ANGEVIN M ONTE C ARLO

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Langevin Monte Carlo (LMC) is a popular Markov chain Monte Carlo sampling method.
Revised: Langevin Monte Carlo (LMC) is a popular Markov chain Monte Carlo sampling method.

Original: One drawback is that it requires the computation of the full gradient at each iteration, an expensive operation if the dimension of the problem is high.
Revised: One drawback is that it requires the computation of the full gradient at each iteration, an expensive operation if the dimension of the problem is high.

Original: We propose a new sampling method: Random Coordinate LMC (RC-LMC).
Revised: We propose a new sampling method: Random Coordinate LMC (RC-LMC).

Original: At each iteration, a single coordinate is randomly selected to be updated by a multiple of the partial derivative along this direction plus noise, and all other coordinates remain untouched.
Revised: At each iteration, a single coordinate is randomly selected to be updated by a multiple of the partial derivative along this direction plus noise, and all other coordinates remain untouched.

Original: We investigate the total complexity of RC-LMC and compare it with the classical LMC for log-concave probability distributions.
Revised: We investigate the total complexity of RC-LMC and compare it with the classical LMC for log-concave probability distributions.

Original: When the gradient of the log-density is Lipschitz, RC-LMC is less expensive than the classical
Revised: When the gradient of the log-density is Lipschitz, RC-LMC is less expensive than the classical

Original: LMC if the log-density is highly skewed for high dimensional problems, and when both the gradient and the Hessian of the log-density are Lipschitz, RC-LMC is always cheaper than the classical LMC, by a factor proportional to the square root of the problem dimension.
Revised: LMC if the log-density is highly skewed for high dimensional problems, and when both the gradient and the Hessian of the log-density are Lipschitz, RC-LMC is always cheaper than the classical LMC, by a factor proportional to the square root of the problem dimension.

Original: In the latter case, our estimate of complexity is sharp with respect to the dimension.
Revised: In the latter case, our estimate of complexity is sharp with respect to the dimension.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Monte Carlo sampling plays an important role in machine learning (Andrieu et al., 2003) and
Revised: Monte Carlo sampling plays an important role in machine learning (Andrieu et al., 2003) and

Original: Bayesian statistics.
Revised: Bayesian statistics.

Original: In applications, the need for sampling is found in atmospheric science (Fabian, 1981), epidemiology (Li et al., 2020), petroleum engineering (Nagarajan et al., 2007), in the form of data assimilation (Reich, 2011), volume computation (Vempala, 2010) and bandit optimization (Russo et al., 2018).
Revised: In applications, the need for sampling is found in atmospheric science (Fabian, 1981), epidemiology (Li et al., 2020), petroleum engineering (Nagarajan et al., 2007), in the form of data assimilation (Reich, 2011), volume computation (Vempala, 2010) and bandit optimization (Russo et al., 2018).

Original: In many of these applications, the dimension of the problem is extremely high.
Revised: In many of these applications, the dimension of the problem is extremely high.

Original: For example, for weather prediction, one measures the current state temperature and moisture level, to infer the ﬂow in the air, before running the Navier–Stokes equations into the near future (Evensen, 2009).
Revised: For example, for weather prediction, one measures the current state temperature and moisture level, to infer the ﬂow in the air, before running the Navier–Stokes equations into the near future (Evensen, 2009).

Original: In a global numerical weather prediction model, the degrees of freedom in the air ﬂow can be as high as 10 9 .
Revised: In a global numerical weather prediction model, the degrees of freedom in the air ﬂow can be as high as 10 9 .

Original: Another example is from epidemiology:
Revised: Another example is from epidemiology:

Original: When a disease is spreading, one measures the everyday new infection cases to infer the transmission rate in different regions.
Revised: When a disease is spreading, one measures the everyday new infection cases to infer the transmission rate in different regions.

Original: On a county-level modeling, one treats 3 , 141 different counties in the US separately, and the parameter to be inferred has a dimension of at least 3 , 

Revised: On a county-level modeling, one treats 3 , 141 different counties in the US separately, and the parameter to be inferred has a dimension of at least 3 ,

Original: (Li et al., 2020).
Revised: (Li et al., 2020).

Original: In this work, we focus on Monte Carlo sampling of log-concave probability distributions on R d , meaning the probability density can be written as p ( x ) ∝ e − f ( x ) where a f ( x ) is a convex function.
Revised: In this work, we focus on Monte Carlo sampling of log-concave probability distributions on R d , meaning the probability density can be written as p ( x ) ∝ e − f ( x ) where a f ( x ) is a convex function.

Original: The goal is to generate (approximately) i.i.d.
Revised: The goal is to generate (approximately) i.i.d.

Original: samples according to the target probability distribution with density p ( x ) .
Revised: samples according to the target probability distribution with density p ( x ) .

Original: Several sampling frameworks have been proposed in the literature, including importance sampling and sequential Monte Carlo (Geweke, 1989; Neal, 2001;
Revised: Several sampling frameworks have been proposed in the literature, including importance sampling and sequential Monte Carlo (Geweke, 1989; Neal, 2001;

Original: Del Moral et al., 2006); ensemble methods (Reich, 2011; Iglesias et al., 2013); Markov chain Monte
Revised: Del Moral et al., 2006); ensemble methods (Reich, 2011; Iglesias et al., 2013); Markov chain Monte

Original: Carlo (MCMC) (Roberts and Rosenthal, 2004), including Metropolis-Hasting based MCMC (MHMCMC) (Metropolis et al., 1953; Hastings, 1970; Roberts and Tweedie, 1996); Gibbs samplers (Geman and Geman, 1984; Casella and George, 1992); and Hamiltonian Monte Carlo (Neal, 1993;
Revised: Carlo (MCMC) (Roberts and Rosenthal, 2004), including Metropolis-Hasting based MCMC (MHMCMC) (Metropolis et al., 1953; Hastings, 1970; Roberts and Tweedie, 1996); Gibbs samplers (Geman and Geman, 1984; Casella and George, 1992); and Hamiltonian Monte Carlo (Neal, 1993;

Original: Duane et al., 1987).
Revised: Duane et al., 1987).

Original: Langevin Monte Carlo (LMC) (Rossky et al., 1978; Parisi, 1981; Roberts and
Revised: Langevin Monte Carlo (LMC) (Rossky et al., 1978; Parisi, 1981; Roberts and

Original: Tweedie, 1996) is a popular MCMC method that has received intense attention in recent years due to progress in the non-asymptotic analysis of its convergence properties (Durmus and Moulines, 2017;
Revised: Tweedie, 1996) is a popular MCMC method that has received intense attention in recent years due to progress in the non-asymptotic analysis of its convergence properties (Durmus and Moulines, 2017;

Original: Dalalyan, 2017; Dalalyan and Karagulyan, 2019; Durmus et al., 2019).
Revised: Dalalyan, 2017; Dalalyan and Karagulyan, 2019; Durmus et al., 2019).

Original: Denoting by x m the location of the sample at m -th iteration, LMC obtains the next location as follows:
Revised: Denoting by x m the location of the sample at m -th iteration, LMC obtains the next location as follows:

Original: [Equation]
Revised: [Equation]

Original: where h is the time stepsize, and ξ md is drawn i.i.d.
Revised: where h is the time stepsize, and ξ md is drawn i.i.d.

Original: from N (0 , I d ) , where I d denotes identity matrix of size d × d .
Revised: from N (0 , I d ) , where I d denotes identity matrix of size d × d .

Original: LMC can be viewed as the Euler-Maruyama discretization of the following stochastic differential equation (SDE):
Revised: LMC can be viewed as the Euler-Maruyama discretization of the following stochastic differential equation (SDE):

Original: [Equation]
Revised: [Equation]

Original: where B t,d is a d -dimensional Brownian motion.
Revised: where B t,d is a d -dimensional Brownian motion.

Original: It is well known that under suitable conditions, the distribution of X t converges exponentially fast to the target distribution (see e.g., (Markowich and
Revised: It is well known that under suitable conditions, the distribution of X t converges exponentially fast to the target distribution (see e.g., (Markowich and

Original: Villani, 1999)).
Revised: Villani, 1999)).

Original: Since (1) approximates the SDE (2) with an O ( h ) discretization error, the probability distribution of x m produced by LMC (1) converges exponentially to the target distribution up to a discretization error (Dalalyan and Karagulyan, 2019).
Revised: Since (1) approximates the SDE (2) with an O ( h ) discretization error, the probability distribution of x m produced by LMC (1) converges exponentially to the target distribution up to a discretization error (Dalalyan and Karagulyan, 2019).

Original: A signiﬁcant drawback of LMC is that the algorithm requires the evaluation of the full gradient at each iteration.
Revised: A signiﬁcant drawback of LMC is that the algorithm requires the evaluation of the full gradient at each iteration.

Original: This could be potentially very expensive in most practical problems.
Revised: This could be potentially very expensive in most practical problems.

Original: Indeed, when the analytical expression of the gradient is not available, each partial derivative component in the gradient needs to be computed separately, either through ﬁnite differencing or automatic differentiation (Baydin et al., 2017), so that the total number of such evaluations can be as many as d times the number of required iterations.
Revised: Indeed, when the analytical expression of the gradient is not available, each partial derivative component in the gradient needs to be computed separately, either through ﬁnite differencing or automatic differentiation (Baydin et al., 2017), so that the total number of such evaluations can be as many as d times the number of required iterations.

Original: In the weather prediction and epidemiology problems discussed above, f stands for the map from the parameter space of measured quantities via the underlying partial differential equations (PDEs), and each dimensional partial derivative calls for one forward and one adjoint PDE solve.
Revised: In the weather prediction and epidemiology problems discussed above, f stands for the map from the parameter space of measured quantities via the underlying partial differential equations (PDEs), and each dimensional partial derivative calls for one forward and one adjoint PDE solve.

Original: Thus, 2 d PDE solves are required in general at each iteration.
Revised: Thus, 2 d PDE solves are required in general at each iteration.

Original: Another example comes from the study of directed graphs with multiple nodes.
Revised: Another example comes from the study of directed graphs with multiple nodes.

Original: Denote the nodes by N = { 1 , 2 , . . .
Revised: Denote the nodes by N = { 1 , 2 , . . .

Original: , d } and directed edges by E ⊂ { ( i, j ) : i, j ∈ N} , and suppose there is a scalar variable x i associated with each node.
Revised: , d } and directed edges by E ⊂ { ( i, j ) : i, j ∈ N} , and suppose there is a scalar variable x i associated with each node.

Original: When the function f has the form f ( x ) = (cid:80) ( i,j ) ∈E f ij ( x i , x j ) , the partial derivative of f with respect to x i is given by
Revised: When the function f has the form f ( x ) = (cid:80) ( i,j ) ∈E f ij ( x i , x j ) , the partial derivative of f with respect to x i is given by

Original: [Equation]
Revised: [Equation]

Original: Note that the number of terms in the summations equals the number of edges that touch node i , the expected value of which is about 2 /d times the total number of edges in the graph.
Revised: Note that the number of terms in the summations equals the number of edges that touch node i , the expected value of which is about 2 /d times the total number of edges in the graph.

Original: Meanwhile, evaluation of the full gradient would require evaluation of both partial derivatives of each f ij for all edges in the graph.
Revised: Meanwhile, evaluation of the full gradient would require evaluation of both partial derivatives of each f ij for all edges in the graph.

Original: Hence, the cost difference between these two operations is a factor of order d .
Revised: Hence, the cost difference between these two operations is a factor of order d .

Original: In this paper, we study how to modify the updating strategies of LMC to reduce the numerical cost, with the focus on reducing dependence on d .
Revised: In this paper, we study how to modify the updating strategies of LMC to reduce the numerical cost, with the focus on reducing dependence on d .

Original: In particular, we will develop and analyze a method called Random Coordinate Langevin Monte Carlo (RC-LMC).
Revised: In particular, we will develop and analyze a method called Random Coordinate Langevin Monte Carlo (RC-LMC).

Original: This idea is inspired by the random coordinate descent (RCD) algorithm from optimization (Nesterov, 2012; Wright, 2015).
Revised: This idea is inspired by the random coordinate descent (RCD) algorithm from optimization (Nesterov, 2012; Wright, 2015).

Original: RCD is a version of Gradient Descent (GD) in which one coordinate (or a block of coordinates) is selected at random for updating along its negative gradient direction.
Revised: RCD is a version of Gradient Descent (GD) in which one coordinate (or a block of coordinates) is selected at random for updating along its negative gradient direction.

Original: In optimization, RCD can be signiﬁcantly cheaper than GD, especially when the objective function is skewed and the dimensionality of the problem is high.
Revised: In optimization, RCD can be signiﬁcantly cheaper than GD, especially when the objective function is skewed and the dimensionality of the problem is high.

Original: In RC-LMC, we use the same basic strategy:
Revised: In RC-LMC, we use the same basic strategy:

Original: At iteration m , a single coordinate of x m is randomly selected for updating, while all others are left unchanged.
Revised: At iteration m , a single coordinate of x m is randomly selected for updating, while all others are left unchanged.

Original: Although each iteration of RC-LMC is cheaper than conventional LMC, more iterations are required to achieve the target accuracy, and delicate analysis is required to obtain bounds on the total cost.
Revised: Although each iteration of RC-LMC is cheaper than conventional LMC, more iterations are required to achieve the target accuracy, and delicate analysis is required to obtain bounds on the total cost.

Original: Analogous to optimization, the savings of RC-LMC by comparison with LMC depend on the structure of the dimensional Lipschitz constants.
Revised: Analogous to optimization, the savings of RC-LMC by comparison with LMC depend on the structure of the dimensional Lipschitz constants.

Original: Under the assumption that there is a factor-of-
Revised: Under the assumption that there is a factor-of-

Original: d difference in per-iteration costs, we compare our results with current results for classical LMC (Dalalyan and Karagulyan, 2019; Durmus et al., 2019) and conclude the following:
Revised: d difference in per-iteration costs, we compare our results with current results for classical LMC (Dalalyan and Karagulyan, 2019; Durmus et al., 2019) and conclude the following:

Original: (Theorem 4.1)
Revised: (Theorem 4.1)

Original: When the gradient of f is Lipschitz but the Hessian is not, RC-LMC costs (cid:101)
Revised: When the gradient of f is Lipschitz but the Hessian is not, RC-LMC costs (cid:101)

Original: O ( d 2 /(cid:15) 2 )
Revised: O ( d 2 /(cid:15) 2 )

Original: to get an (cid:15) -accurate solution.
Revised: to get an (cid:15) -accurate solution.

Original: Therefore, RC-LMC outperforms LMC, in terms of the computational cost, if f is skewed and the dimension of the problem is high, as discussed in Remark 4.1.
Revised: Therefore, RC-LMC outperforms LMC, in terms of the computational cost, if f is skewed and the dimension of the problem is high, as discussed in Remark 4.1.

Original: The optimal numerical cost in this setting is achieved when the probability of choosing the i -th direction is proportional to the i -th directional Lipschitz constant. 2.
Revised: The optimal numerical cost in this setting is achieved when the probability of choosing the i -th direction is proportional to the i -th directional Lipschitz constant. 2.

Original: (Theorem 4.2)
Revised: (Theorem 4.2)

Original: When both the gradient and the Hessian of f are Lipschitz, RC-LMC requires (cid:101) O
Revised: When both the gradient and the Hessian of f are Lipschitz, RC-LMC requires (cid:101) O

Original: ( d 3 / 2 /(cid:15) ) iterations to achieve (cid:15) accuracy.
Revised: ( d 3 / 2 /(cid:15) ) iterations to achieve (cid:15) accuracy.

Original: On the other hand, the currently available result indicates that the classical LMC costs (cid:101)
Revised: On the other hand, the currently available result indicates that the classical LMC costs (cid:101)

Original: O
Revised: O

Original: ( d 2 /(cid:15) ) .
Revised: ( d 2 /(cid:15) ) .

Original: Thus, RC-LMC saves a factor of at least d 1 /regardless of the stiffness structure of f , as discussed in Remark 4.2.
Revised: Thus, RC-LMC saves a factor of at least d 1 /regardless of the stiffness structure of f , as discussed in Remark 4.2.

Original: (Proposition 4.2)
Revised: (Proposition 4.2)

Original: The (cid:101) O
Revised: The (cid:101) O

Original: ( d 3 / 2 /(cid:15) ) complexity bound for RC-LMC is sharp when both the gradient and the Hessian of f are Lipschitz.
Revised: ( d 3 / 2 /(cid:15) ) complexity bound for RC-LMC is sharp when both the gradient and the Hessian of f are Lipschitz.

Original: (The notation (cid:101) O ( · ) omits possible log terms.)
Revised: (The notation (cid:101) O ( · ) omits possible log terms.)

Original: We make three additional remarks.
Revised: We make three additional remarks.

Original: (a) Throughout the paper we assume that one element of the gradient is available at an expected cost of approximately 1 /d of the cost of the full gradient evaluation.
Revised: (a) Throughout the paper we assume that one element of the gradient is available at an expected cost of approximately 1 /d of the cost of the full gradient evaluation.

Original: Although this property is intuitive, and often holds in many situations (such as the graph-based example presented above), it does not hold for all problems (Wright, 2015).
Revised: Although this property is intuitive, and often holds in many situations (such as the graph-based example presented above), it does not hold for all problems (Wright, 2015).

Original: (b) Besides replacing gradient evaluation by coordinate algorithms, one might also improve the dimension dependence of LMC by utilizing a more rapidly convergent method for the underlying SDEs than (2).
Revised: (b) Besides replacing gradient evaluation by coordinate algorithms, one might also improve the dimension dependence of LMC by utilizing a more rapidly convergent method for the underlying SDEs than (2).

Original: One such possibility is to use underdamped Langevin dynamics, see e.g., (Rossky et al., 1978; Dalalyan and Riou-Durand, 2018;
Revised: One such possibility is to use underdamped Langevin dynamics, see e.g., (Rossky et al., 1978; Dalalyan and Riou-Durand, 2018;

Original: Cheng et al., 2018; Eberle et al., 2019; Shen and Lee, 2019; Cao et al., 2019), which can also be combined with coordinate sampling.
Revised: Cheng et al., 2018; Eberle et al., 2019; Shen and Lee, 2019; Cao et al., 2019), which can also be combined with coordinate sampling.

Original: For the clarity of presentation, we will focus only on LMC in this work and leave the extension to underdampped samplers to a future work.
Revised: For the clarity of presentation, we will focus only on LMC in this work and leave the extension to underdampped samplers to a future work.

Original: (c) It is also possible to reduce the cost of full gradient evaluation using stochastic gradient (Welling and Teh, 2011) or MALA-in-Gibbs
Revised: (c) It is also possible to reduce the cost of full gradient evaluation using stochastic gradient (Welling and Teh, 2011) or MALA-in-Gibbs

Original: sampling
Revised: sampling

Original: (Tong et al., 2020).
Revised: (Tong et al., 2020).

Original: However, both methods require speciﬁc forms of the objective function that are not considered in our work.
Revised: However, both methods require speciﬁc forms of the objective function that are not considered in our work.

Original: The paper is organized as follows.
Revised: The paper is organized as follows.

Original: We present the RC-LMC algorithm in Section 2.
Revised: We present the RC-LMC algorithm in Section 2.

Original: Notations and assumptions on f are listed in Section 3, where we also recall theoretical results for the classical
Revised: Notations and assumptions on f are listed in Section 3, where we also recall theoretical results for the classical

Original: LMC method.
Revised: LMC method.

Original: We present our main results regarding the numerical cost in Section 4 and numerical experiments in Section 5.
Revised: We present our main results regarding the numerical cost in Section 4 and numerical experiments in Section 5.

Original: Proofs of the main results are deferred to the Appendix.
Revised: Proofs of the main results are deferred to the Appendix.

Original:  R ANDOM C OORDINATE L ANGEVIN M ONTE C ARLO
Revised:  R ANDOM C OORDINATE L ANGEVIN M ONTE C ARLO

Original: We introduce the Random Coordinate Langevin Monte Carlo (RC-LMC) method in this section.
Revised: We introduce the Random Coordinate Langevin Monte Carlo (RC-LMC) method in this section.

Original: At each iteration, one coordinate is chosen at random and updated, while the other components of x are unchanged.
Revised: At each iteration, one coordinate is chosen at random and updated, while the other components of x are unchanged.

Original: Speciﬁcally, denoting by r m the index of the random coordinate chosen at m -th iteration, we obtain x m +1 r m according to a single-coordinate version of (1) and set x m +1 i = x mi for i (cid:54) =
Revised: Speciﬁcally, denoting by r m the index of the random coordinate chosen at m -th iteration, we obtain x m +1 r m according to a single-coordinate version of (1) and set x m +1 i = x mi for i (cid:54) =

Original: r m .
Revised: r m .

Original: The coordinate index r m can be chosen uniformly from { 1 , 2 , . . .
Revised: The coordinate index r m can be chosen uniformly from { 1 , 2 , . . .

Original: , d } ; but we will consider more general possibilities.
Revised: , d } ; but we will consider more general possibilities.

Original: Let φ i be the probability of component i being chosen, we denote the distribution from which r m is drawn by Φ , where
Revised: Let φ i be the probability of component i being chosen, we denote the distribution from which r m is drawn by Φ , where

Original: [Equation]
Revised: [Equation]

Original: φ
Revised: φ

Original: [Equation]
Revised: [Equation]

Original: all
Revised: all

Original: [Equation]
Revised: [Equation]

Original: The stepsize may depend on the choice of coordinate; we denote the stepsizes by { h 1 , h 2 , . . .
Revised: The stepsize may depend on the choice of coordinate; we denote the stepsizes by { h 1 , h 2 , . . .

Original: , h d } and assume that they do not change across iterations.
Revised: , h d } and assume that they do not change across iterations.

Original: In this paper, we choose h i to be inversely dependent on probabilities φ
Revised: In this paper, we choose h i to be inversely dependent on probabilities φ

Original: i , as follows:
Revised: i , as follows:

Original: [Equation]
Revised: [Equation]

Original: where h > 0 is a parameter that can be viewed as the expected stepsize.
Revised: where h > 0 is a parameter that can be viewed as the expected stepsize.

Original: In Section 4.2-4.3, we will ﬁnd the optimal form of Φ under different scenarios.
Revised: In Section 4.2-4.3, we will ﬁnd the optimal form of Φ under different scenarios.

Original: The initial iterate x 0 is drawn from a distribution q 0 , which can be any distribution that is easy to draw from (the normal distribution, for example).
Revised: The initial iterate x 0 is drawn from a distribution q 0 , which can be any distribution that is easy to draw from (the normal distribution, for example).

Original: We present the complete method in Algorithm 1.
Revised: We present the complete method in Algorithm 1.

Original: When we compare (5) with the classical LMC (1), we see that only one random coordinate is updated per iteration, meaning:
Revised: When we compare (5) with the classical LMC (1), we see that only one random coordinate is updated per iteration, meaning:

Original: [Equation]
Revised: [Equation]

Original: where e i is the unit vector for i -th direction and ξ m is drawn from N (0 , 1) .
Revised: where e i is the unit vector for i -th direction and ξ m is drawn from N (0 , 1) .

Original: Deﬁne the elapsed time at m -th iteration as
Revised: Deﬁne the elapsed time at m -th iteration as

Original: [Equation]
Revised: [Equation]

Original: n , and T
Revised: n , and T

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: [Equation]
Revised: [Equation]

Original: then for t ∈ ( T m , T m +1 ] , the updating formula (5) can be viewed as the Euler approximation to the following SDE:
Revised: then for t ∈ ( T m , T m +1 ] , the updating formula (5) can be viewed as the Euler approximation to the following SDE:

Original: [Equation]
Revised: [Equation]

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: where B t is a 1 -dimensional Brownian motion.
Revised: where B t is a 1 -dimensional Brownian motion.

Original: We will show in Section 4.1 that the SDE preserves the invariant measure, that is, X ( t ) ∼ p for any t > 0 if X (0) ∼ p , and it is ergodic.
Revised: We will show in Section 4.1 that the SDE preserves the invariant measure, that is, X ( t ) ∼ p for any t > 0 if X (0) ∼ p , and it is ergodic.

Original: The invariant measure of RC-LMC, which can be viewed as a discretized version of the SDE, is not exactly p , due to the unavoidable discretization error.
Revised: The invariant measure of RC-LMC, which can be viewed as a discretized version of the SDE, is not exactly p , due to the unavoidable discretization error.

Original:  N OTATIONS , ASSUMPTIONS AND CLASSICAL RESULTS
Revised:  N OTATIONS , ASSUMPTIONS AND CLASSICAL RESULTS

Original: We unify notations and assumptions in this section, and summarize and discuss the classical results on LMC.
Revised: We unify notations and assumptions in this section, and summarize and discuss the classical results on LMC.

Original: Throughout the paper, to quantify the distance between two probability distributions, we use the Wasserstein distance deﬁned by
Revised: Throughout the paper, to quantify the distance between two probability distributions, we use the Wasserstein distance deﬁned by

Original: [Equation]
Revised: [Equation]

Original: where Γ( µ, ν ) is the set of distribution of ( X, Y ) ∈ R 2 d whose marginal distributions, for X and Y respectively, are µ and ν .
Revised: where Γ( µ, ν ) is the set of distribution of ( X, Y ) ∈ R 2 d whose marginal distributions, for X and Y respectively, are µ and ν .

Original: The distributions in Γ( µ, ν ) are called the couplings of µ and ν .
Revised: The distributions in Γ( µ, ν ) are called the couplings of µ and ν .

Original: Due to the use of power 2 in the deﬁnition, this is sometimes called the Wasserstein- 2 distance.
Revised: Due to the use of power 2 in the deﬁnition, this is sometimes called the Wasserstein- 2 distance.

Original: Here and in the sequel, we use |·| to denote the Euclidean norm of a vector.
Revised: Here and in the sequel, we use |·| to denote the Euclidean norm of a vector.

Original: We assume that f is strongly convex, so that p is strongly log-concave.
Revised: We assume that f is strongly convex, so that p is strongly log-concave.

Original: We obtain results under two different assumptions:
Revised: We obtain results under two different assumptions:

Original: First, Lipschitz continuity of the gradient of f (Assumption 3.1) and second,
Revised: First, Lipschitz continuity of the gradient of f (Assumption 3.1) and second,

Original: Lipschitz continuity of the Hessian of f (Assumption 3.2 together with Assumption 3.1).
Revised: Lipschitz continuity of the Hessian of f (Assumption 3.2 together with Assumption 3.1).

Original: Assumption 3.1.
Revised: Assumption 3.1.

Original: The function f is twice differentiable, f is µ -strongly convex for some µ > 0 and its gradient ∇ f is L -Lipschitz.
Revised: The function f is twice differentiable, f is µ -strongly convex for some µ > 0 and its gradient ∇ f is L -Lipschitz.

Original: That is, for all x, x (cid:48) ∈ R d , we have
Revised: That is, for all x, x (cid:48) ∈ R d , we have

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: It is an elementary consequence of (8) that
Revised: It is an elementary consequence of (8) that

Original: [Equation]
Revised: [Equation]

Original: for all x
Revised: for all x

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Since each coordinate direction plays a distinct role in RC-LMC, we distinguish the Lipschitz constants in each such direction.
Revised: Since each coordinate direction plays a distinct role in RC-LMC, we distinguish the Lipschitz constants in each such direction.

Original: When Assumption 3.1 holds, partial derivatives in all coordinate directions are also Lipschitz.
Revised: When Assumption 3.1 holds, partial derivatives in all coordinate directions are also Lipschitz.

Original: Denoting them as L i for each i = 1 , 2 , . .
Revised: Denoting them as L i for each i = 1 , 2 , . .

Original: , d , we have
Revised: , d , we have

Original: [Equation]
Revised: [Equation]

Original: 11) for any x ∈ R d and any t ∈ R .
Revised: 11) for any x ∈ R d and any t ∈ R .

Original: We further denote L max := max i L i and deﬁne condition numbers as follows:
Revised: We further denote L max := max i L i and deﬁne condition numbers as follows:

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: As shown in (Wright, 2015), we have
Revised: As shown in (Wright, 2015), we have

Original: [Equation]
Revised: [Equation]

Original: These assumptions together imply that the spectrum of the Hessian is bounded above and below for all x , speciﬁcally, µI d (cid:22) ∇ 2 f ( x ) (cid:22) LI d and [ ∇ 2 f ( x )] ii ≤ L i ≤ L max for all x ∈ R d .
Revised: These assumptions together imply that the spectrum of the Hessian is bounded above and below for all x , speciﬁcally, µI d (cid:22) ∇ 2 f ( x ) (cid:22) LI d and [ ∇ 2 f ( x )] ii ≤ L i ≤ L max for all x ∈ R d .

Original: Both upper and lower bounds of L in term of L max in (13) are tight.
Revised: Both upper and lower bounds of L in term of L max in (13) are tight.

Original: If ∇ 2 f is a diagonal matrix, then L max =
Revised: If ∇ 2 f is a diagonal matrix, then L max =

Original: L , both being the biggest eigenvalue of ∇ 2 f .
Revised: L , both being the biggest eigenvalue of ∇ 2 f .

Original: Thus, κ max = κ in this case.
Revised: Thus, κ max = κ in this case.

Original: This is the case in which all coordinates are independent of each other, for example f = (cid:80) i λ i x 2 i .
Revised: This is the case in which all coordinates are independent of each other, for example f = (cid:80) i λ i x 2 i .

Original: On the other hand, if ∇ 2 f = e · e (cid:62) where e ∈ R d satisﬁes e i = 1 for all i , then L = dL max and κ = dκ max .
Revised: On the other hand, if ∇ 2 f = e · e (cid:62) where e ∈ R d satisﬁes e i = 1 for all i , then L = dL max and κ = dκ max .

Original: This is a situation in which f is highly skewed, that is, f
Revised: This is a situation in which f is highly skewed, that is, f

Original: =
Revised: =

Original: ( (cid:80) i x i ) 2 / 2 .
Revised: ( (cid:80) i x i ) 2 / 2 .

Original: The next assumption concerns higher regularity for f .
Revised: The next assumption concerns higher regularity for f .

Original: Assumption 3.2.
Revised: Assumption 3.2.

Original: The function f is three times differentiable and ∇ 2 f is H-Lipschitz, that is
Revised: The function f is three times differentiable and ∇ 2 f is H-Lipschitz, that is

Original: [Equation]
Revised: [Equation]

Original: for
Revised: for

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: When this assumption holds, we further deﬁne H i to satisfy
Revised: When this assumption holds, we further deﬁne H i to satisfy

Original: [Equation]
Revised: [Equation]

Original: 15) for any i = 1 , 2 , . . .
Revised: 15) for any i = 1 , 2 , . . .

Original: , d , all x ∈ R d , and all t ∈ R , where ∂ ii f is [ ∇ 2 f ( x )] ii , the ( i, i ) diagonal entry of the Hessian matrix ∇ 2 f .
Revised: , d , all x ∈ R d , and all t ∈ R , where ∂ ii f is [ ∇ 2 f ( x )] ii , the ( i, i ) diagonal entry of the Hessian matrix ∇ 2 f .

Original: We summarize existing results for the classical LMC in the following theorem.
Revised: We summarize existing results for the classical LMC in the following theorem.

Original: Theorem 3.1 ((Durmus et al., 2019, Theorem 9), (Dalalyan and Karagulyan, 2019, Theorem 5)) .
Revised: Theorem 3.1 ((Durmus et al., 2019, Theorem 9), (Dalalyan and Karagulyan, 2019, Theorem 5)) .

Original: Let q m be the probability distribution of the m -th iteration of LMC (1) , and p be the target distribution.
Revised: Let q m be the probability distribution of the m -th iteration of LMC (1) , and p be the target distribution.

Original: Using the notation W m :
Revised: Using the notation W m :

Original: = W ( q m , p ) , we have the following:
Revised: = W ( q m , p ) , we have the following:

Original: • Under Assumption 3.1, let h ≤ 1 /L , we have
Revised: • Under Assumption 3.1, let h ≤ 1 /L , we have

Original: [Equation]
Revised: [Equation]

Original: • Under Assumptions 3.1 and 3.2, let h
Revised: • Under Assumptions 3.1 and 3.2, let h

Original: < 2 / ( µ + L ) , we have
Revised: < 2 / ( µ + L ) , we have

Original: [Equation]
Revised: [Equation]

Original: This theorem yields stopping criteria for the number of iterations M to achieve a user-deﬁned accuracy of (cid:15) .
Revised: This theorem yields stopping criteria for the number of iterations M to achieve a user-deﬁned accuracy of (cid:15) .

Original: When the gradient of f is Lipschitz, to achieve (cid:15) -accuracy, we can require both terms on the right hand side of (16) to be smaller than (cid:15)/ 2 , which occurs when
Revised: When the gradient of f is Lipschitz, to achieve (cid:15) -accuracy, we can require both terms on the right hand side of (16) to be smaller than (cid:15)/ 2 , which occurs when

Original: [Equation]
Revised: [Equation]

Original: leading to a cost of (cid:101)
Revised: leading to a cost of (cid:101)

Original: O
Revised: O

Original: ( d 2 κ/ ( µ(cid:15) 2 )) evaluations of gradient components (when we assume that each full gradient can be obtained at the cost of d individual components of the gradient).
Revised: ( d 2 κ/ ( µ(cid:15) 2 )) evaluations of gradient components (when we assume that each full gradient can be obtained at the cost of d individual components of the gradient).

Original: When both the gradient and the Hessian are Lipschitz, to achieve (cid:15) -accuracy, we require all three terms on the right hand side of (17) to be smaller than (cid:15)/ 3 .
Revised: When both the gradient and the Hessian are Lipschitz, to achieve (cid:15) -accuracy, we require all three terms on the right hand side of (17) to be smaller than (cid:15)/ 3 .

Original: Assuming d (cid:29) 1 and all other constants are O (1) , we thus obtain h
Revised: Assuming d (cid:29) 1 and all other constants are O (1) , we thus obtain h

Original: = Θ(
Revised: = Θ(

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: which yields a cost of (cid:101)
Revised: which yields a cost of (cid:101)

Original: O
Revised: O

Original: ( d 2 H/ ( µ 2 (cid:15) )) evaluations of gradient components.
Revised: ( d 2 H/ ( µ 2 (cid:15) )) evaluations of gradient components.

Original: Here A = Θ( B ) denotes cB ≤ A ≤ CB for some absolute constant c and C .
Revised: Here A = Θ( B ) denotes cB ≤ A ≤ CB for some absolute constant c and C .

Original:  M AIN RESULTS
Revised:  M AIN RESULTS

Original: We discuss the main results from two perspectives.
Revised: We discuss the main results from two perspectives.

Original: In Section 4.1 we examine the convergence of the underlying SDE (7), laying the foundation for the convergence in the discrete setting.
Revised: In Section 4.1 we examine the convergence of the underlying SDE (7), laying the foundation for the convergence in the discrete setting.

Original: We then build upon this result and show the convergence of the RC-LMC algorithm in Section 4.2 and 4.
Revised: We then build upon this result and show the convergence of the RC-LMC algorithm in Section 4.2 and 4.

Original: under two different assumptions.
Revised: under two different assumptions.

Original: We show in Section 4.4 that when both Assumption 3.1 and 3.
Revised: We show in Section 4.4 that when both Assumption 3.1 and 3.

Original: are satisﬁed, our bound is tight with respect to d and (cid:15) .
Revised: are satisﬁed, our bound is tight with respect to d and (cid:15) .

Original: 4. C ONVERGENCE OF THE SDE (7)
Revised: 4. C ONVERGENCE OF THE SDE (7)

Original: To study the convergence of (7), we ﬁrst let X m =
Revised: To study the convergence of (7), we ﬁrst let X m =

Original: X ( T m ) and denote the probability ﬁltration by F m = (cid:8) x 0 , r n ≤ m , B s ≤ T m (cid:9) .
Revised: X ( T m ) and denote the probability ﬁltration by F m = (cid:8) x 0 , r n ≤ m , B s ≤ T m (cid:9) .

Original: Then { X m } ∞ m =0 is a Markov chain and the following proposition shows its geometric ergodicity.
Revised: Then { X m } ∞ m =0 is a Markov chain and the following proposition shows its geometric ergodicity.

Original: Proposition 4.1.
Revised: Proposition 4.1.

Original: Let X m =
Revised: Let X m =

Original: X ( T m ) solve (7) .
Revised: X ( T m ) solve (7) .

Original: If f satisﬁes Assumption 3.1 and h ≤ µ min { φ i } 4+8 L 2 +32 L 4 , then p ( x ) is the stationary probability density of the Markov chain {
Revised: If f satisﬁes Assumption 3.1 and h ≤ µ min { φ i } 4+8 L 2 +32 L 4 , then p ( x ) is the stationary probability density of the Markov chain {

Original: X m } ∞ m =0 .
Revised: X m } ∞ m =0 .

Original: See proof in Appendix A. Under some mild conditions, we can further prove that the solution to the SDE converges to the target distribution exponentially (Proposition A.1).
Revised: See proof in Appendix A. Under some mild conditions, we can further prove that the solution to the SDE converges to the target distribution exponentially (Proposition A.1).

Original: Since x m , the samples generated by the algorithm can be viewed as discrete version of X m , the algorithm then is expected to converge up to a discretization error as well.
Revised: Since x m , the samples generated by the algorithm can be viewed as discrete version of X m , the algorithm then is expected to converge up to a discretization error as well.

Original: This is indeed shown in the upcoming two subsections, where we document the non-asymptotic convergence rate, and calculate the complexity of the algorithm.
Revised: This is indeed shown in the upcoming two subsections, where we document the non-asymptotic convergence rate, and calculate the complexity of the algorithm.

Original: 4. C ONVERGENCE OF RC-LMC. C ASE 1: L IPSCHITZ GRADIENT
Revised: 4. C ONVERGENCE OF RC-LMC. C ASE 1: L IPSCHITZ GRADIENT

Original: Under Assumption 3.1, we have the following result.
Revised: Under Assumption 3.1, we have the following result.

Original: The proof can be found in Appendix B.
Revised: The proof can be found in Appendix B.

Original: Theorem 4.1.
Revised: Theorem 4.1.

Original: Assume f satisﬁes Assumption 3.1, and h i
Revised: Assume f satisﬁes Assumption 3.1, and h i

Original: = h/φ i with h ≤ µ min { φ i } 8 L. Let q m be the probability distribution of x m computed in (5) , let p be the target distribution, and denote
Revised: = h/φ i with h ≤ µ min { φ i } 8 L. Let q m be the probability distribution of x m computed in (5) , let p be the target distribution, and denote

Original: W m := W ( q m , p ) .
Revised: W m := W ( q m , p ) .

Original: Then we have
Revised: Then we have

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: We make a few comments here: (1) the requirement on h is rather weak.
Revised: We make a few comments here: (1) the requirement on h is rather weak.

Original: When both µ and L are moderate (both O (1) constants), the requirement is essentially h (cid:46) 1 /d .
Revised: When both µ and L are moderate (both O (1) constants), the requirement is essentially h (cid:46) 1 /d .

Original: The estimate (20) consists of two terms.
Revised: The estimate (20) consists of two terms.

Original: The ﬁrst is an exponentially decaying term and the second comes from the variance of random coordinate selection.
Revised: The ﬁrst is an exponentially decaying term and the second comes from the variance of random coordinate selection.

Original: If we assume all Lipschitz constants L i are of O (1) , this remainder term is roughly O ( h 1 / 2 d ) .
Revised: If we assume all Lipschitz constants L i are of O (1) , this remainder term is roughly O ( h 1 / 2 d ) .

Original: The theorem suggests a stopping criterion: to have
Revised: The theorem suggests a stopping criterion: to have

Original: W M ≤ (cid:15) , we roughly need h < (cid:15) 2 /d 2 , and M = (cid:101) O
Revised: W M ≤ (cid:15) , we roughly need h < (cid:15) 2 /d 2 , and M = (cid:101) O

Original: ( d 2 /(cid:15) 2 ) , assuming L i
Revised: ( d 2 /(cid:15) 2 ) , assuming L i

Original: = O (1) .
Revised: = O (1) .

Original: In terms of (cid:15) and d dependence, this puts M at the same order as (18), as required by the classical LMC.
Revised: In terms of (cid:15) and d dependence, this puts M at the same order as (18), as required by the classical LMC.

Original: Theorem 4.1 holds for all choices of { φ i } satisfying (3).
Revised: Theorem 4.1 holds for all choices of { φ i } satisfying (3).

Original: From the explicit formula (20) we can choose { φ i } to minimize the right-hand side of the bound.
Revised: From the explicit formula (20) we can choose { φ i } to minimize the right-hand side of the bound.

Original: Nesterov (2012) proposed distributions Φ that depend on the dimensional Lipschitz constants L i , i = 1 , 2 , . . .
Revised: Nesterov (2012) proposed distributions Φ that depend on the dimensional Lipschitz constants L i , i = 1 , 2 , . . .

Original: , d from (11).
Revised: , d from (11).

Original: For α ∈ R , we can let φ i ( α ) ∝ L αi , speciﬁcally,
Revised: For α ∈ R , we can let φ i ( α ) ∝ L αi , speciﬁcally,

Original: [Equation]
Revised: [Equation]

Original: Note that when α = 0 , φ i (0) =
Revised: Note that when α = 0 , φ i (0) =

Original: 1 /d for all i : the uniform distribution among all coordinates.
Revised: 1 /d for all i : the uniform distribution among all coordinates.

Original: When α > 0 , the directions that with larger Lipschitz constants have higher probability to be chosen.
Revised: When α > 0 , the directions that with larger Lipschitz constants have higher probability to be chosen.

Original: Since h i
Revised: Since h i

Original: = h/φ i , one uses smaller stepsizes for stiffer directions.
Revised: = h/φ i , one uses smaller stepsizes for stiffer directions.

Original: (On the other hand, when α < 0 , the directions with larger Lipschitz constants are less likely to be chosen, and the stepsizes are larger in stiffer directions, a situation that is not favorable and should be avoided.)
Revised: (On the other hand, when α < 0 , the directions with larger Lipschitz constants are less likely to be chosen, and the stepsizes are larger in stiffer directions, a situation that is not favorable and should be avoided.)

Original: The following corollary discusses various choices of α and the corresponding computational cost.
Revised: The following corollary discusses various choices of α and the corresponding computational cost.

Original: Corollary 4.1.
Revised: Corollary 4.1.

Original: Under the same conditions as in Theorem 4.1, with φ i =
Revised: Under the same conditions as in Theorem 4.1, with φ i =

Original: φ i ( α ) deﬁned in (21) , the number of iterations M required to attain W M ≤ (cid:15) is M = Θ (cid:16) K 2 − α K α µ(cid:15)log (cid:0) W 0 (cid:15) (cid:1)(cid:17) , where
Revised: φ i ( α ) deﬁned in (21) , the number of iterations M required to attain W M ≤ (cid:15) is M = Θ (cid:16) K 2 − α K α µ(cid:15)log (cid:0) W 0 (cid:15) (cid:1)(cid:17) , where

Original: K α = (cid:80) d i =1 κ αi .
Revised: K α = (cid:80) d i =1 κ αi .

Original: This cost is optimized when α = 1 , for which we have
Revised: This cost is optimized when α = 1 , for which we have

Original: [Equation]
Revised: [Equation]

Original: See proof in Appendix B.
Revised: See proof in Appendix B.

Original: We note that the initial error W 0 enters through a log term and is essentially negligible.
Revised: We note that the initial error W 0 enters through a log term and is essentially negligible.

Original: Remark 4.1.
Revised: Remark 4.1.

Original: We now compare the numerical cost of RC-LMC and LMC in Case 1.
Revised: We now compare the numerical cost of RC-LMC and LMC in Case 1.

Original: We separate the discussion on uniform sampling ( φ i = 1 /d ) and the optimal sampling ( φ i ∝ L i ) below.
Revised: We separate the discussion on uniform sampling ( φ i = 1 /d ) and the optimal sampling ( φ i ∝ L i ) below.

Original: – Optimal sampling:
Revised: – Optimal sampling:

Original: According to Corollary 4.1, the optimal sampling strategy is achieved when α = 1 , meaning φ i ∝ L i .
Revised: According to Corollary 4.1, the optimal sampling strategy is achieved when α = 1 , meaning φ i ∝ L i .

Original: In this case, we compare (22) with (18) , adjusting (18) by a factor of d to account for the higher cost per iteration.
Revised: In this case, we compare (22) with (18) , adjusting (18) by a factor of d to account for the higher cost per iteration.

Original: RC-LMC has more favorable computational cost if
Revised: RC-LMC has more favorable computational cost if

Original: [Equation]
Revised: [Equation]

Original: Considering κ i ≤ κ max ≤ κ ≤ dκ max , as presented in (13) , this is guaranteed if κ ≥ κ 2max .
Revised: Considering κ i ≤ κ max ≤ κ ≤ dκ max , as presented in (13) , this is guaranteed if κ ≥ κ 2max .

Original: In the regime when κ ∼ dκ max this holds so long as d > κ max , meaning the dimension of the problem is high.
Revised: In the regime when κ ∼ dκ max this holds so long as d > κ max , meaning the dimension of the problem is high.

Original: And in the regime when κ max ∼ κ , RC-LMC still outperforms when κ i decreases fast.
Revised: And in the regime when κ max ∼ κ , RC-LMC still outperforms when κ i decreases fast.

Original: One example is to set f ( x ) =
Revised: One example is to set f ( x ) =

Original: dx 21 + (cid:80) di
Revised: dx 21 + (cid:80) di

Original: =2 x 2 i with d (cid:29) 1 .
Revised: =2 x 2 i with d (cid:29) 1 .

Original: – Uniform sampling: Uniform sampling means φ i = 

Revised: – Uniform sampling: Uniform sampling means φ i =

Original: /d
Revised: /d

Original: for all i , with α = 

Revised: for all i , with α =

Original: in Corollary 4.1.
Revised: in Corollary 4.1.

Original: This leads to a cost of Θ (cid:16)
Revised: This leads to a cost of Θ (cid:16)

Original: [Equation]
Revised: [Equation]

Original: κ
Revised: κ

Original: [Equation]
Revised: [Equation]

Original: Comparing with (18) adjusted by a factor of d , we see that RC-LMC still has a more favorable computational cost if
Revised: Comparing with (18) adjusted by a factor of d , we see that RC-LMC still has a more favorable computational cost if

Original: [Equation]
Revised: [Equation]

Original: As
Revised: As

Original: in the optimal case this
Revised: in the optimal case this

Original: happens
Revised: happens

Original: when
Revised: when

Original: f is highly skewed.
Revised: f is highly skewed.

Original: Our proof of Theorem 4.1 follows from a coupling approach similar to that used by Dalalyan and
Revised: Our proof of Theorem 4.1 follows from a coupling approach similar to that used by Dalalyan and

Original: Karagulyan (2019) for LMC.
Revised: Karagulyan (2019) for LMC.

Original: We emphasize that for the coordinate algorithm, we need to overcome the additional difﬁculty that the process of each coordinate is not contracting on the SDE (7) level.
Revised: We emphasize that for the coordinate algorithm, we need to overcome the additional difﬁculty that the process of each coordinate is not contracting on the SDE (7) level.

Original: This is a different situation from the classical LMC (Dalalyan and Karagulyan, 2019) whose corresponding SDE (2) already provides the contraction property and thus only the discretization error needs to be considered.
Revised: This is a different situation from the classical LMC (Dalalyan and Karagulyan, 2019) whose corresponding SDE (2) already provides the contraction property and thus only the discretization error needs to be considered.

Original: Despite this, the algorithm RC-LMC still enjoys the contraction property that ensures that the distance between two different trajectories following the algorithm contract.
Revised: Despite this, the algorithm RC-LMC still enjoys the contraction property that ensures that the distance between two different trajectories following the algorithm contract.

Original: However, this contraction property is not component-wise, so we need to choose Young’s constant wisely and take summation of every coordinate.
Revised: However, this contraction property is not component-wise, so we need to choose Young’s constant wisely and take summation of every coordinate.

Original: The summation will also produce some extra terms, which we need to bound.
Revised: The summation will also produce some extra terms, which we need to bound.

Original: Dalalyan and Karagulyan (2019) obtains an estimate for the cost of the classical LMC of (cid:101)
Revised: Dalalyan and Karagulyan (2019) obtains an estimate for the cost of the classical LMC of (cid:101)

Original: O ( d 2 κ 2 / ( µ(cid:15) 2 )) .
Revised: O ( d 2 κ 2 / ( µ(cid:15) 2 )) .

Original: Compared with this estimate, our estimate for the cost of RCLMC is always cheaper (since κ 2 ≥ κ 2max ).
Revised: Compared with this estimate, our estimate for the cost of RCLMC is always cheaper (since κ 2 ≥ κ 2max ).

Original: The improved estimate of the cost of LMC (18) was obtained by Durmus et al.
Revised: The improved estimate of the cost of LMC (18) was obtained by Durmus et al.

Original: (2019) using a quite different approach based on optimal transportation.
Revised: (2019) using a quite different approach based on optimal transportation.

Original: It is not clear whether their technique can be adapted to the coordinate setting to obtain an improved estimate.
Revised: It is not clear whether their technique can be adapted to the coordinate setting to obtain an improved estimate.

Original: 4. C ONVERGENCE OF RC-LMC. C ASE 2: L IPSCHITZ H ESSIAN
Revised: 4. C ONVERGENCE OF RC-LMC. C ASE 2: L IPSCHITZ H ESSIAN

Original: We now assume that Assumption 3.1 and 3.2 hold, that is, both the gradient and the Hessian of f are
Revised: We now assume that Assumption 3.1 and 3.2 hold, that is, both the gradient and the Hessian of f are

Original: Lipschitz continuous.
Revised: Lipschitz continuous.

Original: In this setting, we obtain the following improved convergence estimate.
Revised: In this setting, we obtain the following improved convergence estimate.

Original: The proof can be found in Appendix C.
Revised: The proof can be found in Appendix C.

Original: Theorem 4.2.
Revised: Theorem 4.2.

Original: Assume f satisﬁes Assumptions 3.1 and 3.2 and let h i
Revised: Assume f satisﬁes Assumptions 3.1 and 3.2 and let h i

Original: = h/φ i , with h ≤ µ min { φ i } 8 L.
Revised: = h/φ i , with h ≤ µ min { φ i } 8 L.

Original: Denoting by q m ( x ) the probability density function of x m computed from (5) and by p the target distribution, and letting W m := W ( q m , p ) , we have:
Revised: Denoting by q m ( x ) the probability density function of x m computed from (5) and by p the target distribution, and letting W m := W ( q m , p ) , we have:

Original: [Equation]
Revised: [Equation]

Original: We see again two terms in the bound, an exponentially decaying term and a variance term.
Revised: We see again two terms in the bound, an exponentially decaying term and a variance term.

Original: Assuming all Lipschitz constants are O (1) , the variance term is of O ( hd 3 / 2 ) .
Revised: Assuming all Lipschitz constants are O (1) , the variance term is of O ( hd 3 / 2 ) .

Original: By comparing with Theorem 4.1, we see that (cid:15) error can be achieved with the looser stepsize requirement h (cid:46) (cid:15)
Revised: By comparing with Theorem 4.1, we see that (cid:15) error can be achieved with the looser stepsize requirement h (cid:46) (cid:15)

Original: d 3 / 2 .
Revised: d 3 / 2 .

Original: By choosing { φ i } to optimize the bound in Theorem 4.2, we obtain the following corollary.
Revised: By choosing { φ i } to optimize the bound in Theorem 4.2, we obtain the following corollary.

Original: Corollary 4.2.
Revised: Corollary 4.2.

Original: Under the same conditions as in Theorem 4.2, the optimal choice of { φ i } is to set:
Revised: Under the same conditions as in Theorem 4.2, the optimal choice of { φ i } is to set:

Original: [Equation]
Revised: [Equation]

Original: For this choice, the number of iterations M required to guarantee W M ≤ (cid:15) satisﬁes
Revised: For this choice, the number of iterations M required to guarantee W M ≤ (cid:15) satisﬁes

Original: [Equation]
Revised: [Equation]

Original: If µ , κ i and H i are all constants of O (1) , then the total cost is (cid:101) O
Revised: If µ , κ i and H i are all constants of O (1) , then the total cost is (cid:101) O

Original: ( d 3 / 2 /(cid:15) )
Revised: ( d 3 / 2 /(cid:15) )

Original: regardless of the choice of { φ i } .
Revised: regardless of the choice of { φ i } .

Original: Remark 4.2.
Revised: Remark 4.2.

Original: We now compare RC-LMC with LMC in Case 2 using Theorem 4.2 and Corollary 4.2.
Revised: We now compare RC-LMC with LMC in Case 2 using Theorem 4.2 and Corollary 4.2.

Original: We still separate the discussion on optimal sampling and uniform sampling strategy.
Revised: We still separate the discussion on optimal sampling and uniform sampling strategy.

Original: – Optimal sampling: This is to set φ i as stated in Corollary 4.
Revised: – Optimal sampling: This is to set φ i as stated in Corollary 4.

Original: Comparing the
Revised: Comparing the

Original: cost
Revised: cost

Original: shown in (
Revised: shown in (

Original: and the cost of LMC ( (19) adjusted by a factor of d to account for the higher cost per iteration)
Revised: and the cost of LMC ( (19) adjusted by a factor of d to account for the higher cost per iteration)

Original: , we see
Revised: , we see

Original: that RC-LMC always has a more favorable computational cost since
Revised: that RC-LMC always has a more favorable computational cost since

Original: [Equation]
Revised: [Equation]

Original: (Here we relaxed (24) using L i ≤ L, H i ≤ H .)
Revised: (Here we relaxed (24) using L i ≤ L, H i ≤ H .)

Original: – Uniform sampling: This is to set φ i = 1 /d in (23) .
Revised: – Uniform sampling: This is to set φ i = 1 /d in (23) .

Original: Then the total cost of RC-LMC is
Revised: Then the total cost of RC-LMC is

Original: [Equation]
Revised: [Equation]

Original: according to Corollary4.2.
Revised: according to Corollary4.2.

Original: Comparing with (19) adjusted by a factor of d as the cost for LMC, and use the fact that L i ≤ L, H i ≤ H , it is clear that RC-LMC is always cheaper, similar to the optimal case.
Revised: Comparing with (19) adjusted by a factor of d as the cost for LMC, and use the fact that L i ≤ L, H i ≤ H , it is clear that RC-LMC is always cheaper, similar to the optimal case.

Original: Suppose L and H are all constants of O (1) , then the cost of RC-LMC is roughly (cid:101)
Revised: Suppose L and H are all constants of O (1) , then the cost of RC-LMC is roughly (cid:101)

Original: O ( d 3 / 2 /(cid:15) ) , while the classical LMC requires (cid:101)
Revised: O ( d 3 / 2 /(cid:15) ) , while the classical LMC requires (cid:101)

Original: O ( d 2 /(cid:15) )
Revised: O ( d 2 /(cid:15) )

Original: , according to Dalalyan and Riou-Durand (2018).
Revised: , according to Dalalyan and Riou-Durand (2018).

Original: This represents a savings factor of d 1 / 2 , regardless of the structure of f .
Revised: This represents a savings factor of d 1 / 2 , regardless of the structure of f .

Original: 4. T IGHTNESS OF THE COMPLEXITY BOUND
Revised: 4. T IGHTNESS OF THE COMPLEXITY BOUND

Original: When both the gradient and the Hessian are Lipschitz, we claim that estimate (cid:101) O
Revised: When both the gradient and the Hessian are Lipschitz, we claim that estimate (cid:101) O

Original: ( d 3 / 2 /(cid:15) ) obtained in Corollary 4.2 is tight.
Revised: ( d 3 / 2 /(cid:15) ) obtained in Corollary 4.2 is tight.

Original: An example is presented in the following proposition.
Revised: An example is presented in the following proposition.

Original: Proposition 4.2.
Revised: Proposition 4.2.

Original: Let φ i = 1 /d for all i , and set the initial distribution and the target distribution to be:
Revised: Let φ i = 1 /d for all i , and set the initial distribution and the target distribution to be:

Original: [Equation]
Revised: [Equation]

Original: where e ∈ R d satisﬁes e i = 1 for all i .
Revised: where e ∈ R d satisﬁes e i = 1 for all i .

Original: Let q m be the probability distribution of x m generated by
Revised: Let q m be the probability distribution of x m generated by

Original: Algorithm 1, and denote W m := W ( q m , p ) .
Revised: Algorithm 1, and denote W m := W ( q m , p ) .

Original: Then we have
Revised: Then we have

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: In particular, to have W M ≤ (cid:15) , one needs at least M = (cid:101)
Revised: In particular, to have W M ≤ (cid:15) , one needs at least M = (cid:101)

Original: O ( d 3 / 2 /(cid:15) ) .
Revised: O ( d 3 / 2 /(cid:15) ) .

Original: See proof in Appendix D.
Revised: See proof in Appendix D.

Original:  N UMERICAL RESULTS
Revised:  N UMERICAL RESULTS

Original: We provide some numerical results in this section.
Revised: We provide some numerical results in this section.

Original: Since it is extremely challenging to estimate the Wasserstein distance between two distributions in high dimensions, we demonstrate instead the convergence of estimated expectation for a given observable.
Revised: Since it is extremely challenging to estimate the Wasserstein distance between two distributions in high dimensions, we demonstrate instead the convergence of estimated expectation for a given observable.

Original: Denoting by { x ( i ) ,M } Ni =1 the list of
Revised: Denoting by { x ( i ) ,M } Ni =1 the list of

Original: N samples, with each of them computed through Algorithm 1 independently with M iterations, we deﬁne the error as follows:
Revised: N samples, with each of them computed through Algorithm 1 independently with M iterations, we deﬁne the error as follows:

Original: [Equation]
Revised: [Equation]

Original: where ψ is a matrix function and E p ( ψ ) is the expectation of ψ under the target distribution p .
Revised: where ψ is a matrix function and E p ( ψ ) is the expectation of ψ under the target distribution p .

Original: As h → 0 and Mh → ∞ , we have W M → 0 , and x ( i ) ,M can be regarded as approximately sampled from p .
Revised: As h → 0 and Mh → ∞ , we have W M → 0 , and x ( i ) ,M can be regarded as approximately sampled from p .

Original: According to the central limit theorem, we have lim h → 0 ,Mh →∞ Error M,N = O (1 / √ N ) .
Revised: According to the central limit theorem, we have lim h → 0 ,Mh →∞ Error M,N = O (1 / √ N ) .

Original: In this example, we set the target and initial distributions to be Gaussian p ( x ) ∝ p 1 ( x ) p 2 ( x ) and q 0 ( x ) ∝ p 1 ( x − e ) p 2 ( x ) with
Revised: In this example, we set the target and initial distributions to be Gaussian p ( x ) ∝ p 1 ( x ) p 2 ( x ) and q 0 ( x ) ∝ p 1 ( x − e ) p 2 ( x ) with

Original: [Equation]
Revised: [Equation]

Original: where x = ( x 1 , x 2 , . . .
Revised: where x = ( x 1 , x 2 , . . .

Original: , x 10 ) (cid:62) , e = (1 , 1 , . . .
Revised: , x 10 ) (cid:62) , e = (1 , 1 , . . .

Original: , 1) (cid:62) ∈ R 10 , I is the identity matrix and T is a random matrix with each entry i.i.d.
Revised: , 1) (cid:62) ∈ R 10 , I is the identity matrix and T is a random matrix with each entry i.i.d.

Original: drawn from N (0 , 1) .
Revised: drawn from N (0 , 1) .

Original: We run the simulation with N = 10 5 , and we compute Error M with ψ ( x )
Revised: We run the simulation with N = 10 5 , and we compute Error M with ψ ( x )

Original: =
Revised: =

Original: xx (cid:62) .
Revised: xx (cid:62) .

Original: This measures the spectral norm of the covariance matrix of the ﬁrst 10 entries.
Revised: This measures the spectral norm of the covariance matrix of the ﬁrst 10 entries.

Original: The results are shown in Figure 1.
Revised: The results are shown in Figure 1.

Original: We run RC-LMC with time stepsize h = 10 − 5 and α = 1 , following (21).
Revised: We run RC-LMC with time stepsize h = 10 − 5 and α = 1 , following (21).

Original: It is unclear what stepsize h to choose for LMC to yield a fair comparison.
Revised: It is unclear what stepsize h to choose for LMC to yield a fair comparison.

Original: Bearing in mind that d = 100 in this example, so that the per-iteration cost of LMC is 100 times of that of RC-LMC, we try ﬁrst h = 10 − 3 .
Revised: Bearing in mind that d = 100 in this example, so that the per-iteration cost of LMC is 100 times of that of RC-LMC, we try ﬁrst h = 10 − 3 .

Original: It is clear that RC-LMC, presented by the purple dashed line, achieves lower error than LMC at the same cost, before achieving the error plateau.
Revised: It is clear that RC-LMC, presented by the purple dashed line, achieves lower error than LMC at the same cost, before achieving the error plateau.

Original: Next, we try smaller choices of h in LMC.
Revised: Next, we try smaller choices of h in LMC.

Original: The choices h = .
Revised: The choices h = .

Original: 0008 and h = .
Revised: 0008 and h = .

Original: 0005 yield slower decay rates (see the red (star) and yellow (circle) lines, respectively), but lower error plateaus as well, meaning that the saturation error is smaller.
Revised: 0005 yield slower decay rates (see the red (star) and yellow (circle) lines, respectively), but lower error plateaus as well, meaning that the saturation error is smaller.

Original: However, computation required to reach these plateaus is longer, and the plateaus are still higher than for RC-LMC.
Revised: However, computation required to reach these plateaus is longer, and the plateaus are still higher than for RC-LMC.

Original: [Figure]
Revised: [Figure]

Original: A P ROOF OF P ROPOSITION 4. 
Revised: A P ROOF OF P ROPOSITION 4.

Original: We recall the SDE (7):
Revised: We recall the SDE (7):

Original: [Equation]
Revised: [Equation]

Original: where r m is randomly selected from 1 , . .
Revised: where r m is randomly selected from 1 , . .

Original: , d .
Revised: , d .

Original: Moreover, recall that X m +

Revised: Moreover, recall that X m +

Original: = X (cid:0) T m +1 (cid:1) is a
Revised: = X (cid:0) T m +1 (cid:1) is a

Original: Markovian process.
Revised: Markovian process.

Original: We denote its transition kernel by Ξ , meaning that
Revised: We denote its transition kernel by Ξ , meaning that

Original: [Equation]
Revised: [Equation]

Original: Moreover, we denote Ξ n the n -step transition kernel.
Revised: Moreover, we denote Ξ n the n -step transition kernel.

Original: Proposition 4.1 is a consequence of the following Proposition.
Revised: Proposition 4.1 is a consequence of the following Proposition.

Original: Proposition A.1.
Revised: Proposition A.1.

Original: Denote Π m the probability distribution of X m and Π be the probability distribution induced by p ( x ) , then under the conditions of Proposition 4.1, we have
Revised: Denote Π m the probability distribution of X m and Π be the probability distribution induced by p ( x ) , then under the conditions of Proposition 4.1, we have

Original: • Π is the stationary distribution of the Markov chain {
Revised: • Π is the stationary distribution of the Markov chain {

Original: X m } ∞ m =0 .
Revised: X m } ∞ m =0 .

Original: • If the second moment of Π 0 is ﬁnite and X 0 is drawn from Π 0 , then there are constants R > 0 and r > 1 , independent of m , such that for any m ≥ 

Revised: • If the second moment of Π 0 is ﬁnite and X 0 is drawn from Π 0 , then there are constants R > 0 and r > 1 , independent of m , such that for any m ≥

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: Remark A.1.
Revised: Remark A.1.

Original: According to Mattingly et al.
Revised: According to Mattingly et al.

Original: , the constants R and r do not depend on m , but their dependence on other parameters such as h , d , and L is hard to trace.
Revised: , the constants R and r do not depend on m , but their dependence on other parameters such as h , d , and L is hard to trace.

Original: This contrasts with the results in Dalalyan and Karagulyan (2019) for the classical Langevin dynamics, which are built upon the contraction property.
Revised: This contrasts with the results in Dalalyan and Karagulyan (2019) for the classical Langevin dynamics, which are built upon the contraction property.

Original: The new complication comes mainly from the complicated coordinate selection process, making the contraction property no longer available.
Revised: The new complication comes mainly from the complicated coordinate selection process, making the contraction property no longer available.

Original: Nor can we claim sharpness of the theorem.
Revised: Nor can we claim sharpness of the theorem.

Original: In fact, unlike in Dalalyan and Karagulyan (2019); Xu et al.
Revised: In fact, unlike in Dalalyan and Karagulyan (2019); Xu et al.

Original: (2018), where the authors directly studied LMC, we discuss here only convergence of the SDE, the continuous version of RC-LMC.
Revised: (2018), where the authors directly studied LMC, we discuss here only convergence of the SDE, the continuous version of RC-LMC.

Original: The explicit dependencies of the convergence rate here are unimportant, and we allow the results to be loose.
Revised: The explicit dependencies of the convergence rate here are unimportant, and we allow the results to be loose.

Original: Non-asymptotic convergence results of the algorithms are presented in Section 4.2 and 4.3.
Revised: Non-asymptotic convergence results of the algorithms are presented in Section 4.2 and 4.3.

Original: To prove Proposition A.1, we need to introduce the following lemma:
Revised: To prove Proposition A.1, we need to introduce the following lemma:

Original: Lemma A.1.
Revised: Lemma A.1.

Original: Under conditions of Theorem 4.1, there are constants R 1 > 0 , r 1 > 1 , such that for any x 0 ∈ R d
Revised: Under conditions of Theorem 4.1, there are constants R 1 > 0 , r 1 > 1 , such that for any z 0 ∈ R d

Original: [Equation]
Revised: [Equation]

Original: where x ∗ is the minimal point of f ( x ) and Ξ is the transition kernel for { X m } ∞ m =0 .
Revised: where x ∗ is the minimal point of f ( x ) and Ξ is the transition kernel for { X m } ∞ m =0 .

Original: We postpone the proof of Lemma A.1 to Section A.1.
Revised: We postpone the proof of Lemma A.1 to Section A.1.

Original: Now, we are ready to prove the proposition.
Revised: Now, we are ready to prove the proposition.

Original: Proof of Proposition A.1 (Proposition 4.1).
Revised: Proof of Proposition A.1 (Proposition 4.1).

Original: First, suppose the distribution of X m is Π.
Revised: To prove the ﬁrst bullet point of Proposition A.1, we assume the distribution of X m is Π

Original: It sufﬁces to prove:
Revised: and we need to prove:

Original: For any choice of r m , the conditional distribution of X m +1 is  Π .
Revised: For any choice of r m , the conditional distribution of X m +1 is also Π .

Original: Without loss of generality, we considerthe condition r m = 
Revised: Without loss of generality, we consider r m = 1 .

Original: Under this condition, we have the following.
Revised: Under this condition, we have the following.

Original: • The distribution of X 2 ≤ j ≤ d ( t ) between [ T m , T m +1 ] is preserved.
Revised: • The distribution of X 2 ≤ j ≤ d ( t ) between [ T m , T m +1 ] is preserved.

Original: • For ﬁxed z 2 , z 3 , . . .
Revised: • For ﬁxed z 2 , z 3 , . . .

Original: , z d , the stationary density of SDE
Revised: , z d , the stationary density of SDE

Original: [Equation]
Revised: [Equation]

Original: is
Revised: is

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: z
Revised: z

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: )d z .
Revised: )d z .

Original: This implies that the conditional distribution of X 1 ( t ) with ﬁxed
Revised: This implies that the conditional distribution of X 1 ( t ) with ﬁxed

Original: X 2 ≤ j ≤ d ( t ) is  preserved.
Revised: X 2 ≤ j ≤ d ( t ) is also preserved.

Original: By combining these two points, we ﬁnd that under condition r m = 1 , the conditional distribution of
Revised: Combining these two points, we ﬁnd that under condition r m = 1 , the conditional distribution of

Original: X m +1 is Π , which further proves that Π is the stationary distribution and Proposition 4.1 holds.
Revised: X m +1 is Π , which further proves that Π is the stationary distribution and Proposition 4.1 holds.

Original: Second, to prove (29), we take the expectation of (30) using Π 0 , then we can obtain that for any A ∈ B R d hat
Revised: To show (29), we take the expectation of (30) using Π 0 , then we can obtain that for any A ∈ B R d hat

Original: [Equation]
Revised: [Equation]

Original: where we use X md d = Ξ md ( X 0 , · ) and Π( A ) =
Revised: where we use X md d = Ξ md ( X 0 , · ) and Π( A ) =

Original: (cid:82) A p ( x ) d x in ( I ) , Π 0 is a non-negative measure in ( II ) and (30) in ( III ) .
Revised: (cid:82) A p ( x ) d x in ( I ) , Π 0 is a non-negative measure in ( II ) and (30) in ( III ) .

Original: Since this is true for all A ∈ B R d , we have
Revised: Since this is true for all A ∈ B R d , we have

Original: [Equation]
Revised: [Equation]

Original: By using (28) with It ˆ o’s formula, we have
Revised: By using (28) with Itˆo’s formula, we have

Original: [Equation]
Revised: [Equation]

Original: where C 1 ,r m and C 2 ,r m are constants that depend only on x ∗ and L r m .
Revised: where C 1 ,r m and C 2 ,r m are constants that depend only on x ∗ and L r m .

Original: From Gr ¨ onwall’s inequality, we obtain
Revised: From Gr ¨ onwall’s inequality, we obtain

Revised: E

Original: [Equation]
Revised: [Equation]

Original: for all i = 1 , 2 , . . .
Revised: for all i = 1 , 2 , . . .

Original: , d .
Revised: , d .

Original: Then, if E | X m | 2 < ∞ , we have for any i = 1 , 2 , . . .
Revised: Then, if E | X m | 2 < ∞ , we have for any i = 1 , 2 , . . .

Original: , d that
Revised: , d that

Original: [Equation]
Revised: [Equation]

Original: which implies E |
Revised: which implies E |

Original: X m +

Revised: X m +

Original: | 2 < ∞ .
Revised: | 2 < ∞ .

Original: Therefore, if Π 0 has ﬁnite second moment, then Π i all have ﬁnite second moments for i = 1 , . . .
Revised: Therefore, if Π 0 has ﬁnite second moment, then Π i all have ﬁnite second moments for i = 1 , . . .

Original: , d − 1 .
Revised: , d − 1 .

Original: Letting x 0 ∼ Π i  , we take the expectation of (30)using Π i and  obtain
Revised: Suppose the initial data is drawn from Π i for i < d , then taking the expectation of (30) and using (32), we obtain

Original: [Equation]
Revised: [Equation]

Original: where C i is a constant.
Revised: where C i is a constant.

Original: Since this bound holds  for all 0 ≤ i ≤ d − 1 , we set R =
Revised: This bound holds true for all 0 ≤ i ≤ d − 1 , we set R =

Original: (max i C i )
Revised: (max i C i )

Original: r 1 and r = r 1 /dto obtain (29).
Revised: r 1 and r = r 1 /dto obtain (29).

Original: A. P ROOF OF L EMMA A. 
Revised: A. P ROOF OF L EMMA A.

Original: Before we prove the Lemma, we ﬁrst recall a result from (Mattingly et al., 2002) for the convergence of Markov chain using Lyapunov condition together with minorization condition.
Revised: Before we prove the Lemma, we ﬁrst recall a result from (Mattingly et al., 2002) for the convergence of Markov chain using Lyapunov condition together with minorization condition.

Original: Theorem A.1.
Revised: Theorem A.1.

Original: [(Mattingly et al., 2002, Theorem 2.5)] Let { X n } ∞ n =0 denote the Markov chain on R d with transition kernel Ξ and ﬁltration F n .
Revised: [(Mattingly et al., 2002, Theorem 2.5)] Let { X n } ∞ n =0 denote the Markov chain on R d with transition kernel Ξ and ﬁltration F n .

Original: Let { X n } ∞ n =

Revised: Let { X n } ∞ n =

Original: satisfy the following two conditions:
Revised: satisfy the following two conditions:

Original: Lyapunov condition: There is a function L : R d → [1 , ∞ ) , with lim x →∞ L ( x ) = ∞ , and real numbers α ∈ (0 , 1) , and β ∈ [0 , ∞ ) such that
Revised: Lyapunov condition: There is a function L : R d → [1 , ∞ ) , with lim x →∞ L ( x ) = ∞ , and real numbers α ∈ (0 , 1) , and β ∈ [0 , ∞ ) such that

Original: [Equation]
Revised: [Equation]

Original: Minorization condition:
Revised: Minorization condition:

Original: For L from the Lyqpunov condition, deﬁne the set C ⊂ R d as follows:
Revised: For L from the Lyqpunov condition, deﬁne the set C ⊂ R d as follows:

Original: [Equation]
Revised: [Equation]

Original: for some γ ∈ ( α 1 / 2 , 1) .
Revised: for some γ ∈ ( α 1 / 2 , 1) .

Original: Then there exists an η > 0 and a probability measure M supported on C (that is, M ( C ) = 1 ), such that
Revised: Then there exists an η > 0 and a probability measure M supported on C (that is, M ( C ) = 1 ), such that

Original: [Equation]
Revised: [Equation]

Original: Under these conditions, the Markov chain { X n } ∞ n =0 has a unique invariant measure π .
Revised: Under these conditions, the Markov chain { X n } ∞ n =0 has a unique invariant measure π .

Original: Furthermore, there are constants r ∈ (0 , 1) and R ∈ (0 , ∞ ) such that, for any x 0 ∈ R d , we have
Revised: Furthermore, there are constants r ∈ (0 , 1) and R ∈ (0 , ∞ ) such that, for any z 0 ∈ R d , we have

Original: [Equation]
Revised: [Equation]

Original: To use this result to prove Lemma A.1, we will consider the d -step chain of { X n } and verify the two conditions, as in the following two lemmas for the Lyapunov function and the minorization over a small set, respectively.
Revised: To use this result to prove Lemma A.1, we will consider the d -step chain of { X n } and verify the two conditions, as in the following two lemmas for the Lyapunov function and the minorization over a small set, respectively.

Original: Lemma A.2.
Revised: Lemma A.2.

Original: Assume f satisﬁes Assumption 3.1 and
Revised: Assume f satisﬁes Assumption 3.1 and

Original: [Equation]
Revised: [Equation]

Original: where L is the Lipschitz constant deﬁned in (9) .
Revised: where L is the Lipschitz constant deﬁned in (9) .

Original: Let the Lyapunov function be L ( x ) =
Revised: Let the Lyapunov function be L ( x ) =

Original: | x − x ∗ | 2 +1 , then we have:
Revised: | x − x ∗ | 2 +1 , then we have:

Original: [Equation]
Revised: [Equation]

Original: with α
Revised: with α

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation]
Revised: [Equation]

Original: −
Revised: −

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: Lemma A.3.
Revised: Lemma A.3.

Original: Under conditions of Lemma A.2, with L ( x ) =
Revised: Under conditions of Lemma A.2, with L ( x ) =

Original: | x − x ∗ | 2 +1 , let Ξ denote the transition kernel.
Revised: | x − x ∗ | 2 +1 , let Ξ denote the transition kernel.

Original: Deﬁne the set C ⊂ R d as in (33) , for some γ ∈ ( α 1 / 2 , 1) .
Revised: Deﬁne the set C ⊂ R d as in (33) , for some γ ∈ ( α 1 / 2 , 1) .

Original: Then there exists an η > 0 and a probability measure M with M ( C ) = 1 , such that
Revised: Then there exists an η > 0 and a probability measure M with M ( C ) = 1 , such that

Original: [Equation]
Revised: [Equation]

Original: Lemma A.1 follows easily from these results.
Revised: Lemma A.1 follows easily from these results.

Original: Proof of Lemma A.1.
Revised: Proof of Lemma A.1.

Original: It sufﬁces to show d -step chain (cid:8) X md (cid:9)
Revised: It sufﬁces to show d -step chain (cid:8) X md (cid:9)

Original: ∞ m =0 satisﬁes the conditions in Theorem A.1 with L ( x ) =
Revised: ∞ m =0 satisﬁes the conditions in Theorem A.1 with L ( x ) =

Original: | x − x ∗ | 2 + 1 , α = α d 1 and β
Revised: | x − x ∗ | 2 + 1 , α = α d 1 and β

Original: = dβ 1 , and π is induced by p .
Revised: = dβ 1 , and π is induced by p .

Original: We apply (36) from Lemma A.2 iteratively, d times, to obtain
Revised: We apply (36) from Lemma A.2 iteratively, d times, to obtain

Original: [Equation]
Revised: [Equation]

Original: , which implies that (cid:8) X md (cid:9) ∞ m
Revised: , which implies that (cid:8) X md (cid:9) ∞ m

Original: =0 satisﬁes
Revised: =0 satisﬁes

Original: Lyapunov condition in Theorem A.1 with α = α d 1 .
Revised: Lyapunov condition in Theorem A.1 with α = α d 1 .

Original: Moreover, Lemma A.3 directly implies that the d -step transition kernel satisﬁes the minorization condition.
Revised: Moreover, Lemma A.3 directly implies that the d -step transition kernel satisﬁes the minorization condition.

Original: Therefore, by Theorem A.1, we have
Revised: Therefore, by Theorem A.1, we have

Original: [Equation]
Revised: [Equation]

Original: which concludes the proof of the lemma when we substitute π ( A )
Revised: which concludes the proof of the lemma when we substitute π ( A )

Original: = (cid:82) A p ( x ) d x .
Revised: = (cid:82) A p ( x ) d x .

Original: Proof of Lemma A.2.
Revised: Proof of Lemma A.2.

Original: We assume without loss of generality that x ∗ = 0 ∈ R d
Revised: We assume without loss of generality that x ∗ = 0 ∈ R d

Original: (so that L ( x ) =
Revised: (so that L ( x ) =

Original: | x | 2 + 1 ) and drop the ﬁltration F m in the formula for simplicity of notation.
Revised: | x | 2 + 1 ) and drop the ﬁltration F m in the formula for simplicity of notation.

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: Since
Revised: Since

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: To deal with second term and third term in (39), we ﬁrst note that, under condition r m =
Revised: To deal with second term and third term in (39), we ﬁrst note that, under condition r m =

Original: i :
Revised: i :

Original: [Equation]
Revised: [Equation]

Original: This means
Revised: This means

Original: [Equation]
Revised: [Equation]

Original: We further bound the second term of (41):
Revised: We further bound the second term of (41):

Original: [Equation]
Revised: [Equation]

Original: where we used Young’s inequality in (I), the Lipschitz condition in (II), Lemma A.4 below (specifically, inequality (45)) in (III), and the Lipschitz condition again in (IV).
Revised: where we used Young’s inequality in (I), the Lipschitz condition in (II), Lemma A.4 below (specifically, inequality (45)) in (III), and the Lipschitz condition again in (IV).

Original: This, when substituted into (41), gives
Revised: This, when substituted into (41), gives

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: where we used Young’s inequality in (I), Lemma A.4 below (speciﬁcally, inequality (44)) in (II), and Lipschitz continuity in (III), together with 88 h 2 i L 2 i ≤ 20 by (35).
Revised: where we used Young’s inequality in (I), Lemma A.4 below (speciﬁcally, inequality (44)) in (II), and Lipschitz continuity in (III), together with 88 h 2 i L 2 i ≤ 20 by (35).

Original: Finally, we have
Revised: Finally, we have

Original: [Equation]
Revised: [Equation]

Original: By summing according to (38), and using (4) and L i ≤ L for all i = 1 , 2 , . . .
Revised: By summing according to (38), and using (4) and L i ≤ L for all i = 1 , 2 , . . .

Original: , d , we obtain
Revised: , d , we obtain

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: Finally, using (cid:104) X m , ∇ f ( X m ) (cid:105) ≥ µ ( L ( X m ) − 1) (from (10) with x (cid:48) =
Revised: Finally, using (cid:104) X m , ∇ f ( X m ) (cid:105) ≥ µ ( L ( X m ) − 1) (from (10) with x (cid:48) =

Original: X m and x = x ∗ = 0 ) and (35), we obtain (36).
Revised: X m and x = x ∗ = 0 ) and (35), we obtain (36).

Original: Proof of Lemma A.3.
Revised: Proof of Lemma A.3.

Original: To prove (37), we construct a new Markov process (cid:101) X m .
Revised: To prove (37), we construct a new Markov process (cid:101) X m .

Original: Deﬁning (cid:101) X 

Revised: Deﬁning (cid:101) X

Original: = x 0 , we obtain (cid:101)
Revised: = x 0 , we obtain (cid:101)

Original: X m +1 from (cid:101)
Revised: X m +1 from (cid:101)

Original: X m by running the following process:
Revised: X m by running the following process:

Original: [Equation]
Revised: [Equation]

Original: i =
Revised: i =

Original: h i , (cid:101) T
Revised: h i , (cid:101) T

Original: [Equation]
Revised: [Equation]

Original: = 0 ,
Revised: = 0 ,

Original: Z
Revised: Z

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: = (cid:101) X m .
Revised: = (cid:101) X m .

Original: Then for (cid:101) T n − 1 ≤ t ≤ (cid:101) T n and n ≤ d , let  
Revised: Then for (cid:101) T n − 1 ≤ t ≤ (cid:101) T n and n ≤ d , let  

Original: [Equation]
Revised: [Equation]

Original: Z
Revised: Z

Original: [Equation]
Revised: [Equation]

Original: (cid:17) , i (cid:54) = n , and set (cid:101)
Revised: (cid:17) , i (cid:54) = n , and set (cid:101)

Original: X m +

Revised: X m +

Original: =
Revised: =

Original: Z (cid:16) (cid:101) T d (cid:17) .
Revised: Z (cid:16) (cid:101) T d (cid:17) .

Original: Denote the transition kernel by Ξ cyc (corresponding to one round of a cyclic version of the coordinate algorithm).
Revised: Denote the transition kernel by Ξ cyc (corresponding to one round of a cyclic version of the coordinate algorithm).

Original: We then have the following properties:
Revised: We then have the following properties:

Original: • For any x ∈ C and A ∈ B ( R d ) , we have Ξ d ( x, A ) ≥
Revised: • For any x ∈ C and A ∈ B ( R d ) , we have Ξ d ( x, A ) ≥

Original: Π di =1 φ i Ξ cyc ( x, A ) > 0 .
Revised: Π di =1 φ i Ξ cyc ( x, A ) > 0 .

Original: • Ξ cyc possesses a positive jointly continuous density.
Revised: • Ξ cyc possesses a positive jointly continuous density.

Original: According to (Mattingly et al., 2002, Lemma 2.3), since Ξ cyc has a positive jointly continuous density, there exists an η (cid:48) > 0 and a probability measure M with M ( C ) = 1 , such that
Revised: According to (Mattingly et al., 2002, Lemma 2.3), since Ξ cyc has a positive jointly continuous density, there exists an η (cid:48) > 0 and a probability measure M with M ( C ) = 1 , such that

Original: [Equation]
Revised: [Equation]

Original: which implies
Revised: which implies

Original: [Equation]
Revised: [Equation]

Original: This proves (37) by setting η = Π di =1 φ i η (cid:48) .
Revised: This proves (37) by setting η = Π di =1 φ i η (cid:48) .

Original: In the proof of Lemma A.2, we used several estimates in inequalities (42) and (43).
Revised: In the proof of Lemma A.2, we used several estimates in inequalities (42) and (43).

Original: We prove these estimates in the following lemma.
Revised: We prove these estimates in the following lemma.

Original: Lemma A.4.
Revised: Lemma A.4.

Original: Suppose that the assumptions of Lemma A.2 hold
Revised: Suppose that the assumptions of Lemma A.2 hold

Original: [Equation]
Revised: [Equation]

Original: and let X i evolve according to (40) .
Revised: and let X i evolve according to (40) .

Original: Then we have the following bounds:
Revised: Then we have the following bounds:

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: To obtain (44)
Revised: To obtain (44)

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: To bound the second term, we use (40) again:
Revised: To bound the second term, we use (40) again:

Original: [Equation]
Revised: [Equation]

Original: where we use Young’s inequality and
Revised: where we use Young’s inequality and

Original: [Equation]
Revised: [Equation]

Original: by Doob’s maximal inequality.
Revised: by Doob’s maximal inequality.

Original: By substituting (47) into (46), we obtain
Revised: By substituting (47) into (46), we obtain

Original: E (cid:18) sup
Revised: E (cid:18) sup

Original: [Equation]
Revised: [Equation]

Original: ≤
Revised: ≤

Original: [Equation]
Revised: [Equation]

Original: ∂
Revised: ∂

Original: [Equation]
Revised: [Equation]

Original: Using h i L i ≤ 14 , we move the ﬁrst term on the right to the left to obtain
Revised: Using h i L i ≤ 14 , we move the ﬁrst term on the right to the left to obtain

Original: [Equation]
Revised: [Equation]

Original: leading to (44).
Revised: leading to (44).

Original: Then we obtain (45) by plugging this in (47) and using the fact that 88 h 3 i L 2 i < 14 h i by (35).
Revised: Then we obtain (45) by plugging this in (47) and using the fact that 88 h 3 i L 2 i < 14 h i by (35).

Original: B P ROOF OF T HEOREM 4. 
Revised: B P ROOF OF T HEOREM 4.

Original: The proof of this theorem requires us to design a reference solution to explicitly bound W ( q m , p ) .
Revised: The proof of this theorem requires us to design a reference solution to explicitly bound W ( q m , p ) .

Original: Let ˜ x 0 be a random vector drawn from target distribution induced by p , so that W 22 ( q 0 , p ) =
Revised: Let ˜ x 0 be a random vector drawn from target distribution induced by p , so that W 22 ( q 0 , p ) =

Original: E | x 0 − ˜ x 0 | 2 .
Revised: E | x 0 − ˜ x 0 | 2 .

Original: We then require ˜ x to solve the following SDE: for t ∈ ( T m , T m +1 ] , with T m deﬁned in (6):
Revised: We then require ˜ x to solve the following SDE: for t ∈ ( T m , T m +1 ] , with T m deﬁned in (6):

Original: [Equation]
Revised: [Equation]

Original: If we use the same Brownian motion as in (5), we have
Revised: If we use the same Brownian motion as in (5), we have

Original: [Equation]
Revised: [Equation]

Original: where e r m is the unit vector in r m direction.
Revised: where e r m is the unit vector in r m direction.

Original: Since the r m -th marginal distribution of ˜ x ( t ) is preserved in each time step according to (48), the whole distribution of ˜ x ( t ) is preserved to be p for all t .
Revised: Since the r m -th marginal distribution of ˜ x ( t ) is preserved in each time step according to (48), the whole distribution of ˜ x ( t ) is preserved to be p for all t .

Original: Therefore, by the deﬁnition W m = W ( q m , p ) , we have
Revised: Therefore, by the deﬁnition W m = W ( q m , p ) , we have

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: This means bounding W m amounts to evaluating E | ∆ m | 2 .
Revised: This means bounding W m amounts to evaluating E | ∆ m | 2 .

Original: Under Assumption 3.1, we have the following result.
Revised: Under Assumption 3.1, we have the following result.

Original: Proposition B.1.
Revised: Proposition B.1.

Original: Suppose the assumptions of Theorem 4.1 are satisﬁed and let { x m } , { ˜ x m } , and { ∆ m } be deﬁned in (5) , (48) , and (50) , respectively.
Revised: Suppose the assumptions of Theorem 4.1 are satisﬁed and let { x m } , { ˜ x m } , and { ∆ m } be deﬁned in (5) , (48) , and (50) , respectively.

Original: Then, we have
Revised: Then, we have

Original: [Equation]
Revised: [Equation]

Original: The proof of this result appears in Appendix B.1.
Revised: The proof of this result appears in Appendix B.1.

Original: The proof for Theorem 4.1 is now immediate.
Revised: The proof for Theorem 4.1 is now immediate.

Original: Proof of Theorem 4.1.
Revised: Proof of Theorem 4.1.

Original: By iterating (51), we obtain
Revised: By iterating (51), we obtain

Original: [Equation]
Revised: [Equation]

Original: and since hµ/ 2 ∈ (0 , 1) , we have
Revised: and since hµ/ 2 ∈ (0 , 1) , we have

Original: [Equation]
Revised: [Equation]

Original: By construction, we have W 2 ( q 0 , p ) =
Revised: By construction, we have W 2 ( q 0 , p ) =

Original: E | ∆ 0 | 2 and W 2 ( q m , p ) ≤ E | ∆ m | 2 .
Revised: E | ∆ 0 | 2 and W 2 ( q m , p ) ≤ E | ∆ m | 2 .

Original: By taking the square root of both sides and using a 2 ≤ b 2 + c 2 ⇒ a ≤ b + c for any nonnegative a , b , and c , we arrive at (20).
Revised: By taking the square root of both sides and using a 2 ≤ b 2 + c 2 ⇒ a ≤ b + c for any nonnegative a , b , and c , we arrive at (20).

Original: The proof for Corollary 4.1 is also obvious.
Revised: The proof for Corollary 4.1 is also obvious.

Original: Proof of Corollary 4.1.
Revised: Proof of Corollary 4.1.

Original: To ensure that W m ≤ (cid:15) , we set the two terms on the right hand side of (20) to be smaller than (cid:15)/ 2 , which implies that
Revised: To ensure that W m ≤ (cid:15) , we set the two terms on the right hand side of (20) to be smaller than (cid:15)/ 2 , which implies that

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: By using the deﬁnition of φ i ( α ) according to (21), we obtain
Revised: By using the deﬁnition of φ i ( α ) according to (21), we obtain

Original: [Equation]
Revised: [Equation]

Original: which implies that m = (cid:101) O (cid:0)
Revised: which implies that m = (cid:101) O (cid:0)

Original: ( K 2 − α K α ) / ( µ(cid:15) 2 ) (cid:1) .
Revised: ( K 2 − α K α ) / ( µ(cid:15) 2 ) (cid:1) .

Original: Furthermore, α = 1 gives the optimal cost, because:
Revised: Furthermore, α = 1 gives the optimal cost, because:

Original: [Equation]
Revised: [Equation]

Original: due to H¨older’s inequality.
Revised: due to H¨older’s inequality.

Original: B. P ROOF OF P ROPOSITION B. 
Revised: B. P ROOF OF P ROPOSITION B.

Original: We prove the Proposition by means of the following lemma.
Revised: We prove the Proposition by means of the following lemma.

Original: Lemma B
Revised: Lemma B

Original: [Equation]
Revised: [Equation]

Original: Under the conditions of Proposition B.1, for m ≥ 0 and i = 1 , 2 , . . .
Revised: Under the conditions of Proposition B.1, for m ≥ 0 and i = 1 , 2 , . . .

Original: , d , we have
Revised: , d , we have

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: In the m -th time step
Revised: In the m -th time step

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: so that
Revised: so that

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: We now analyze the ﬁrst term on the right hand side under condition r m = i .
Revised: We now analyze the ﬁrst term on the right hand side under condition r m = i .

Original: By deﬁnition of ∆ m +

Revised: By deﬁnition of ∆ m +

Original: i , we have
Revised: i , we have

Original: [Equation]
Revised: [Equation]

Original: where we have deﬁned
Revised: where we have deﬁned

Original: [Equation]
Revised: [Equation]

Original: By Young’s inequality, we have
Revised: By Young’s inequality, we have

Original: [Equation]
Revised: [Equation]

Original: where a > 0 is a parameter to be speciﬁed later.
Revised: where a > 0 is a parameter to be speciﬁed later.

Original: For the ﬁrst term on the right hand side of (58), we have
Revised: For the ﬁrst term on the right hand side of (58), we have

Original: [Equation]
Revised: [Equation]

Original: Note term will become the third line in (54) (upon the proper choice of a ).
Revised: Note term will become the third line in (54) (upon the proper choice of a ).

Original: For very small h , this term is negligible.
Revised: For very small h , this term is negligible.

Original: For the second term on the right-hand side of (58), we recall the deﬁnition (57) and obtain
Revised: For the second term on the right-hand side of (58), we recall the deﬁnition (57) and obtain

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: where (II) comes from L -Lipschitz condition (11), (I) and (III) come from the use of Young’s inequality and Jensen’s inequality when we move the | · | 2 from outside to inside of the integral, and (IV) and (V) hold true because ˜ x ( t ) ∼ p for all t .
Revised: where (II) comes from L -Lipschitz condition (11), (I) and (III) come from the use of Young’s inequality and Jensen’s inequality when we move the | · | 2 from outside to inside of the integral, and (IV) and (V) hold true because ˜ x ( t ) ∼ p for all t .

Original: In (VI) we use E p | ∂ i f | 2 ≤ L i using (Dalalyan and Karagulyan, 2019, Lemma 3).
Revised: In (VI) we use E p | ∂ i f | 2 ≤ L i using (Dalalyan and Karagulyan, 2019, Lemma 3).

Original: By substituting (59) and (60) into the right hand side of (58)
Revised: By substituting (59) and (60) into the right hand side of (58)

Original: [Equation]
Revised: [Equation]

Original: we obtain
Revised: we obtain

Original: [Equation]
Revised: [Equation]

Original: By substituting (61) into (55)
Revised: By substituting (61) into (55)

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: E |
Revised: E |

Original: [Equation]
Revised: [Equation]

Original: m +

Revised: m +

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: where we have used h i φ i
Revised: where we have used h i φ i

Original: = h .
Revised: = h .

Original: Now, we need to choose a value of a > 0 appropriate to establish (54).
Revised: Now, we need to choose a value of a > 0 appropriate to establish (54).

Original: By comparing the two formulas, we see the need to set
Revised: By comparing the two formulas, we see the need to set

Original: [Equation]
Revised: [Equation]

Original: since h ≤ min { φ i } /µ .
Revised: since h ≤ min { φ i } /µ .

Original: It follows that 1 + 1 a ≤ 2 φ i hµ .
Revised: It follows that 1 + 1 a ≤ 2 φ i hµ .

Original: By substituting into (62), we obtain
Revised: By substituting into (62), we obtain

Original: [Equation]
Revised: [Equation]

Original: We conclude the lemma by using the following Cauchy-Schwartz inequality to control the third term on the right hand side of this expression:
Revised: We conclude the lemma by using the following Cauchy-Schwartz inequality to control the third term on the right hand side of this expression:

Original: [Equation]
Revised: [Equation]

Original: Proposition B.1 is obtained by simply summing all components in the lemma.
Revised: Proposition B.1 is obtained by simply summing all components in the lemma.

Original: Proof of Proposion B.1.
Revised: Proof of Proposion B.1.

Original: Noting
Revised: Noting

Original: [Equation]
Revised: [Equation]

Original: we bound the right hand side by (54) and get
Revised: we bound the right hand side by (54) and get

Original: [Equation]
Revised: [Equation]

Original: The second and third terms on the right-hand side can be bounded in terms of E | ∆ m | 2 :
Revised: The second and third terms on the right-hand side can be bounded in terms of E | ∆ m | 2 :

Original: • By
Revised: • By

Original: convexity
Revised: convexity

Original: , we
Revised: , we

Original: have
Revised: have

Original: [Equation]
Revised: [Equation]

Original: • As the gradient is L -Lipschitz,
Revised: • As the gradient is L -Lipschitz,

Original: we
Revised: we

Original: have
Revised: have

Original: [Equation]
Revised: [Equation]

Original: By substituting (65) and (66) into (64) and using µ ≤ L
Revised: By substituting (65) and (66) into (64) and using µ ≤ L

Original: [Equation]
Revised: [Equation]

Original: we obtain
Revised: we obtain

Original: [Equation]
Revised: [Equation]

Original: If we take h sufﬁciently small, the coefﬁcient in front of E | ∆ m | 2 is strictly smaller than 1 , ensuring the decay of the error.
Revised: If we take h sufﬁciently small, the coefﬁcient in front of E | ∆ m | 2 is strictly smaller than 1 , ensuring the decay of the error.

Original: Indeed, by setting h ≤ µ min { φ i } 8 L, we have 4 h 2 Lmin { φ i } ≤ hµ 2 , and hL i φ i ≤ µ 8 L ≤ 1 , which leads to the iteration formula (51).
Revised: Indeed, by setting h ≤ µ min { φ i } 8 L, we have 4 h 2 Lmin { φ i } ≤ hµ 2 , and hL i φ i ≤ µ 8 L ≤ 1 , which leads to the iteration formula (51).

Original: C P ROOF OF T HEOREM 4. 
Revised: C P ROOF OF T HEOREM 4.

Original: Theorem 4.2 is based on the following proposition.
Revised: Theorem 4.2 is based on the following proposition.

Original: Proposition C.1.
Revised: Proposition C.1.

Original: Suppose the assumptions of Theorem 4.2 and let { x m } , { ˜ x m } , and { ∆ m } be deﬁned as in (5) , (48) , and (50) , respectively.
Revised: Suppose the assumptions of Theorem 4.2 and let { x m } , { ˜ x m } , and { ∆ m } be deﬁned as in (5) , (48) , and (50) , respectively.

Original: Then we have
Revised: Then we have

Original: [Equation]
Revised: [Equation]

Original: We prove this result in Appendix C.1.
Revised: We prove this result in Appendix C.1.

Original: The proof of the theorem is now immediate.
Revised: The proof of the theorem is now immediate.

Original: Proof of Theorem 4.2.
Revised: Proof of Theorem 4.2.

Original: Use (68) iteratively, we have
Revised: Use (68) iteratively, we have

Original: [Equation]
Revised: [Equation]

Original: Using W 2 ( q 0 , p ) =
Revised: Using W 2 ( q 0 , p ) =

Original: E | ∆ 0 | 2 and W 2 ( q m , p ) ≤ E | ∆ m | 2 , we take the square root on both sides, we obtain (23).
Revised: E | ∆ 0 | 2 and W 2 ( q m , p ) ≤ E | ∆ m | 2 , we take the square root on both sides, we obtain (23).

Original: The proof of Corollary 4.2 is also immediate.
Revised: The proof of Corollary 4.2 is also immediate.

Original: Proof of Corollary 4.2.
Revised: Proof of Corollary 4.2.

Original: Use (23), to ensure W m ≤ (cid:15) , we set two terms on the right hand side of (23) to be smaller than (cid:15)/ 2 , which implies that
Revised: Use (23), to ensure W m ≤ (cid:15) , we set two terms on the right hand side of (23) to be smaller than (cid:15)/ 2 , which implies that

Original: [Equation]
Revised: [Equation]

Original: To ﬁnd optimal choice of φ i , we need to minimize
Revised: To ﬁnd optimal choice of φ i , we need to minimize

Original: [Equation]
Revised: [Equation]

Original: under constraint (cid:80) di φ i = 1 and φ i > 0 .
Revised: under constraint (cid:80) di φ i = 1 and φ i > 0 .

Original: Introducing a Lagrange multiplier λ ∈ R , deﬁne the
Revised: Introducing a Lagrange multiplier λ ∈ R , deﬁne the

Original: Lagrangian function as follows:
Revised: Lagrangian function as follows:

Original: [Equation]
Revised: [Equation]

Original: By setting ∂F/∂φ i = 0 for all i , and substituting into the constraint (cid:80) di φ i = 1 to ﬁnd the appropriate value of λ , we ﬁnd that the optimal ( φ 1 , φ 2 , . .
Revised: By setting ∂F/∂φ i = 0 for all i , and substituting into the constraint (cid:80) di φ i = 1 to ﬁnd the appropriate value of λ , we ﬁnd that the optimal ( φ 1 , φ 2 , . .

Original: . , φ d ) satisﬁes
Revised: . , φ d ) satisﬁes

Original: [Equation]
Revised: [Equation]

Original: By substituting into (69), we obtain (24).
Revised: By substituting into (69), we obtain (24).

Original: C. P ROOF OF P ROPOSITION C. 
Revised: C. P ROOF OF P ROPOSITION C.

Original: The strategy of the proof for this proposition is almost identical to that of the previous section.
Revised: The strategy of the proof for this proposition is almost identical to that of the previous section.

Original: The reference solution ˜ x is deﬁned as in (48).
Revised: The reference solution ˜ x is deﬁned as in (48).

Original: We will use the following lemma:
Revised: We will use the following lemma:

Original: Lemma C.1.
Revised: Lemma C.1.

Original: Under the conditions of Proposition C.1, for m ≥ 0 and i = 1 , 2 , . . .
Revised: Under the conditions of Proposition C.1, for m ≥ 0 and i = 1 , 2 , . . .

Original: , d , we have
Revised: , d , we have

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: In the m -th time step
Revised: In the m -th time step

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: meaning that
Revised: meaning that

Original: [Equation]
Revised: [Equation]

Original: To bound the ﬁrst term in (55) we use the deﬁnition of ∆ m
Revised: To bound the ﬁrst term in (55) we use the deﬁnition of ∆ m

Original: +1 i .
Revised: +1 i .

Original: Under the condition r m =
Revised: Under the condition r m =

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: we have, with the same derivation as in (56):
Revised: we have, with the same derivation as in (56):

Original: [Equation]
Revised: [Equation]

Original: where we denoted V m = (cid:82) T m
Revised: where we denoted V m = (cid:82) T m

Original: + h
Revised: + h

Original: i T m
Revised: i T m

Original: [Equation]
Revised: [Equation]

Original: f
Revised: f

Original: [Equation]
Revised: [Equation]

Original: i f
Revised: i f

Original: [Equation]
Revised: [Equation]

Original: d s .
Revised: d s .

Original: However, different from (60), since f has higher regularity, we can ﬁnd a tighter bound for the integral.
Revised: However, different from (60), since f has higher regularity, we can ﬁnd a tighter bound for the integral.

Original: Denote
Revised: Denote

Original: [Equation]
Revised: [Equation]

Original: and
Revised: and

Original: [Equation]
Revised: [Equation]

Original: Then (72) can be written as
Revised: Then (72) can be written as

Original: [Equation]
Revised: [Equation]

Original: which implies, according to Young’s inequality
Revised: which implies, according to Young’s inequality

Original: [Equation]
Revised: [Equation]

Original: that, for any a :
Revised: that, for any a :

Original: [Equation]
Revised: [Equation]

Original: Both terms on the right-hand side of (76) are small.
Revised: Both terms on the right-hand side of (76) are small.

Original: We now control the ﬁrst term.
Revised: We now control the ﬁrst term.

Original: Plug in the deﬁnition (75), we have:
Revised: Plug in the deﬁnition (75), we have:

Original: [Equation]
Revised: [Equation]

Original: Noting that
Revised: Noting that

Original: [Equation]
Revised: [Equation]

Original: because
Revised: because

Original: [Equation]
Revised: [Equation]

Original: according to the property of It ˆ o’s integral
Revised: according to the property of It ˆ o’s integral

Original: [Equation]
Revised: [Equation]

Original: we can discard the cross terms with Φ m in (77) to obtain
Revised: we can discard the cross terms with Φ m in (77) to obtain

Original: E (cid:0) |
Revised: E (cid:0) |

Original: [Equation]
Revised: [Equation]

Original: m
Revised: m

Original: [Equation]
Revised: [Equation]

Original: For the last term of (78)
Revised: For the last term of (78)

Original: [Equation]
Revised: [Equation]

Original: we have the following control:
Revised: we have the following control:

Original: [Equation]
Revised: [Equation]

Original: 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12) r m = i 
Revised: 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12) r m = i 

Original:  (I) ≤ 2 E (cid:34)(cid:32)(cid:90) T m
Revised:  (I) ≤ 2 E (cid:34)(cid:32)(cid:90) T m

Original: + h
Revised: + h

Original: i
Revised: i

Original: T m d s (cid:33) (cid:32)(cid:90) T m
Revised: T m d s (cid:33) (cid:32)(cid:90) T m

Original: + h
Revised: + h

Original: i
Revised: i

Original: T m (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s
Revised: T m (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s

Original: T m ∂ ii f
Revised: T m ∂ ii f

Original: (˜ x ( z ))
Revised: (˜ x ( z ))

Original: d B z (cid:12)(cid:12)(cid:12)(cid:12)
Revised: d B z (cid:12)(cid:12)(cid:12)(cid:12)

Original: d s (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r m =
Revised: d s (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r m =

Original: i (cid:35) ≤ 2 h i (cid:90) T m + h
Revised: i (cid:35) ≤ 2 h i (cid:90) T m + h

Original: i
Revised: i

Original: T m
Revised: T m

Original: E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s
Revised: E (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s

Original: T m ∂ ii f
Revised: T m ∂ ii f

Original: (˜ x ( z ))
Revised: (˜ x ( z ))

Original: d B z (cid:12)(cid:12)(cid:12)(cid:12) 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r m =
Revised: d B z (cid:12)(cid:12)(cid:12)(cid:12) 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r m =

Original: i (cid:33) d s (II)
Revised: i (cid:33) d s (II)

Original: = 2 h i (cid:90) T m
Revised: = 2 h i (cid:90) T m

Original: + h
Revised: + h

Original: i
Revised: i

Original: T m (cid:90) s
Revised: T m (cid:90) s

Original: T m
Revised: T m

Original: E (cid:16) | ∂ ii f (˜ x ( z )) | 2 (cid:12)(cid:12)(cid:12) r m = i (cid:17)
Revised: E (cid:16) | ∂ ii f (˜ x ( z )) | 2 (cid:12)(cid:12)(cid:12) r m = i (cid:17)

Original: d z d s (III) =
Revised: d z d s (III) =

Original: h 3 i E p | ∂ ii f | 2 =
Revised: h 3 i E p | ∂ ii f | 2 =

Original: h 3 i L 2 i , where we use H ¨ older’s inequality in I and ˜ x ( t ) ∼ p for all t in III .
Revised: h 3 i L 2 i , where we use H ¨ older’s inequality in I and ˜ x ( t ) ∼ p for all t in III .

Original: In II , we use the following property of Itˆo’s integral:
Revised: In II , we use the following property of Itˆo’s integral:

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: By substituting into (78), we obtain
Revised: By substituting into (78), we obtain

Original: [Equation]
Revised: [Equation]

Original: To bound the second term on the right-hand side of (76), we ﬁrst note that f is three times continuously differentiable, and (15) implies (cid:107) ∂ iii f (cid:107) ∞ ≤ H i .
Revised: To bound the second term on the right-hand side of (76), we ﬁrst note that f is three times continuously differentiable, and (15) implies (cid:107) ∂ iii f (cid:107) ∞ ≤ H i .

Original: Take d t on both sides of (48), under condition r m =
Revised: Take d t on both sides of (48), under condition r m =

Original: i , we ﬁrst have
Revised: i , we ﬁrst have

Original: [Equation]
Revised: [Equation]

Original: According to Itˆo’s formula
Revised: According to Itˆo’s formula

Original: [Equation]
Revised: [Equation]

Original: we obtain
Revised: we obtain

Original: [Equation]
Revised: [Equation]

Original: Substituting (80) into (81), we have
Revised: Substituting (80) into (81), we have

Original: [Equation]
Revised: [Equation]

Original: By substituting into (73), we obtain
Revised: By substituting into (73), we obtain

Original: [Equation]
Revised: [Equation]

Revised: [Table]

Revised: [Equation]

Original: [Table] 
Revised: [Table] [Equation] [Table] [Equation]

Original:  [Equation]
Revised: [Table] [Equation] [Table] [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation] [Table] 
Revised: [Equation] [Table] [Equation]

Original: [Equation] [Table] [Equation] [Table]
Revised: [Table]

Original: [Equation][Table] [Equation] [Table]
Revised: [Equation]

Original: [Equation] [Table] [Equation] [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: [Table]
Revised: [Table]

Original: [Equation]
Revised: [Equation]

Original: In the derivation, (II) comes from plugging in (82), and (I) and (III) come from the use of Jensen’s inequality, (V) comes from the use of Lipschitz continuity in the ﬁrst and the second derivative ((11) and (15) in particular), and the fact that ˜ x ( t ) ∼ p for all t .
Revised: In the derivation, (II) comes from plugging in (82), and (I) and (III) come from the use of Jensen’s inequality, (V) comes from the use of Lipschitz continuity in the ﬁrst and the second derivative ((11) and (15) in particular), and the fact that ˜ x ( t ) ∼ p for all t .

Original: Note also E p | ∂ i f | 2 ≤ L i by (Dalalyan and Karagulyan, 2019, Lemma 3).
Revised: Note also E p | ∂ i f | 2 ≤ L i by (Dalalyan and Karagulyan, 2019, Lemma 3).

Original: By plugging (79) and (83) into (71) and (76), we obtain
Revised: By plugging (79) and (83) into (71) and (76), we obtain

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: where we use h i φ i
Revised: where we use h i φ i

Original: = h .
Revised: = h .

Original: Comparing with (70)
Revised: Comparing with (70)

Original: [Equation]
Revised: [Equation]

Original: we need to set
Revised: we need to set

Original: [Equation]
Revised: [Equation]

Original: where we use h < µ min { φ i } 8 L. This leads to 

Revised: where we use h < µ min { φ i } 8 L. This leads to

Original: + 1 a ≤ 2 φ
Revised: + 1 a ≤ 2 φ

Original: i hµ .
Revised: i hµ .

Original: By substituting into (62), we obtain
Revised: By substituting into (62), we obtain

Original: [Equation]
Revised: [Equation]

Original: Noting L i /µ > 1 , we conclude the lemma.
Revised: Noting L i /µ > 1 , we conclude the lemma.

Original: The proof of Proposition C.1 is obtained by summing up all components and applying Lemma C.1.
Revised: The proof of Proposition C.1 is obtained by summing up all components and applying Lemma C.1.

Original: Proof of Proposition C.1.
Revised: Proof of Proposition C.1.

Original: Noting that
Revised: Noting that

Original: [Equation]
Revised: [Equation]

Original: we substitute using (70) to obtain
Revised: we substitute using (70) to obtain

Original: [Equation]
Revised: [Equation]

Original: The second and third terms in the right-hand side of this bound can be controlled by E | ∆ m | 2 , as follows.
Revised: The second and third terms in the right-hand side of this bound can be controlled by E | ∆ m | 2 , as follows.

Original: By convexity, we have
Revised: By convexity, we have

Original: [Equation]
Revised: [Equation]

Original: By the L -Lipschitz property
Revised: By the L -Lipschitz property

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: By substituting (86) and (87) into (64)
Revised: By substituting (86) and (87) into (64)

Original: [Equation]
Revised: [Equation]

Original: and using µ < L
Revised: and using µ < L

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: Since h < µ min { φ i } 8 L, we obtain (68).
Revised: Since h < µ min { φ i } 8 L, we obtain (68).

Original: D P ROOF OF P ROPOSITION 4. 
Revised: D P ROOF OF P ROPOSITION 4.

Original: Proof of Proposition 4.2.
Revised: Proof of Proposition 4.2.

Original: For this special target distribution p , the objective function is f ( x ) = (cid:80) di =1 | x i | 2 2 .
Revised: For this special target distribution p , the objective function is f ( x ) = (cid:80) di =1 | x i | 2 2 .

Original: With α = 0 and φ i = 1 /d , we have: x m +1 i
Revised: With α = 0 and φ i = 1 /d , we have: x m +1 i

Original: = x mi for all i (cid:54) =
Revised: = x mi for all i (cid:54) =

Original: r m and x m +1 r m
Revised: r m and x m +1 r m

Original: [Equation]
Revised: [Equation]

Original: Therefore for all i = 1 , 2 , . . .
Revised: Therefore for all i = 1 , 2 , . . .

Original: , d , we have
Revised: , d , we have

Original: E
Revised: E

Original: [Equation]
Revised: [Equation]

Original: x m +

Revised: x m +

Original: i
Revised: i

Original: [Equation]
Revised: [Equation]

Original: (89) where we use E ξ (cid:12)(cid:12)(cid:12) x mi − dhx mi + √ 2 dhξ m (cid:12)(cid:12)(cid:12) 2 = (1 − dh ) 2 | x mi | 2 + 2 dh in the last equation.
Revised: (89) where we use E ξ (cid:12)(cid:12)(cid:12) x mi − dhx mi + √ 2 dhξ m (cid:12)(cid:12)(cid:12) 2 = (1 − dh ) 2 | x mi | 2 + 2 dh in the last equation.

Original: By summing (89) over i , we obtain
Revised: By summing (89) over i , we obtain

Original: E | x m +
Revised: E | x m +

Original: [Equation]
Revised: [Equation]

Original: |
Revised: |

Original: [Equation]
Revised: [Equation]

Original: =
Revised: =

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: 2 h + dh 2 (cid:1) E | x m | 2 + 2 dh .
Revised: 2 h + dh 2 (cid:1) E | x m | 2 + 2 dh .

Original: Using it iteratively, and considering E | x 0 | 2 = 3 d , we have:
Revised: Using it iteratively, and considering E | x 0 | 2 = 3 d , we have:

Original: [Equation]
Revised: [Equation]

Original: where we use dh ≤ 1 in the last inequality.
Revised: where we use dh ≤ 1 in the last inequality.

Original: Since
Revised: Since

Original: [Equation]
Revised: [Equation]

Original: we have
Revised: we have

Original: [Equation]
Revised: [Equation]

Original: where in the last inequality we use
Revised: where in the last inequality we use

Original: [Equation]
Revised: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: we ﬁnally prove (26)
Revised: we ﬁnally prove (26)

Original: [Equation]
Revised: [Equation]