SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Xiang Cheng*
x.cheng@berkeley.edu
Jingzhao Zhang*
IIIS, Tsinghua; Shanghai Qizhi Institute
jingzhaoz@mail.tsinghua.edu.cn
Bohan Wang*
USTC
Yusong Zhu*
zhuys19@ mails.tsinghua.edu.cn
MCMC algorithms offer empirically efficient tools for sampling from a target distribution  $\pi(x) \propto \exp(-V(x))$ . However, on the theory side, MCMC algorithms suffer from slow mixing rate when  $\pi(x)$  is non-log-concave. Our work examines this gap and shows that when Poincaré-style inequality holds on a subset  $\mathcal{X}$  of the state space, the conditional distribution of MCMC iterates over  $\mathcal{X}$  mixes fast to the true conditional distribution. This fast mixing guarantee can hold in cases when global mixing is provably slow. We formalize the statement and quantify the conditional mixing rate. We further show that conditional mixing can have interesting implications for sampling from mixtures of Gaussians, parameter estimation for Gaussian mixture models and Gibbs-sampling with well-connected local minima.
Sampling from a given target distribution of the form  $\pi(x) \propto e^{-V(x)}$  plays a central role in many machine learning problems, such as Bayesian inference, optimization, and generative modeling [9, 14, 15]. The Langevin MCMC algorithm in particular has received a lot of recent attention; it makes use of the first-order gradient information  $\nabla V(x)$ , and can be viewed as the sampling analog of gradient descent.
Langevin MCMC has been shown to converge quickly when  $\pi(x)$  is log-concave [6, 10]. More recently, similar guarantees have been established for  $p(x)$  satisfying weaker conditions such as log-Sobolev inequality (LSI) [23], for instance,  $\pi(x) \propto e^{-V(x)}$  can have a good LSI constant when  $V(x)$  is a small perturbation of a convex function. However, few guarantees exist for general nonlog-concave distributions. One simple example is when  $p(x)$  is a well-separated mixture of two Gaussian distributions; in this case, one verifies that Langevin MCMC mixes at a rate proportional to the inverse of the exponential of the separation distance.
Many important modern machine-learning problems are highly non-convex, one prominent class being functions arising from neural networks. Though finding the global minimum of a non-convex function can be difficult, gradient descent can often be shown to converge rather quickly to a local optimum [3]. This raises the important question:
What is the sampling analog of a local minimum? And how can we sample efficiently from such a minimum?
*Alphabetical author order
In [3], authors provide a partial answer to this question by adopting the view of Langevin Diffusion as the gradient flow of KL divergence in probability space. Under this perspective, the gradient of KL divergence is given by the Fisher Information (FI). Authors of [3] show that LMC achieves  $\epsilon$  error in FI in  $O(1 / \epsilon^2)$  steps.
However, one crucial question remains unanswered: how does the local optimality of FI help us sample from non-convex distributions? Intuitively, small FI is useful for characterizing local convergence when  $\pi$  is multi-modal. Authors of [3] suggested this connection, but it remains unclear how local mixing is defined and how small FI can quantitatively lead to good local mixing. Thus motivated, one of the goals of this paper is to provide a useful interpretation of the FI bound.
To this end, we propose a rigorous quantitative measure of "local mixing". We also provide a more general notion of "stationary point" for the sampling problem. Under these definitions, we show that LMC can achieve  $\epsilon$  error in our proposed "measure of local mixing", in polynomial time. Finally, we consider discrete-time analog of the aforementioned ideas, and prove local convergence for random walk on a hypercube. Below is a detailed description of our theoretical contributions.
1. We define a notion of conditional convergence: Let  $\mathcal{X} \subseteq \Omega$  denote a subset of the state space. We study the convergence of  $\pi_t | \mathcal{X}$  to  $\pi | \mathcal{X}$ , where  $\pi_t$  denotes distributions of LMC iterates and  $\pi$  denotes the target distribution. This definition of convergence is much weaker than the standard global convergence, but in exchange, LMC can achieve fast conditional convergence in settings where global convergence is known to be exponentially slow.
2. We define local Logarithmic Sobolev Inequality and show how to combine it with existing results on the convergence of Fisher information to derive the conditional convergence of LMC.
3. When local Logarithmic Sobolev Inequality does not hold, we define local Poincaré Inequality and Poincaré Fisher information (which is an analogy of Fisher information). We show the convergence of Poincaré Fisher information assuming strong dissipativity and show the conditional convergence of LMC when local Poincaré Inequality is present.
4. To showcase the applications of our results, we respectively study sampling from Gaussian mixture model with the same covariance and sampling from the power posterior distribution of symmetric two-component Gaussian mixtures. The global isometric constants of these examples are exponential in dimension and may have slow global convergence. We show that the local isoperimetric constants of these examples are polynomial in dimension, and prove fast conditional convergence for these examples.
5. In Theorem 1, we consider an application of our result to Gibbs sampling on discrete state space. We show that fast conditional mixing happens when the spectral gap is not small. We further show in Theorem 2 that a subset has large spectral gap if it contains only one local minimum and is well connected.
When the target distribution  $\pi$  is strongly log-concave, the entropy  $\mathrm{Ent}_{\pi}\left[\frac{\mu}{\pi}\right]$  is known to be strongly convex with respect to  $\mu$ , and thus Langevin Dynamics converges exponentially fast [2]. Such a result is later extended to LMC [8, 4, 9, 7, 21]. Several works further loosen the assumption by using isoperimetric inequalities such as Logarithmic Sobolev Inequality and Poincaré Inequality instead of strong log-concavity [24, 5, 25, 11]. However, there are few existing works on general non-log-concave sampling. Recently, Balasubramanian [3] defines convergence in relative Fisher information as a kind of "weak convergence" for sampling and proves a polynomial guarantee in general non-log-concave case only assuming global Lipschitz condition. However, this paper doesn't give any rigorous statistical interpretation of this weak convergence; Majka et al. [18] and Erdogdu et al. [11] study the cases when the target distribution is non-log-concave but has some good tail growth or curvature; Ma et al. [17] analyze the situation when the target distribution is non-log-concave inside the region but log-concave outside. Although these works give strict proofs of polynomial time guarantee in their setting, their results only hold for a small branch of non-log-concave distributions. It is still hardly possible to obtain a polynomial guarantee in general
non-log-concave cases. Multimodality, as a special case of non-log-concavity, has attracted lots of attention due to its prevalence in applied science. Many modified versions of MCMC were proposed to try to tackle the sampling of these distributions, such as Darting MC [1], Wormhole HMC [16], and etc. However, these algorithms require explicit knowledge of the location of the modes.
3 Conditional Mixing for MCMC
In this section, we provide a formal definition of "conditional mixing". Specifically, let  $\mu_t$  denote the distribution of a Markov chain  $\{Z_t\}_t$  at time  $t$ . We assume that the Markov chain dynamics is reversible with a unique stationary distribution  $\pi$ . Existing analyses mostly focus on understanding the rate of convergence measured by  $d(\mu_t, \pi)$  where  $d$  is some probability distance.
However, unless the stationary distribution  $\pi$  satisfies certain restrictive properties (e.g., log-concavity), the rate of convergence can be exponentially slow in the problem dimension or the distribution moments even for simple distributions such as the mixture of Gaussians. For this reason, we consider a weaker notion of convergence below.
Definition 1 (Conditional mixing). Given a distribution  $\mu_t$  supported on the state space  $\Omega$ . We say  $\mu_t$  converges conditioned on set  $\mathcal{X} \subseteq \Omega$  with respect to the divergence  $d$  if
$$
d (\mu_ {t} | \mathcal {X}, \mu | \mathcal {X}) \leq \epsilon ,
$$
where we have the conditional distribution
$$
\mu_ {t} | \mathcal {X} (x) = \frac {\mu_ {t} (x) \mathbb {1} \{x \in \mathcal {X} \}}{\mu_ {t} (\mathcal {X})}.
$$
For now we can think of the distance  $d$  as the total variation distance. Later we will discuss stronger divergence such as KL-divergence or Chi-squared divergence.
The focus of our work is on identifying several sufficient conditions for fast convergence, and quantitatively bounding the convergence rate under these conditions. We focus on two MCMC algorithms: the Langevin Monte Carlo in continuous space, and the Gibbs sampling algorithm in discrete space. We further discuss the implications of conditional convergence for the two algorithms.
4 Conditional Mixing for Langevin Monte Carlo
In this section, we study the conditional mixing of the Langevin Monte Carlo (LMC) algorithm. This section is organized as follows: in Subsection 4.1, we first introduce the Langevin Monte Carlo algorithm, Langevin Dynamics, functional inequalities, and the Fisher information; in Subsection 4.2, we provide our main results characterizing the conditional convergence of LMC; finally, in Subsection 4.3, we showcase two applications of our main results.
Langevin Monte Carlo. We are interested in the convergence of Langevin Monte Carlo (LMC), which is a standard algorithm employed to sample from a target probability density  $\pi \propto e^{-V}:\mathbb{R}^d\to \mathbb{R}$ , where  $V$  is called the potential function. The pseudocode of LMC is given in Algorithm 1.
Algorithm 1 Langevin Monte Carlo
Input: Initial parameter  $z$  , potential function  $V$  , step size  $h$  , number of iteration  $T$    
1: Initialization  $z_0\gets z$    
2: For  $t = 0\to T$  .   
3: Generate Gaussian random vector  $\xi_t\sim \mathcal{N}(0,\mathbb{I}_d)$    
4: Update  $z_{(t + 1)h}\leftarrow z_{th} - h\nabla V(z_{th}) + \sqrt{2h}\xi_t$    
5: EndFor
LMC can be viewed as a time-discretization of Langevin Dynamics (LD), described by the following stochastic differential equation:
$$
\mathrm {d} Z _ {t} = - \nabla V (Z _ {t}) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}. \tag {1}
$$
We can interpolate LMC following the similar manner of LD as
$$
\mathrm {d} Z _ {t} = - \nabla V \left(Z _ {k h}\right) \mathrm {d} t + \sqrt {2} \mathrm {d} B _ {t}, t \in [ k h, k h + 1), k \in \mathbb {N}. \tag {2}
$$
One can easily observe that  $\{Z_{kh}\}_{k = 0}^{\infty}$  in Eq. (2) has the same joint distribution as  $\{z_{kh}\}_{k = 0}^{\infty}$  in Algorithm 1, and thus Eq. (2) is a continuous-time interpolation of Algorithm 1.
Poincaré Inequality & Logarithmic Sobolev Inequalities. The convergence of LMC does not necessarily hold for all potential functions, and quantitative bounds require specific conditions over the potential function  $V$ . Poincaré Inequality (PI) and Logarithmic Sobolev Inequality (LSI) are two commonly used conditions in the analysis of LMC convergence. We present these below:
Definition 2 (Poincaré Inequality). A probability measure  $\pi$  on  $\mathbb{R}^d$  satisfies the Poincaré inequality with constant  $\rho > 0$  (abbreviated as  $\mathrm{PI}(\rho)$ ), if for all functions  $f: \mathbb{R}^d \to \mathbb{R}$ ,
$$
\int \| \nabla f (x) \| ^ {2} \mathrm {d} \pi (x) \geq \rho \operatorname {V a r} _ {\pi} [ f ]. \tag {PI}
$$
Definition 3 (Logarithmic Sobolev Inequality). Given a function  $f: \mathbb{R}^d \to \mathbb{R}^+$  and a probability measure  $\pi$  on  $\mathbb{R}^d$ , define  $\mathrm{Ent}_{\pi}(f) \triangleq \int f \log f \, \mathrm{d}\pi(x) - \int f \, \mathrm{d}\pi(x) \left( \log \int f \, \mathrm{d}\pi(x) \right)$ . We say that a distribution  $\pi$  on  $\mathbb{R}^d$  satisfies the Logarithmic Sobolev Inequality (abbreviated as LSI(α)) with some constant  $\alpha$ , if for all functions  $f: \mathbb{R}^d \to \mathbb{R}^+$ ,
$$
\int_ {\mathcal {X}} \frac {\left\| \nabla f (x) \right\| ^ {2}}{f} d \pi (x) \geq \alpha \operatorname {E n t} _ {\pi} (f). \tag {LSI}
$$
Both PI and LSI can imply the convergence of LMC when the step size  $h$  is small. Specifically, denote  $\pi_{th}$  as the distribution of  $z_{th}$  in LMC (Algorithm 1) and  $\tilde{\pi}_t$  as the distribution of  $Z_{t}$  in Langevin Dynamics (Eq.(1)). We further denote  $\pi_t$  as the distribution of  $Z_{t}$  interpolating LMC. The left-hand-side of Eq.(PI) with  $f = \frac{\tilde{\pi}_t}{\pi}$  is then the derivative (w.r.t. time) of the entropy of  $f$ , i.e.,  $\mathrm{Ent}_{\pi}\left[\frac{\tilde{\pi}_t}{\pi}\right]$ , and thus we directly establish the exponential convergence of  $\mathrm{Var}_p\left[\frac{\tilde{\pi}_t}{\pi}\right]$ . The convergence of LMC can then be induced by bounding the discretization error between LMC and Langevin Dynamics. The methodology is similar when we have LSI.
Fisher information. It is well-known that if  $V$  is either strongly convex or convex with bounded support, then it obeys both PI and LSI, and the convergence of LMC follows immediately according to the arguments above. However, real-world sampling problems usually have non-convex potential functions, and for these problems we may no longer have either PI or LSI. If we revisit the above methodology to establish the convergence of LMC under LSI, we find that we still have
$$
\frac {\mathrm {d}}{\mathrm {d} t} \operatorname {E n t} _ {\pi} \left[ \frac {\tilde {\pi} _ {t}}{\pi} \right] = - \int \left\| \nabla \ln \frac {\tilde {\pi} _ {t}}{\pi} (x) \right\| ^ {2} \mathrm {d} \pi_ {t} (x), \operatorname {E n t} _ {\pi} \left[ \frac {\tilde {\pi} _ {0}}{\pi} \right] - \operatorname {E n t} _ {\pi} \left[ \frac {\tilde {\pi} _ {T}}{\pi} \right] = \int_ {0} ^ {T} \int \left\| \nabla \ln \frac {\tilde {\pi} _ {t}}{\pi} (x) \right\| ^ {2} \mathrm {d} \pi_ {t} (x) \mathrm {d} t,
$$
and thus  $\lim_{T\to \infty}\min_{t\in [0,T]}\int \left\| \nabla \frac{\tilde{\pi}_t}{\pi} (x)\right\| ^2\mathrm{d}\pi (x) = 0$ . Following this methodology, [3] uses the considers a notion of convergence which is defined using Fisher Information  $\operatorname {FI}(\mu ||\pi)\triangleq$ $\int \| \nabla \ln \mu /\pi \| ^2\mathrm{d}\mu$  , which they use to analyze LMC convergence. We present the result of [3] for completeness:
Proposition 1 (Theorem 2, [3]). Assume  $\nabla V$  is  $L$ -lipschitz. Then, for any step size  $h \in (0, \frac{1}{6L})$ ,
$$
\frac {1}{T h} \int_ {0} ^ {T h} \operatorname {F I} _ {\pi} (\pi_ {t} \| \pi) \mathrm {d} t \leq \frac {2 \operatorname {E n t} \left(\frac {\pi_ {0}}{\pi}\right)}{T h} + 8 L ^ {2} d h.
$$
4.2 Rates of convergence
4.2.1 Conditional mixing under local Logarithmic Sobolev Inequality
We first show that if the target distribution  $\pi$  obeys a local Logarithmic Sobolev Inequality (defined below), then convergence of Fisher information implies conditional convergence.
Definition 4 (Local Logarithmic Sobolev Inequality). We say that a distribution  $\pi$  on  $\mathbb{R}^d$  satisfies the local Logarithmic Sobolev Inequality over a subset  $S\subset \mathbb{R}^d$  with some constant  $\alpha$  (abbreviated as  $\mathrm{LSI}_S(\alpha)$ ), if  $\pi |S$  satisfies  $LSI(\alpha)$ .
Definition 4 characterizes the local property of one distribution. It is considerably weaker than global LSI (i.e., Definition 3) when  $S$  is convex, and recovers LSI when  $S = \mathbb{R}^d$ . The intuition is that when the distribution is multi-modal, it is quite possible that LSI does not hold (or hold but with an exponentially small constant). In contrast, we can reasonably expect local LSI hold within each mode. In Subsection 4.3, we will show that a Gaussian mixture model with the same covariance satisfies Local Logarithmic Sobolev Inequality. We show in the following lemma that, whenever we have an estimation on the Fisher information between  $\mu$  and  $\pi$ , we can turn it to an estimation of the entropy between  $\mu |S$  and  $\pi |S$  given  $\mathrm{LSI}_S(\alpha)$ .
Lemma 1. Let  $S \subset \mathcal{X}$  and assume that  $\pi$  obeys  $\mathrm{LSI}_S(\alpha)$ . For any distribution  $\mu$  such that  $\mathrm{FI}(\mu ||\pi) \leq \varepsilon$ , we have that either  $\mu(S) \leq \frac{\sqrt{\varepsilon}}{\sqrt{\alpha}}$ , or  $\mathrm{Ent}_{\pi|S}[\frac{\mu|S}{\pi|S}] \leq \frac{\sqrt{\varepsilon}}{\sqrt{\alpha}}$ .
We defer the proof of Lemma 1 to Appendix A.1. Intuitively, Lemma 1 states that if the distribution  $\pi$  satisfies local LSI over the region  $S$  and has small global Fisher information with respect to a distribution  $\mu$ , one can expect that either the probability mass of  $\mu$  over  $S$  is minor, or  $\mu$  is close to  $\pi$  both conditional over  $S$ . In other words,  $\mu$  is close to  $\pi$  conditional over  $S$  whenever  $\mu$  has a moderate mass over  $S$ . Note that Proposition 1 has already guaranteed global FI of LMC with little assumption, which together with Lemma 1 leads to the following corollary:
Corollary 1. Assume  $\nabla V$  is  $L$ -lipschitz and  $S \subset \mathcal{X}$  satisfies that  $\pi$  obeys  $\mathrm{LSI}_S(\alpha)$ . Define  $\bar{\pi}_{Th} = \frac{\int_0^{Th}\pi_t\mathrm{d}t}{Th}$ . Choosing step size  $h = \frac{1}{\sqrt{T}}$ , we have that either  $\bar{\pi}_{Th}(S) \leq \mathcal{O}\left(\frac{\sqrt{d}\sqrt[4]{\operatorname{Ent}\left(\frac{\pi_0}{\pi}\right)}}{\sqrt{\alpha}\sqrt[4]{T}}\right)$ , or  $\mathrm{Ent}_{\pi |S}\left[\frac{\bar{\pi}_{Th}|S}{\pi|S}\right] \leq \mathcal{O}\left(\frac{\sqrt{d}\sqrt[4]{\mathrm{Ent}\left(\frac{\pi_0}{\pi}\right)}}{\sqrt{\alpha}\sqrt[4]{T}}\right)$ .
Corollary 1 is one of our key results. Intuitively, it shows that LMC can achieve local mixing when only local LSI is assumed. As we discussed under Definition 4, we can reasonably expect that local LSI holds within each mode when the distribution is multi-modal, in which case Corollary 1 can be further applied to show local mixing with polynomial rate. Note here the dependence of the local mixing rate is polynomial over the dimension  $d$ , meaning that considering local mixing helps to get rid of the exponential dependence over the dimension  $d$  in the global mixing rate when studying multi-modal distributions.
4.2.2 Conditional Convergence under local Poincaré Inequality
We have established the conditional convergence of LMC under local Logarithmic Sobolev Inequality above. However, there are cases where even local LSI fails to hold. To tackle these cases and inspired by Poincaré Inequality, we introduce a weaker notion than local LSI called local Poincaré Inequality:
Definition 5 (Local Poincaré Inequality). A distribution  $\pi$  on  $\mathbb{R}^d$  satisfies the local Poincaré Inequality over a subset  $S\subset \mathbb{R}^d$  with constant  $\rho$  (abbreviated as  $\mathrm{PI}_S(\rho)$ ), if  $\pi |S$  satisfies  $\mathrm{PI}(\rho)$ .
As Poincaré Inequality is weaker than Logarithmic Sobolev Inequality, one can easily see that local Poincaré Inequality is also weaker than local Logarithmic Sobolev Inequality. Based on local Poincaré Inequality, we have the following Poincaré version of Lemma 1, which converts an estimation of Poincaré Fisher information to an estimation of chi-squared divergence of the conditional distributions.
Lemma 2. Define Poincaré Fisher information as  $\mathrm{PFI}(\mu ||\pi)\triangleq \int \| \nabla (\mu /\pi)\| ^2\mathrm{d}\mu$  . Let  $S\subset \mathcal{X}$  and assume that  $\pi$  obeys  $\mathrm{PI}_S(\rho)$  . For any distribution  $\mu$  such that  $\mathrm{PFI}(\mu ||\pi)\leq \varepsilon$  , we have that either  $\mu (S)^2\mathrm{Var}_{\pi |_S}[\frac{\mu|_S}{\pi|_S} ]\leq \frac{\varepsilon\pi(S)}{\rho}$
To derive the conditional convergence of LMC under local Poincaré inequality, we further need to bound Poincaré Fisher information of LMC. Such a result, however, does not exist in the literature to our best knowledge. In analogy to the analysis in [3], we develop the following lemma to control the Fisher information of LD (recall that  $\tilde{\pi}_t$  is the distribution of Langevin Dynamics in time  $t$ ).
Proposition 2. Denote  $\bar{\tilde{\pi}}_{Th} = \frac{\int_0^{Th}\tilde{\pi}_t\mathrm{d}t}{Th}$ . Then, we have the following estimation of the PFI between  $\bar{\tilde{\pi}}_{Th}$  and  $\pi$ .
$$
\operatorname {P F I} \left(\tilde {\bar {\pi}} _ {T h} \| \pi\right) \mathrm {d} t \leq \frac {\operatorname {V a r} _ {\pi} \left[ \tilde {\bar {\pi}} _ {0} \right]}{2 T h}.
$$
The proof can be directly derived by taking the derivative of  $\mathrm{Var}_{\pi}[\frac{\tilde{\pi}_t}{\pi}]$  with respect to  $t$ . We defer the formal proof to Appendix A.2. Combining Lemma 2, Proposition 2, and recent advances in discrete error analysis of LD [19], we obtain the following result showing conditional convergence under local Poincaré inequality.
Corollary 2. Assume  $\nabla V$  and  $\nabla^2 V$  is  $L$ -lipschitz,  $V$  satisfies strong dissipativity, i.e.,  $\langle \nabla V(x), x \rangle \geq m \| x \|^2 - b$ , and  $S \subset \mathcal{X}$  satisfies that  $\pi|_S$  obeys  $\mathrm{PI}(\alpha)$ . Initialize  $z_0 \sim \mathcal{N}(0, \sigma^2 \mathbb{I})$  and select  $h = \tilde{\Theta}\left(\frac{\mathrm{Var}_{\pi}[\bar{\pi}_0]^{1/3}}{T^{\frac{2}{3}} \rho^{\frac{1}{3}} d^{\frac{2}{3}}}\right)$ . Recall that  $\bar{\pi}_{Th} = \frac{\int_0^{Th} \pi_t \, \mathrm{d}t}{Th}$ . Then either  $\bar{\pi}_{Th}(S) = \mathcal{O}\left(\frac{d^{\frac{1}{6}} \mathrm{Var}_{\pi}[\bar{\pi}_0]^{\frac{1}{6}}}{\rho^{\frac{1}{6}} T^{\frac{1}{12}}}\right)$ , or  $D_{TV}[\bar{\pi}_{Th}|S|\pi|S] = \mathcal{O}\left(\frac{d^{\frac{1}{6}} \mathrm{Var}_{\pi}[\bar{\pi}_0]^{\frac{1}{6}}}{\rho^{\frac{1}{6}} T^{\frac{1}{12}}}\right)$ .
Corollary 2 shows that conditional mixing can be established when local PI holds, which is a weaker assumption than local LSI. However, the assumption of Corollary 2 is stronger than its LSI counterpart, and the mixing rate depends on  $\mathrm{Var}_{\pi}[\frac{\tilde{\pi}_0}{\pi}]$  instead of  $\mathrm{Ent}\left(\frac{\tilde{\pi}_0}{\pi}\right)$ , which can be considerably large. These drawbacks are mainly due to that the dynamics of chi-square distance are much harder than that of KL divergence, even when global PI holds [12].
Extension to Rényi divergence. In some prior works, e.g. [5, 23], Rényi divergence is considered a natural "measure" of convergence, due to the analytical simplicity of the resulting expressions. One may wonder if our methodology above can be extended to establish a local mixing rate of Rényi divergence. Unfortunately, there are technical difficulties which for now we have not found a way to tackle or bypass, which we list in Appendix A.3.
4.3 Applications
In this subsection, we apply Corollary 1 and Corollary 2 to two concrete examples to demonstrate their usage in obtaining local mixing rates. We start by analyzing the example of sampling from Gaussian Mixture Models with uniform covariance and then turn to the example of sampling from the power posterior distribution of symmetric two-component Gaussian mixtures.
4.3.1 Case Study: Sampling from Gaussian Mixture Model with the uniform covariance
Target distribution and potential function. We are interested in the gaussian mixture model formed by gaussians with the same variance but different means. Specifically, the target distribution is defined as  $\pi = \sum_{i=1}^{n} w_i p_i \in \Delta(\mathbb{R}^d)$ , where  $w_i > 0$ ,  $\sum_{i=1}^{n} w_i = 1$ ,  $p_i \sim \mathcal{N}(\mu_i, \Sigma)$  and  $\Sigma \succ \sigma^2 \mathbb{I}_d$ . The potential function is then defined as  $V(x) = -\log \left( \sum_{i=1}^{n} w_i p_i(x) \right)$ .