Title: METROPOLIS-HASTINGS VIEW ON VARIATIONAL INFERENCE AND ADVERSARIAL TRAINING

Abstract: In this paper we propose to view the acceptance rate of the Metropolis-Hastings algorithm as a universal objective for learning to sample from target distribution -given either as a set of samples or in the form of unnormalized density. This point of view unifies the goals of such approaches as Markov Chain Monte Carlo (MCMC), Generative Adversarial Networks (GANs), variational inference. To reveal the connection we derive the lower bound on the acceptance rate and treat it as the objective for learning explicit and implicit samplers. The form of the lower bound allows for doubly stochastic gradient optimization in case the target distribution factorizes (i.e. over data points). We empirically validate our approach on Bayesian inference for neural networks and generative models for images.

1. INTRODUCTION:

Bayesian framework and deep learning have become more and more interrelated during recent years. Recently Bayesian deep neural networks were used for estimating uncertainty (Gal & Ghahramani, 2016) , ensembling (Gal & Ghahramani, 2016) and model compression (Molchanov et al., 2017) . On the other hand, deep neural networks may be used to improve approximate inference in Bayesian models (Kingma & Welling, 2014) .

Learning modern Bayesian neural networks requires inference in the spaces with dimension up to several million by conditioning the weights of DNN on hundreds of thousands of objects. For such applications, one has to perform the approximate inference -predominantly by either sampling from the posterior with Markov Chain Monte Carlo (MCMC) methods or approximating the posterior with variational inference (VI) methods.

MCMC methods provide the unbiased (in the limit) estimate but require careful hyperparameter tuning especially for big datasets and high dimensional problems. The large dataset problem has been addressed for different MCMC algorithms: stochastic gradient Langevin dynamics (Welling & Teh, 2011) , stochastic gradient Hamiltonian Monte Carlo (Chen et al., 2014) , minibatch Metropolis-Hastings algorithms (Korattikara et al., 2014; Chen et al., 2016) . One way to address the problem of high dimension is the design of a proposal distribution. For example, for the Metropolis-Hastings (MH) algorithm there exists a theoretical guideline for scaling the variance of a Gaussian proposal (Roberts et al., 1997; 2001) . More complex proposal designs include adaptive updates of the proposal distribution during iterations of the MH algorithm (Holden et al., 2009; Giordani & Kohn, 2010) . Another way to adapt the MH algorithm for high dimensions is combination of adaptive direction sampling and the multiple-try Metropolis algorithm as proposed in (Liu et al., 2000) . Thorough overview of different extensions of the MH algorithm is presented in (Martino, 2018) .

Variational inference is extremely scalable but provides a biased estimate of the target distribution. Using the doubly stochastic procedure (Titsias & Lázaro-Gredilla, 2014; Hoffman et al., 2013) VI can be applied to extremely large datasets and high dimensional spaces, such as a space of neural network weights (Kingma et al., 2015; Gal & Ghahramani, 2015; 2016) . The bias introduced by variational approximation can be mitigated by using flexible approximations (Rezende & Mohamed, 2015) and resampling (Grover et al., 2018) .

Generative Adversarial Networks (Goodfellow et al., 2014) (GANs) is a different approach to learn samplers. Under the framework of adversarial training different optimization problems could be solved efficiently (Arjovsky et al., 2017; Nowozin et al., 2016) . The shared goal of "learning to sample" inspired the connection of GANs with VI (Mescheder et al., 2017) and MCMC (Song et al., 2017) .

In this paper, we propose a novel perspective on learning to sample from a target distribution by optimizing parameters of either explicit or implicit probabilistic model. Our objective is inspired by the view on the acceptance rate of the Metropolis-Hastings algorithm as a quality measure of the sampler. We derive a lower bound on the acceptance rate and maximize it with respect to parameters of the sampler, treating the sampler as a proposal distribution in the Metropolis-Hastings scheme.

We consider two possible forms of the target distribution: unnormalized density (density-based setting) and a set of samples (sample-based setting). Each of these settings reveals a unifying property of the proposed perspective and the derived lower bound. In the density-based setting, the lower bound is the sum of forward and reverse KL-divergences between the true posterior and its approximation, connecting our approach to VI. In the sample-based setting, the lower bound admit a form of an adversarial game between the sampler and a discriminator, connecting our approach to GANs.

The closest work to ours is of Song et al. (2017) . In contrast to their paper our approach (1) is free from hyperparameters; (2) is able to optimize the acceptance rate directly; (3) avoids minimax problem in the density based setting.

Our main contributions are as follows:

1. We introduce a novel perspective on learning to sample from the target distribution by treating the acceptance rate in the Metropolis-Hastings algorithm as a measure of sampler quality. 2. We derive the lower bound on the acceptance rate allowing for doubly stochastic optimization of the proposal distribution in case when the target distribution factorizes (i.e. over data points). 3. For sample-based and density-based forms of target distribution we show the connection of the proposed algorithm to variational inference and GANs.

The rest of the paper is organized as follows. In Section 2 we introduce the lower bound on the AR. Special forms of target distribution are addressed in Section 3. We validate our approach on the problems of approximate Bayesian inference in the space of high dimensional neural network weights and generative modeling in the space of images in Section 4. We discuss results and directions of the future work in Section 5.

2. ACCEPTANCE RATE FOR THE METROPOLIS-HASTINGS ALGORITHM:



2.1. PRELIMINARIES:

In MH algorithm we need to sample from target distribution p(x) while we are only able to sample from proposal distribution q(x | x). One step of the MH algorithm can be described as follows.

1. sample proposal point x ∼ q(x | x), given previously accepted point x

2. accept x , if p(x )q(x | x ) p(x)q(x | x) > u, u ∼ Uniform[0, 1] x, otherwise

If the proposal distribution q(x | x) does not depend on x, i.e. q(x | x) = q(x ), the algorithm is called independent MH algorithm.

The quality of the proposal distribution is measured by acceptance rate and mixing time. Mixing time defines the speed of convergence of the Markov chain to the stationary distribution. The acceptance rate of the MH algorithm is defined as

EQUATION

where

EQUATION

In case of independent proposal distribution we show that the acceptance rate defines a semimetric in distribution space between p and q (see Appendix A.2).

2.2. OPTIMIZING THE LOWER BOUND ON ACCEPTANCE RATE:

Although, we can maximize the acceptance rate of the MH algorithm (Eq. 1) directly w.r.t. parameters φ of the proposal distribution q φ (x | x), we propose to maximize the lower bound on the acceptance rate. As our experiments show (see Section 4) the optimization of the lower bound compares favorably to the direct optimization of the acceptance rate. To introduce this lower bound we first express the acceptance rate in terms of total variation distance.

Theorem 1 For random variable ξ

EQUATION

)

where TV is the total variation distance.

The proof of Theorem 1 can be found in Appendix A.1. This reinterpretation in terms of total variation allows us to lower bound the acceptance rate through the Pinsker's inequality

EQUATION

The maximization of this lower bound can be equivalently formulated as the following optimization problem

EQUATION

In the following sections, we show the benefits of this optimization problem in two different settings -when the target distribution is given in a form of unnormalized density and as a set of samples. In Appendix C.5 and C.1 we provide the empirical evidence that maximization of the proposed lower bound results in the maximization of the acceptance rate.

3. OPTIMIZATION OF PROPOSAL DISTRIBUTION:

From now on we consider only optimization problem Eq. 5 but the proposed algorithms can be also used for the direct optimization of the acceptance rate (Eq. 1).

To estimate the loss function (Eq. 5) we need to evaluate the density ratio. In the density-based setting unnormalized density of the target distribution is given, so we suggest to use explicit proposal distribution to compute the density ratio explicitly. In the sample-based setting, however, we cannot compute the density ratio, so we propose to approximate it via adversarial training (Goodfellow et al., 2014) . The brief summary of constraints for both settings is shown in Table 1 . The following subsections describe the algorithms in detail.

3.1. DENSITY-BASED SETTING:

In the density-based setting, we assume the proposal to be an explicit probabilistic model, i.e. the model that we can sample from and evaluate its density at any point up to the normalization constant. We also assume that the proposal is reparameterisable (Kingma & Welling, 2014; Rezende et al., 2014; Gal, 2016) .

During the optimization of the acceptance rate we might face a situation when proposal collapses to the delta-function. This problem usually arises when we use Markov chain proposal, for example, q φ (x | x) = N (x | x, σ). For such proposal we can obtain arbitrary high acceptance rate, making the σ small enough. However, this is not the case for the independent proposal distribution q φ (x | x) = q φ (x ). In Appendix B.1 we provide more details and intuition on this property of acceptance rate maximization. We also provide empirical evidence in Section 4 that collapsing to the delta-function does not happen for the independent proposal.

In this paper, we consider two types of explicit proposals: simple parametric family (Section 4.2) and normalizing flows (Rezende & Mohamed, 2015; Dinh et al., 2016 ) (Section 4.1). Rich family of normalizing flows allows to learn expressive proposal and evaluate its density in any point of target distribution space. Moreover, an invertible model (such as normalizing flow) is a natural choice for the independent proposal due to its ergodicity. Indeed, choosing the arbitrary point in the target distribution space, we can obtain the corresponding point in the latent space using the inverse function. Since every point in the latent space has positive density, then every point in the target space also has positive density.

Considering q φ (x ) as the proposal, the objective of optimization problem 5 takes the form

EQUATION

Explicit form of the proposal q φ (x ) and the target p(x) distributions allows us to obtain density ratios q φ (x)/q φ (x ) and p(x )/p(x) for any points x, x . But to estimate the loss in Eq. 6 we also need to obtain samples from the target distribution x ∼ p(x) during training. For this purpose, we use the current proposal q φ and run the independent MH algorithm. After obtaining samples from the target distribution it is possible to perform optimization step by taking stochastic gradients w.r.t. φ. Pseudo-code for the obtained procedure is shown in Algorithm 1.

Algorithm 1 Optimization of proposal distribution in density-based case:

Require: explicit probabilistic model q φ (x ) Require:

density of target distribution p(x) ∝ p(x) while φ not converged do sample {x k } K k=1 ∼ q φ (x ) sample {x k } K k=1 ∼ p(x) using independent MH with current proposal q φ L(p, q φ ) 1 K K k=1 log p(x k )q φ (x k ) p(x k )q φ (x k )

approximate loss with finite number of samples φ ← φ -α∇ φ L(p, q φ ) perform gradient descent step end while return optimal parameters φ Algorithm 1 could also be employed for the direct optimization of the acceptance rate (Eq. 1). Now we apply this algorithm for Bayesian inference problem and show that during optimization of the lower bound we can use minibatches of data, while it is not the case for direct optimization of the acceptance rate. We consider Bayesian inference problem for discriminative model on dataset

D = {(x i , y i )} N i=1

, where x i is the feature vector of ith object and y i is its label. For the discriminative model we know likelihood p(y i | x i , θ) and prior distribution p(θ). In order to obtain predictions for some object x i , we need to evaluate the predictive distribution p(

y i | x i ) = E p(θ | D) p(y i | x i , θ).

(7) To obtain samples from posterior distribution p(θ | D) we suggest to learn proposal distribution q φ (θ) and perform independent MH algorithm. Thus the objective 6 can be rewritten as

EQUATION

Note that due to the usage of independent proposal, the minimized KL-divergence splits up into the sum of two KL-divergences.

EQUATION

Minimization of the first KL-divergence corresponds to the variational inference procedure.

EQUATION

The second KL-divergence has the only term that depends on φ. Thus we obtain the following optimization problem

EQUATION

The first summand here contains the sum over all objects in dataset D. We follow doubly stochastic variational inference and suggest to perform unbiased estimation of the gradient in problem 11 using only minibatches of data. Moreover, we can use recently proposed techniques (Korattikara et al., 2014; Chen et al., 2016 ) that perform the independent MH algorithm using only minibatches of data.

Combination of these two techniques allows us to use only minibatches of data during iterations of algorithm 1. In the case of the direct optimization of the acceptance rate, straightforward usage of minibatches results in biased gradients. Indeed, for the direct optimization of the acceptance rate (Eq. 1) we have the product over the all training data inside min function.

3.2. SAMPLE-BASED SETTING:

In the sample-based setting, we assume the proposal to be an implicit probabilistic model, i.e. the model that we can only sample from. As in the density-based setting, we assume that we are able to perform the reparameterization trick for the proposal.

In this subsection we consider only Markov chain proposal q φ (x | x), but everything can be applied to independent proposal q φ (x ) by simple substitution q φ (x | x) with q φ (x ). From now we will assume our proposal distribution to be a neural network that takes x as its input and outputs x . Considering proposal distribution parameterized by a neural network allows us to easily exclude delta-function from the space of solutions. We avoid learning the identity mapping by using neural networks with the bottleneck and noisy layers. For the detailed description of the architectures see Appendix C.8.

The set of samples from the true distribution X ∼ p(x) allows for the Monte Carlo estimation of the loss

EQUATION

)

To compute the density ratio

p(x)q φ (x | x) p(x )q φ (x | x )

we suggest to use well-known technique of density ratio estimation via training discriminator network. Denoting discriminator output as D(x, x ), we suggest the following optimization problem for the discriminator.

EQUATION

Speaking informally, such discriminator takes two images as input and tries to figure out which image is sampled from true distribution and which one is generated by the one step of proposal distribution. It is easy to show that optimal discriminator in problem 13 will be

EQUATION

)

Note that for optimal discriminator we have D(x, x ) = 1 -D(x , x). In practice, we have no optimal discriminator and these values can differ significantly. Thus, we have four ways for density ratio estimation that may differ significantly.

EQUATION

To avoid the ambiguity we suggest to use the discriminator of a special structure. Let D(x, x ) be a convolutional neural network with scalar output. Then the output of discriminator D(x, x ) is defined as follows.

EQUATION

In other words, such discriminator can be described as the following procedure. For single neural network D(•, •) we evaluate two outputs D(x, x ) and D(x , x). Then we take softmax operation for these values. Summing up all the steps, we obtain algorithm 2.

Algorithm 2 Optimization of proposal distribution in sample-based case

Require: implicit probabilistic model q φ (x | x) Require: large set of samples X ∼ p(x) for n iterations do sample {x k } K k=1 ∼ X sample {x k } K k=1 ∼ q φ (x |x) train discriminator D by optimizing 13 L(p, q φ ) ≈ 1 K K k=1 log D(x k ,x k ) 1-D(x k ,x k )

approximate loss with finite number of samples φ ← φ -α∇ φ L(p, q φ ) perform gradient descent step end for return parameters φ Algorithm 2 could also be employed for direct optimization of the acceptance rate (Eq. 1). But, in Appendix B.2 we provide an intuition for this setting that the direct optimization of the acceptance rate may struggle from vanishing gradients.

4. EXPERIMENTS:

In this section, we provide experiments for both density-based and sample-based settings, showing the proposed procedure is applicable to high dimensional target distributions. Code for reproducing all of the experiments will be published with the camera-ready version of the paper.

4.1. SYNTHETIC DISTRIBUTIONS:

To demonstrate performance of our approach we reproduce the experiment from (Song et al., 2017) . For target distributions we use synthetic 2d distributions (see Appendix C.3 for densities): ring (a ring-shaped density), mog2 (a mixture of 2 Gaussians), mog6 (a mixture of 6 Gaussians), ring5 (a mixture of 5 distinct rings). We measure performance of learned samplers using Effective Sample Size (see Appendix C.4 for formulation). Since the unnormalized densities of target distributions are given, we can learn proposals as suggested in the density-based setting (Section 3.1).

To learn the independent proposal we use RealNVP model (Dinh et al., 2016) (see details in Appendix C.2) and compare the performance of proposals after optimization of different objectives: the acceptance rate (AR), our lower bound on the acceptance rate (ARLB), evidence lower bound that corresponds to the variational inference (VI). We also compare the performance of obtained independent proposals with the performance of Markov chain proposals: A-NICE-MC (Song et al., 2017) , Hamiltonian Monte Carlo (HMC).

In Tables 2, 3 we see that our approach has comparable performance with A-NICE-MC (Song et al., 2017) . However, comparison between A-NICE-MC and learning independent proposal is not the main subject of interest, since A-NICE-MC learns Markov chain proposal. On the one hand, Markov chain proposal uses more information while generating a new sample, hence can learn more expressive stationary distribution, on the other hand, usage of previous sample increase autocorrelation between samples and reduces ESS. Thus, the main point of interest is the comparison of two independent proposals: one is learned by maximization of the acceptance rate (or its lower bound), and the second is learned by variational inference procedure, i.e. maximization of evidence lower bound. In Table 2 we see that both maximization of the acceptance rate and its lower bound outperform variational inference for all target distributions. Moreover, in Fig. 1 we show that variational inference fails to cover all the modes of mog6 in contrast to proposals learned via maximization of acceptance rate or its lower bound. Densities of learned proposals and histograms for all distributions are presented in Appendix C.6. 

In density-based setting, we consider Bayesian inference problem for the weights of a neural network. In our experiments we consider approximation of predictive distribution (Eq. 7) as our main goal. To estimate the goodness of the approximation we measure negative log-likelihood and accuracy on the test set.

In subsection 3.1 we show that lower bound on acceptance rate can be optimized more efficiently than acceptance rate due to the usage of minibatches. But other questions arise.

1. Does the proposed objective in Eq. 11 allow for better estimation of predictive distribution compared to the variational inference?

2. Does the application of the MH correction to the learned proposal distribution allow for better estimation of the predictive distribution (Eq. 7) than estimation via raw samples from the proposal?

To answer these questions we consider reduced LeNet-5 architecture (see Appendix C.7) for classification task on 20k images from MNIST dataset (for test data we use all of the MNIST test set). Even after architecture reduction we still face a challenging task of learning a complex distribution in 8550-dimensional space. For the proposal distribution we use fully-factorized gaussian q φ (θ) = d j=1 N (θ j | µ j , σ j ) and standard normal distribution for prior p(θ) = d j=1 N (θ j | 0, 1). For variational inference, we train the model using different initialization and pick the model according to the best ELBO. For our procedure, we do the same and choose the model by the maximum value of the acceptance rate lower bound. In Algorithm 1 we propose to sample from the posterior distribution using the independent MH and the current proposal. It turns out in practice that it is better to use the currently learned proposal q φ (θ) = N (θ | µ, σ) as the initial state for random-walk MH algorithm. That is, we start with the mean µ as an initial point, and then use random-walk proposal q(θ | θ) = N (θ | θ, σ) with the variances σ of current independent proposal. This should be considered as a heuristic that improves the approximation of the loss function.

The optimization of the acceptance rate lower bound results in the better estimation of predictive distribution than the variational inference (see Fig. 2 ). Optimization of acceptance rate for the same number of epochs results in nearly 30% accuracy on the test set. That is why we do not report results for this procedure in Fig. 2 . In both procedures we apply the independent MH algorithm to estimate the predictive distribution. To answer the second question we estimate predictive distribution in two ways. The first way is to perform 100 accept/reject steps of the independent MH algorithm with the learned proposal q φ (θ)

after each epoch, i.e. perform MH correction of the samples from the proposal. The second way is to take the same number of samples from q φ (θ) without MH correction. For both estimations of predictive distribution, we evaluate negative log-likelihood on the test set and compare them.

The MH correction of the learned proposal improves the estimation of predictive distribution for the variational inference (right plot of Fig. 3 ) but does not do so for the optimization of the acceptance rate lower bound (left plot of Fig. 3 ). This fact may be considered as an implicit evidence that our procedure learns the proposal distribution with higher acceptance rate.

In the sample-based setting, we estimate density ratio using a discriminator. Hence we do not use the minibatching property (see subsection 3.1) of the obtained lower bound, and optimization problems for the acceptance rate and for the lower bound have the same efficiency in terms of using data.

That is why our main goal in this setting is to compare the optimization of the acceptance rate and the optimization of the lower bound. Also, in this setting, we have Markov chain proposal that is interesting to compare with the independent proposal. Summing up, we formulate the following questions:

1. Does the optimization of the lower bound has any benefits compared to the direct optimization of the acceptance rate? 2. Do we have mixing issue while learning Markov chain proposal in practice? 3. Could we improve the visual quality of samples by applying the MH correction to the learned proposal?

We use DCGAN architecture for the proposal and discriminator (see Appendix C.8) and apply our algorithm to MNIST dataset. We consider two optimization problems: direct optimization of the acceptance rate and its lower bound. We also consider two ways to obtain samples from the approximation of the target distribution -use raw samples from the learned proposal, or perform the MH algorithm, where we use the learned discriminator for density ratio estimation.

In Note that we obtain different distributions of the samples because of conditioning of our proposal.

To show that the learned proposal distribution has the Markov property rather than being totally independent, we show samples from the proposal conditioned on two different points in the dataset (see Fig. 6 ). The difference in samples from two these distributions (Fig. 6 (a), 6(a)) reflects the dependence on the conditioning.

Additionally, in Appendix C.9 we present samples from the chain after 10000 accepted images and also samples from the chain that was initialized with noise.

5. DISCUSSION AND FUTURE WORK:

This paper proposes to use the acceptance rate of the MH algorithm as the universal objective for learning to sample from some target distribution. We also propose the lower bound on the acceptance rate that should be preferred over the direct maximization of the acceptance rate in many cases. The proposed approach provides many ways of improvement by the combination with techniques from the recent developments in the field of MCMC, GANs, variational inference. For example

• The proposed loss function can be combined with the loss function from (Levy et al., 2017) , thus allowing to learn the Markov chain proposal in the density-based setting. • We can use stochastic Hamiltonian Monte Carlo (Chen et al., 2014) for the loss estimation in Algorithm 1. • In sample-based setting one can use more advanced techniques of density ratio estimation.

Application of the MH algorithm to improve the quality of generative models also requires exhaustive further exploration and rigorous treatment.

A ACCEPTANCE RATE OF THE MH ALGORITHM:

A.1 PROOF OF THEOREM 1 Remind that we have random variables ξ = p(x )q(x | x ) p(x)q(x | x) , x ∼ p(x), x ∼ q(x | x) and u ∼ Uniform[0, 1], and want to prove the following equalities.

EQUATION

Equality E ξ min{1, ξ} = P{ξ > u} is obvious.

EQUATION

Equality P{ξ > u} = 1 -1 2 E ξ |ξ -1| can be proofed as follows.

EQUATION

where F ξ (u) is CDF of random variable ξ. Note that F ξ (0) = 0 since ξ ∈ (0, +∞]. Eq. 21 can be rewritten in two ways.

EQUATION

To rewrite Eq. 21 in the second way we note that Eξ = 1.

1-F ξ (1)+ 

EQUATION

Using the form of ξ we can rewrite the acceptance rate as

EQUATION

A.2 ACCEPTANCE RATE OF INDEPENDENT MH DEFINES SEMIMETRIC IN DISTRIBUTION:



SPACE:

In independent case we have ξ = p(x )q(x) p(x)q(x ) , x ∼ p(x), x ∼ q(x ) and we want to prove that E ξ |ξ -1| is semimetric (or pseudo-metric) in space of distributions. For this appendix, we denote D(p, q) = E ξ |ξ -1|. The first two axioms for metric obviously holds 1. D(p, q) = 0 ⇐⇒ p = q 2. D(p, q) = D(q, p)

There is an example when triangle inequality does not hold.

For Here N (x | 0, σ, -a, a) is truncated normal distribution. The first KL-divergence can be written as follows.

EQUATION

)

EQUATION

Here Z is normalization constant of truncated log normal distribution and Z = Φ(a/σ) -Φ(-a/σ), where Φ(x) is CDF of standard normal distribution. The second KL-divergence is KL

EQUATION

Summing up two KL-divergencies and taking derivative w.r.t. σ we obtain

∂ ∂σ KL(Uniform[-a, a] N (x | 0, σ, -a, a)) + KL(N (x | 0, σ, -a, a) Uniform[-a, a]) = (44) = - a 2 3σ 3 + a 3 √ 2πσ 4 Z exp - a 2 2σ 2 + a √ 2π exp - a 2 2σ 2 - 1 σ 2 Z - 1 σZ 2 -2a σ 2 √ 2π exp - a 2 2σ 2 = (45) = 1 a - a 3 3σ 3 + a 2 √ 2πσ 2 Z exp - a 2 2σ 2 a 2 σ 2 -1 + 2a √ 2πσZ exp - a 2 2σ 2 (46)

To show that the derivative of the lower bound w.r.t. σ is negative, we need to prove that the following inequality holds for positive x.

-

1 3 x 3 + x 2 √ 2π(Φ(x) -Φ(-x)) exp(-x 2 /2) x 2 -1+ 2x √ 2π(Φ(x) -Φ(-x)) exp(-x 2 /2) < 0, x > 0 (47) Defining φ(x) =

x 0 e -t 2 /2 dt and noting that 2φ(x) = √ 2π(Φ(x) -Φ(-x)) we can rewrite inequality 47 as

1 φ(x) e -x 2 /2 x 2 -1 + 2xe -x 2 /2 φ(x) < 2x 3 , x > 0 (48)

By the fundamental theorem of calculus, we have

xe -x 2 /2 = x 0 e -t 2 /2 (1 -t 2 )dt (49) Hence, φ(x) -xe -x 2 /2 = x 0 e -t 2 /2 t 2 dt ≥ e -x 2 /2 x 0 t 2 dt = e -x 2 /2 x 3 3 (50)

Or equivalently,

EQUATION

Using this inequality twice, we obtain

EQUATION

and

EQUATION

Thus, the target inequality can be verified by the verification of

EQUATION

Thus, we show that partial derivative of our lower bound w.r.t. σ is negative. Using that knowledge we can improve our loss by taking a bigger value of σ. Hence, such proposal does not collapse to delta-function.

B.2 INTUITION FOR BETTER GRADIENTS IN SAMPLE-BASED SETTING:

In this section, we provide an intuition for sample-based setting that the loss function for lower bound has better gradients than the loss function for acceptance rate. Firstly, we remind that in the sample-based setting we use a discriminator for density ratio estimation.

EQUATION

For this purpose we use the discriminator of special structure

EQUATION

We denote d(x, x ) = D(x, x ) -D(x , x) and consider the case when the discriminator can easily distinguish fake pairs from valid pairs. So D(x, x ) is close to 1 and d(x, x ) 0 for x ∼ p(x) and x ∼ q(x | x). To evaluate gradients we consider Monte Carlo estimations of each loss and take gradients w.r.t. x in order to obtain gradients for parameters of proposal distribution. We do not introduce the reparameterization trick to simplify the notation but assume it to be performed. For the optimization of the acceptance rate we have

EQUATION

EQUATION

While for the optimization of the lower bound we have

EQUATION

EQUATION

Now we compare Eq. 59 and Eq. 62. We see that in case of strong discriminator we have vanishing gradients in Eq. 59 due to exp(-d(x, x )), while it is not the case for Eq. 62. This experiment shows that it is possible to optimize the acceptance rate, optimizing its lower bound.

C EXPERIMENTS C.1 TOY PROBLEM:

For the target distribution we consider bimodal Gaussian p(x) = 0.5 • N (x | -2, 0.5) + 0.5 • N (x | 2, 0.7), for the independent proposal we consider unimodal gaussian q(x) = N (x | µ, σ). We perform stochastic gradient optimization using Algorithm 1 from the same initialization for both objectives (Fig. 8 ) and obtain approximately the same local maximums.

C.2 ARCHITECTURE OF THE REALNVP PROPOSAL:

For the proposal distribution we use similar architecture to the NICE proposal. The RealNVP model (Dinh et al., 2016) where the functions s, t can be arbitrary complex, since the structure of the functions doesn't influence the computation of the Jacobian.

For our proposal we use 4 coupling layers with s and t consist of two fully-connected layers with hidden dimension of 256.

C.3 SYNTHETIC DISTRIBUTIONS DENSITIES:

For synthetic distributions we consider the same distributions as in Song et al. (2017) . In this section, we show additional figures for Markov chain proposals. In Fig. 12 we show samples from the chain that was initialized by the noise. In Fig. 13 we show samples from the chain after 10000 accepted samples. 

:



Figures and Tables:

Figure 1: 2d histrogram of samples from the MH algorithm with different proposals. From left to right proposals are learned by: variational inference, the acceptance rate maximization, the acceptance rate lower bound maximization.

Figure 2: Negative log-likelihood (left) and accuracy (right) on test set of MNIST dataset for variational inference (blue lines) and the optimization of the acceptance rate lower bound (orange lines).In both procedures we apply the independent MH algorithm to estimate the predictive distribution.

Figure 3: Test negative log-likelihood for two approximations of the predictive distribution based on samples: from proposal distribution nll q and after MH correction nll M H . Left figure corresponds to the optimization of the acceptance rate lower bound, right figure corresponds to the variational inference.

case of the independent proposal, we show that the MH correction at evaluation step allows to improve visual quality of samples -figures 4(a) and 4(b) for the direct optimization of acceptance rate, figures 4(c) and 4(d) for the optimization of its lower bound. Note that in Algorithm 2 we do not apply the independent MH algorithm during training. Potentially, one can use the MH algorithm considering any generative model as a proposal distribution and learning a discriminator for density ratio estimation. Also, for this proposal, we demonstrate the negligible difference in visual quality of samples obtained by the direct optimization of acceptance rate (see Fig. 4(a)) and by the optimization of the lower bound (see Fig. 4(c)).

Figure 4: Samples from the learned independent proposal obtained via optimization: of acceptance rate (4(a), 4(b)) and its lower bound (4(c), 4(d)). In Fig. 4(b), 4(d) we show raw samples from the learned proposal. In Fig. 4(a), 4(c) we show the samples after applying the independent MH correction to the samples, using the learned discriminator for density ratio estimation. In the case of the Markov chain proposal, we show that the direct optimization of acceptance rate results in slow mixing (see Fig. 5(a)) -most of the time the proposal generates samples from one of the modes (digits) and rarely switches to another mode. When we perform the optimization of the lower bound the proposal switches between modes frequently (see Fig. 5(b)).

Figure 5: Samples from the chain obtained via the MH algorithm with the learned proposal and the learned discriminator for density ratio estimation. Fig. 5(a) corresponds to the direct optimization of the acceptance rate. Fig. 5(b) -to optimization of the lower bound on acceptance rate. Samples in the chain are obtained one by one from left to right from top to bottom.

ξ (u)du (23) Summing equations 22 and 23 results in the following formula

Figure 7: In this figure we show schematic view of approximation of of target distribution with uniform distribution. Red bounding box is made bigger for better comprehension.

Figure 8: Level-plots in parameter space for the toy problem. Left: level-plot for the acceptance rate of the MH algorithm. Right: level-plot for the lower bound of the acceptance rate.

use the same strategy for evaluating the Jacobian as the NICE model does. Each coupling layer define the following function. Given a D dimensional input x and d < D, the output y is evaluated by the formula y 1:d = x 1:d , y d+1:D = x d+1:D exp(s(x 1:d )) + t(x 1:d ),

Figure9: plots for the acceptance rate and the rate lower bound evaluated at every iteration during the optimization of the acceptance rate lower bound. Correlation coefficient is evaluated between the logarithm of the acceptance rate and the acceptance rate lower bound.

Figure 12: Samples from the chain initialized with noise. To obtain samples we use the MH algorithm with the learned proposal and the learned discriminator for density ratio estimation. In Fig. 5(a) we use proposal and discriminator that are learned during optimization of acceptance rate. In Fig. 5(b) we use proposal and discriminator that are learned during the optimization of the acceptance rate lower bound. Samples in the chain are obtained one by one from left to right from top to bottom starting with noise (first image in the figure).

Constraints for two settings of learning sampling algorithms

Table content: <table><tr><td>Setting</td><td>Target distribution</td><td colspan="2">Proposal distribution Density Ratio</td></tr><tr><td colspan="2">Density-based given p(x) ∝ p(x)</td><td>explicit model q(x )</td><td>explicit</td></tr><tr><td colspan="2">Sample-based set of samples X ∼ p(x)</td><td>implicit model q(x )</td><td>learned discriminator</td></tr><tr><td/><td/><td>implicit model q(x |x)</td></tr></table>

Performance of learned proposals as measured by Effective Sample Size (see Appendix C.4 for formulation). Higher is better (1000 maximum). All the numbers are rounded to integers. See description of the compared methods in the text.

Table content: <table><tr><td colspan="6">Target A-NICE-MC HMC AR (ours) ARLB (ours) VI</td></tr><tr><td>ring</td><td>1000</td><td>1000</td><td>851</td><td>850</td><td>702</td></tr><tr><td>mog2</td><td>355</td><td>1</td><td>786</td><td>604</td><td>297</td></tr><tr><td>mog6</td><td>320</td><td>1</td><td>311</td><td>367</td><td>12</td></tr><tr><td>ring5</td><td>156</td><td>1</td><td>336</td><td>249</td><td>170</td></tr></table>



Table content: <table><tr><td>VI</td></tr></table>

Peer review (iclr2019):
 One of the main contributions of the paper is showing how maximizing the acceptance rate in Metropolis Hastings (MH) translates in minimizing the symmetric KL divergence of the target and the proposal distribution. The result aligns with the intuition that if this variance is 0, detailed balanced is satisfied and, hence, we can always accept the proposal. Also Equation 11 nicely fits the intuition that a good (independent) proposal should minimize the KL divergence between proposal and target (as in VI) under the constraint that the proposal has full support compared to the target distribution, which is enforced by the last term. Theorem 1 and its proof are great. However, the proposed algorithms leveraging these findings are problematic. Algorithm 1 suggest independent Metropolis-Hastings in order to avoid the proposal to collapse to a point distribution, that is, a Dirac delta function centered at the current position. However, in the experiments, the authors study a "modified" version using a random walk proposal parameterized with the current (diagonal) covariance estimate. This is surprising as the authors explicitly avoided this proposal when motivating MH acceptance rate maximization. In any case, the correctness of the algorithm is neither shown for an independent proposal nor a Markov Chain proposal. Indeed, I would argue that Algorithm 1 generally does not find a sample distributed according to the target distribution. The algorithm assumes that we can create a sample from the target p(x) that can be used to approximate the loss (a bound on the expected acceptance rate). However, if we could find an unbiased sample in the very first iteration of the algorithm, we could immediately stop and there wouldn*t be a need for the algorithm at all. Hence, we must assume that the sample drawn from the target is biased (in the beginning); which is indeed a realistic assumption as neither independent MH nor random walk MH will yield an unbiased sample in any reasonable time (for any practical problem). However, this would bias the loss and, consequently, the parameter update for the proposal distribution. In particular, for random walk MH, I would expect the covariance to contract such that the final proposal indeed collapses to a point distribution. This is because the proposal is only required to have support around the observed samples and this area will become smaller over time. I would expect a proof that the sample at iteration k is "better" than a sample drawn at iteration k-1, to show that the bias vanishes over time. Though, I assume that this is hard to show as the proposal parameters form a Markov Chain itself. So at least a rigor empirical study is needed. Therefore, I would expect a metric measuring the quality of the final sample. The log-likelihood is not such a measure. While the marginalized log-likelihood could measure the quality of the sample, we cannot compute it for any real data/model (which is why we use sampling in the first place). So we need some artificial settings. However, the 1-dimensional toy example is insufficient as MH primarily suffers in high-dimensional spaces. It would be interesting to also report the acceptance rate depending on the number of dimensions of the target distribution. I would assume an exponential decay; even with learning, which might be the reason why the authors only report random walk MH in Section 4.2. Algorithm 2 does not require to sample from some target distribution but can leverage the observed sample. While the algorithm nicely connects GANs and sampling, the actual value of the algorithm is not fully clear to me. Learning an independent proposal reduces the problem to learning a GAN; and learning a Markov Chain seems only relevant for sampling-based inference; however, we already have a sample from the target distribution, and we can sample more data using a trained GAN. Minor comments: - The prefix/in-fix notation of integrals is mixed, e.g. in Eq 19, "dx" appears before the integrand, but "du" appears after the integrand of the inner integral. UPDATE: The revised version is much better in empirically demonstrating the value of the method; though, there is still some work needed. First, the artificial examples are still rather low-dimensional where independent MH is expected to perform well. Second, ESS does not help to assess the biasedness of the sample; maybe [1] can help with this. Third, NUTS might be a better baseline than standard HMC which is know to be sensitive to the stepsize/number of leapfrog steps. An effective sample size of 1 suggests that the method did not even start to sample - likely because of a bad choice of the stepsize and/or mass matrix. I would suggest using PyMC3*s NUTS implementation. Finally, to show the correctness of the method, I would suggest to 1) let alpha converge to zero such that .phi will be fixed at some point, and 2) ensure that the proposal has full support under the target for any value of .phi. In this case, the sample drawn from the target will be unbiased for large enough n (same arguments as for adaptive MCMC should apply). The idea of reusing the samples from previous iterations for approximating the loss is interesting and worth exploring. [1] Jackson Gorham, Lester Mackey. "Measuring Sample Quality with Kernels", https://arxiv.org/abs/1703.01717