Hamiltonian Variational Auto-Encoder

05/29/2018 ∙ by Anthony L. Caterini, et al. ∙ University of Oxford 0

Variational Auto-Encoders (VAEs) have become very popular techniques to perform inference and learning in latent variable models as they allow us to leverage the rich representational power of neural networks to obtain flexible approximations of the posterior of latent variables as well as tight evidence lower bounds (ELBOs). Combined with stochastic variational inference, this provides a methodology scaling to large datasets. However, for this methodology to be practically efficient, it is necessary to obtain low-variance unbiased estimators of the ELBO and its gradients with respect to the parameters of interest. While the use of Markov chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo (HMC) has been previously suggested to achieve this [23, 26], the proposed methods require specifying reverse kernels which have a large impact on performance. Additionally, the resulting unbiased estimator of the ELBO for most MCMC kernels is typically not amenable to the reparameterization trick. We show here how to optimally select reverse kernels in this setting and, by building upon Hamiltonian Importance Sampling (HIS) [17], we obtain a scheme that provides low-variance unbiased estimators of the ELBO and its gradients using the reparameterization trick. This allows us to develop a Hamiltonian Variational Auto-Encoder (HVAE). This method can be reinterpreted as a target-informed normalizing flow [20] which, within our context, only requires a few evaluations of the gradient of the sampled likelihood and trivial Jacobian calculations at each iteration.

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1 Introduction

Variational Auto-Encoders (VAEs), introduced by Kingma and Welling [2014] and Rezende et al. [2014], are popular techniques to carry out inference and learning in complex latent variable models. However, the standard mean-field parametrization of the approximate posterior distribution can limit its flexibility. Recent work has sought to augment the VAE approach by sampling from the VAE posterior approximation and transforming these samples through mappings with additional trainable parameters to achieve richer posterior approximations. The most popular application of this idea is the Normalizing Flows (NFs) approach Rezende and Mohamed [2015] in which the samples are deterministically evolved through a series of parameterized invertible transformations called a flow. NFs have demonstrated success in various domains Berg et al. [2018], Kingma et al. [2016], but the flows do not explicitly use information about the target posterior. Therefore, it is unclear whether the improved performance is caused by an accurate posterior approximation or simply a result of overparametrization. The related Hamiltonian Variational Inference (HVI) Salimans et al. [2015] instead stochastically evolves the base samples according to Hamiltonian Monte Carlo (HMC) Neal et al. [2011] and thus uses target information, but relies on defining reverse dynamics in the flow, which, as we will see, turns out to be unnecessary and suboptimal.

One of the key components in the formulation of VAEs is the maximization of the evidence lower bound (ELBO). The main idea put forward in Salimans et al. [2015] is that it is possible to use

MCMC iterations to obtain an unbiased estimator of the ELBO and its gradients. This estimator is obtained using an importance sampling argument on an augmented space, with the importance distribution being the joint distribution of the

states of the ‘forward’ Markov chain, while the augmented target distribution is constructed using a sequence of ‘reverse’ Markov kernels such that it admits the original posterior distribution as a marginal. The performance of this estimator is strongly dependent on the selection of these forward and reverse kernels, but no clear guideline for selection has been provided therein. By linking this approach to earlier work by Del Moral et al. [2006], we show how to select these components. We focus, in particular, on the use of time-inhomogeneous Hamiltonian dynamics, proposed originally in Neal [2005]. This method uses reverse kernels which are optimal for reducing variance of the likelihood estimators and allows for simple calculation of the approximate posteriors of the latent variables. Additionally, we can easily use the reparameterization trick to calculate unbiased gradients of the ELBO with respect to the parameters of interest. The resulting method, which we refer to as the Hamiltonian Variational Auto-Encoder (HVAE), can be thought of as a Normalizing Flow scheme in which the flow depends explicitly on the target distribution. This combines the best properties of HVI and NFs, resulting in target-informed and inhomogeneous deterministic Hamiltonian dynamics, while being scalable to large datasets and high dimensions.

2 Evidence Lower Bounds, MCMC and Hamiltonian Importance Sampling

2.1 Unbiased likelihood and evidence lower bound estimators

For data and parameter , consider the likelihood function

where are some latent variables. If we assume we have access to a strictly positive unbiased estimate of , denoted , then

(1)

with ,

denoting all the random variables used to compute

. Here, denotes additional parameters of the sampling distribution. We emphasize that depends on both and potentially as this is not done notationally. By applying Jensen’s inequality to (1), we thus obtain, for all and ,

(2)

It can be shown that decreases as the variance of decreases; see, e.g., Burda et al. [2016], Maddison et al. [2017]. The standard variational framework corresponds to and , while the Importance Weighted Auto-Encoder (IWAE) Burda et al. [2016] with importance samples corresponds to , and

In the general case, we do not have an analytical expression for . When performing stochastic gradient ascent for variational inference, however, we only require an unbiased estimator of . This is given by if the reparameterization trick Glasserman [1991], Kingma and Welling [2014] is used, i.e. , and is a ‘smooth enough’ function of . As a guiding principle, one should attempt to obtain a low-variance estimator of , which typically translates into a low-variance estimator of . We can analogously optimize with respect to through stochastic gradient ascent to obtain tighter bounds.

2.2 Unbiased likelihood estimator using time-inhomogeneous MCMC

Salimans et al. [2015] propose to build an unbiased estimator of by sampling a (potentially time-inhomogeneous) ‘forward’ Markov chain of length using and for . Using artificial ‘reverse’ Markov transition kernels for , it follows easily from an importance sampling argument that

(3)

is an unbiased estimator of as long as the ratio in (3) is well-defined. In the framework of the previous section, we have and is given by the denominator of (3). Although we did not use measure-theoretic notation, the kernels are typically MCMC kernels which do not admit a density with respect to the Lebesgue measure (e.g. the Metropolis–Hastings kernel). This makes it difficult to define reverse kernels for which (3) is well-defined as evidenced in Salimans et al. [2015, Section 4] or Wolf et al. [2016]. The estimator (3) was originally introduced in Del Moral et al. [2006] where generic recommendations are provided for this estimator to admit a low relative variance: select as MCMC kernels which are invariant, or approximately invariant as in Heng et al. [2017], with respect to , where is a sequence of artificial densities bridging to smoothly using . It is also established in Del Moral et al. [2006] that, given any sequence of kernels , the sequence of reverse kernels minimizing the variance of is given by where denotes the marginal density of under the forward dynamics, yielding

(4)

For stochastic forward transitions, it is typically not possible to compute and the corresponding estimator (4) as the marginal densities do not admit closed-form expressions. However this suggests that should be approximating and various schemes are presented in Del Moral et al. [2006]. As noticed by Del Moral et al. [2006] and Salimans et al. [2015], Annealed Importance Sampling (AIS) Neal [2001] – also known as the Jarzynski-Crooks identity (Crooks [1998], Jarzynski [1997]) in physics – is a special case of (3) using, for , a -invariant MCMC kernel and the reversal of this kernel as the reverse transition kernel 111The reversal of a -invariant kernel is given by . If is -reversible then .. This choice of reverse kernels is suboptimal but leads to a simple expression for estimator (3

). AIS provides state-of-the-art estimators of the marginal likelihood and has been widely used in machine learning. Unfortunately, it typically cannot be used in conjunction with the reparameterization trick. Indeed, although it is very often possible to reparameterize the forward simulation of

in terms of the deterministic transformation of some random variables independent of and , this mapping is not continuous because the MCMC kernels it uses typically include singular components. In this context, although (1) holds, is not an unbiased estimator of ; see, e.g., Glasserman [1991] for a careful discussion of these issues.

2.3 Using Hamiltonian dynamics

Given the empirical success of Hamiltonian Monte Carlo (HMC) Hoffman and Gelman [2014], Neal et al. [2011], various contributions have proposed to develop algorithms exploiting Hamiltonian dynamics to obtain unbiased estimates of the ELBO and its gradients when . This was proposed in Salimans et al. [2015]. However, the algorithm suggested therein relies on a time-homogeneous leapfrog where momentum resampling is performed at each step and no Metropolis correction is used. It also relies on learned reverse kernels. To address the limitations of Salimans et al. [2015], Wolf et al. [2016] have proposed to include some Metropolis acceptance steps, but they still limit themselves to homogeneous dynamics and their estimator is not amenable to the reparameterization trick. Finally, in Hoffman [2017], an alternative approach is used where the gradient of the true likelihood, , is directly approximated by using Fisher’s identity and HMC to obtain approximate samples from . However, the MCMC bias can be very significant when one has multimodal latent posteriors and is strongly dependent on both the initial distribution and .

Here, we follow an alternative approach where we use Hamiltonian dynamics that are time-inhomogeneous as in Del Moral et al. [2006] and Neal [2001], and use optimal reverse Markov kernels to compute . This estimator can be used in conjunction with the reparameterization trick to obtain an unbiased estimator of . This method is based on the Hamiltonian Importance Sampling (HIS) scheme proposed in Neal [2005]; one can also find several instances of related ideas in physics Jarzynski [2000], Schöll-Paschinger and Dellago [2006]. We work in an extended space , introducing momentum variables to pair with the position variables , with new target . Essentially, the idea is to sample using deterministic transitions so that , where and , define diffeomorphisms corresponding to a time-discretized and inhomogeneous Hamiltonian dynamics. In this case, it is easy to show that

(5)

It can also be shown that this is nothing but a special case of (3) (on the extended position-momentum space) using the optimal reverse kernels222Since this is a deterministic flow, the density can be evaluated directly. However, direct evaluation corresponds to optimal reverse kernels in the deterministic case. . This setup is similar to the one of Normalizing Flows Rezende and Mohamed [2015], except here we use a flow informed by the target distribution. Salimans et al. [2015] is in fact mentioned in Rezende and Mohamed [2015], but the flow therein is homogeneous and yields a high-variance estimator of the normalizing constants even if is used, as demonstrated in our simulations in section 4.

Under these dynamics, the estimator defined in (5) can be rewritten as

(6)

Hence, if we can simulate using , where and is a smooth mapping, then we can use the reparameterization trick since are also smooth mappings.

In our case, the deterministic transformation has two components: a leapfrog step, which discretizes the Hamiltonian dynamics, and a tempering step, which adds inhomogeneity to the dynamics and allows us to explore isolated modes of the target Neal [2005]. To describe the leapfrog step, we first define the potential energy of the system as for a single datapoint . Leapfrog then takes the system from into via the following transformations:

(7)
(8)
(9)

where are the individual leapfrog step sizes per dimension, denotes elementwise multiplication, and the gradient of is taken with respect to . The composition of equations (7) - (9) has unit Jacobian since each equation describes a shear transformation. For the tempering portion, we multiply the momentum output of each leapfrog step by for where . We consider two methods for setting the values . First, fixed tempering involves allowing an inverse temperature to vary, and then setting , where each is a deterministic function of and . In the second method, known as free tempering, we allow each of the values to be learned, and then set the initial inverse temperature to . For both methods, the tempering operation has Jacobian . We obtain by composing the leapfrog integrator with the cooling operation, which implies that the Jacobian is given by , which in turns implies

The only remaining component to specify is the initial distribution. We will set , where will be referred to as the variational prior over the latent variables and is the canonical momentum distribution at inverse temperature . The full procedure to generate an unbiased estimate of the ELBO from (2) on the extended space for a single point and fixed tempering is given in Algorithm 1. The set of variational parameters to optimize contains the flow parameters and , along with additional parameters of the variational prior.333We avoid reference to a mass matrix throughout this formulation because we can capture the same effect by optimizing individual leapfrog step sizes per dimension as pointed out in [Neal et al., 2011, Section 4.2]. We can see from (6) that we will obtain unbiased gradients with respect to and from our estimate of the ELBO if we write , for and , provided we are not also optimizing with respect to parameters of the variational prior. We will require additional reparameterization when we elect to optimize with respect to the parameters of the variational prior, but this is generally quite easy to implement on a problem-specific basis and is well-known in the literature; see, e.g. Kingma and Welling [2014], Rezende and Mohamed [2015], Rezende et al. [2014] and section 4.

is the unnormalized posterior for and
is the variational prior on
function HIS()
     Sample
     
     for  do Run steps of alternating leapfrog and tempering
          Start of leapfrog; Equation (7)
          Equation (8)
          Equation (9)
          Quadratic tempering scheme
               
     
      Equation (5), left side
      Take the of equation (5), right side
     return Can take unbiased gradients of this estimate wrt
Algorithm 1 Hamiltonian ELBO, Fixed Tempering

3 Stochastic Variational Inference

We will now describe how to use Algorithm 1 within a stochastic variational inference procedure, moving to the setting where we have a dataset and for all . In this case, we are interested in finding

(10)

where is the empirical measure of the data. We must resort to variational methods since cannot generally be calculated exactly and instead maximize the surrogate ELBO objective function

(11)

for defined as in (2). We can now turn to stochastic gradient ascent (or a variant thereof) to jointly maximize (11) with respect to and by approximating the expectation over using minibatches of observed data.

For our specific problem, we can reduce the variance of the ELBO calculation by analytically evaluating some terms in the expectation (i.e. Rao-Blackwellization) as follows:

(12)

where we write under reparameterization. We can now consider the output of Algorithm 1 as taking a sample from the inner expectation for a given sample from the outer expectation. Algorithm 2 provides a full procedure to stochastically optimize (12). In practice, we take the gradients of (12

) using automatic differentation packages. This is achieved by using TensorFlow

Abadi et al. [2015] in our implementation.

is the unnormalized posterior for and
function HVAE() is minibatch size
     Initialize
     while  not converged do Stochastic optimization loop
         Sample independently
          Average ELBO estimators over mini-batch
         for  do
               HIS() +          
         
         

      Optimize the ELBO using gradient-based techniques such as RMSProp, ADAM, etc.

          UpdateTheta()
          UpdatePhi()      
     return
Algorithm 2 Hamiltonian Variational Auto-Encoder

4 Experiments

In this section, we discuss the experiments used to validate our method. We first test HVAE on an example with a tractable full log likelihood (where no neural networks are needed), and then perform larger-scale tests on the MNIST dataset. Code is available online.444https://github.com/anthonycaterini/hvae-nips All models were trained using TensorFlow Abadi et al. [2015].

4.1 Gaussian Model

The generative model that we will consider first is a Gaussian likelihood with an offset and a Gaussian prior on the mean, given by

where is constrained to be diagonal. We will again write to denote an observed dataset under this model, where each . In this example, we have . The goal of the problem is to learn the model parameters , where and .

Here, we have only one latent variable generating the entire set of data. Thus, our variational lower bound is now given by

for the variational posteroir approximation . We note that this is not exactly the same as the auto-encoder setting, in which an individual latent variable is associated with each observation, however it provides a tractable framework to analyze effectiveness of various variational inference methods. We also note that we can calculate the log-likelihood exactly in this case, but we use variational methods for the sake of comparison.

From the model, we see that the logarithm of the unnormalized target is given by

For this example, we will use a HVAE with variational prior equal to the true prior, i.e. , and fixed tempering. The potential, given by , has gradient

The set of variational parameters here is , where contains the per-dimension leapfrog stepsizes and is the initial inverse temperature. We constrain each of the leapfrog step sizes such that for some , for all – this is to prevent the leapfrog discretization from entering unstable regimes. Note that in this example; in particular, we do not optimize any parameters of the variational prior and thus require no further reparameterization.

We will compare HVAE with a basic Variational Bayes (VB) scheme with mean-field approximate posterior , where is diagonal and denotes the set of learned variational parameters. We will also include a planar normalizing flow of the form of equation (10) in Rezende and Mohamed [2015], but with the same flow parameters across iterations to keep the number of variational parameters of the same order as the other methods. The variational prior here is also set to the true prior as in HVAE above. The log variational posterior is given by equation (13) of Rezende and Mohamed [2015], where .

We set our true offset vector to be

, and our scale parameters to range quadratically from , reaching a minimum at , and increasing back to .666When is even, does not exist, although we could still consider

to be the location of the minimum of the parabola defining the true standard deviations.

All experiments have and all training was done using RMSProp Tieleman and Hinton [2012] with a learning rate of .

To compare the results across methods, we train each method ten times on different datasets. For each training run, we calculate , where is the estimated value of given by the variational method on a particular run, and plot the average of this across the 10 runs for various dimensions in (a). We note that, as the dimension increases, HVAE performs best in parameter estimation. The VB method suffers most on prediction of as the dimension increases, whereas the NF method does poorly on predicting .

(a) Comparison across all methods
(b) HVAE with and without tempering
Figure 1: Averages of for several variational methods and choices of dimensionality , where is the estimated maximizer of the ELBO for each method and is the true parameter.

We also compare HVAE with tempering to HVAE without tempering, i.e. where is fixed to in training. This has the effect of making our Hamiltonian dynamics homogeneous in time. We perform the same comparison as above and present the results in (b). We can see that the tempered methods perform better than their non-tempered counterparts; this shows that time-inhomogeneous dynamics are a key ingredient in the effectiveness of the method.

4.2 Generative Model for MNIST

The next experiment that we consider is using HVAE to improve upon a convolutional variational auto-encoder (VAE) for the binarized MNIST handwritten digit dataset. Again, our training data is

, where each for . The generative model is as follows:

for , where is the component of , is the latent variable associated with , and

is a convolutional neural network (i.e. the

generative network, or encoder) parametrized by the model parameters . This is the standard generative model used in VAEs in which each pixel in the image is conditionally independent given the latent variable. The VAE approximate posterior – and the HVAE variational prior across the latent variables in this case – is given by , where and are separate outputs of the same neural network (the inference network, or encoder) parametrized by , and is constrained to be diagonal.

We attempted to match the network structure of Salimans et al. [2015]. The inference network consists of three convolutional layers, each with filters of size

and a stride of 2. The convolutional layers output 16, 32, and 32 feature maps, respectively. The output of the third layer is fed into a fully-connected layer with hidden dimension

, whose output is then fully connected to the output means and standard deviations each of size

. Softplus activation functions are used throughout the network except immediately before the outputted mean. The generative network mirrors this structure in reverse, replacing the stride with upsampling as in

Dosovitskiy et al. [2015] and replicated in Salimans et al. [2015].

We apply HVAE on top of the base convolutional VAE. We evolve samples from the variational prior according to Algorithm 1 and optimize the new objective given in (12). We reparameterize as , for and , to generate unbiased gradients of the ELBO with respect to . We select various values for and set . In contrast with normalizing flows, we do not need our flow parameters and to be outputs of the inference network because our flow is guided by the target. This allows our method to have fewer overall parameters than normalizing flow schemes. We use the standard stochastic binarization of MNIST Salakhutdinov and Murray [2008] as training data, and train using Adamax Kingma and Ba [2014] with learning rate

. We also employ early stopping by halting the training procedure if there is no improvement in the loss on validation data over 100 epochs.

To evaluate HVAE after training is complete, we estimate out-of-sample negative log likelihoods (NLLs) using 1000 importance samples from the HVAE approximate posterior. For each trained model, we estimate NLL three times, noting that the standard deviation over these three estimates is no larger than 0.12 nats. We report the average NLL values over either two or three different initializations (in addition to the three NLL estimates for each trained model) for several choices of tempering and leapfrog steps in Table 1. A full accounting of the tests is given in the supplementary material. We also consider an HVAE scheme in which we allow to vary across layers of the flow and report the results.

From Table 1, we notice that generally increasing the inhomogeneity in the dynamics improves the test NLL values. For example, free tempering is the most successful tempering scheme, and varying the leapfrog step size across layers also improves results. We also notice that increasing the number of leapfrog steps does not always improve the performance, as provides the best results in free tempering schemes. We believe that the improvement in HVAE over the base VAE scheme can be attributed to a more expressive approximate posterior, as we can see that samples from the HVAE approximate posterior exhibit non-negligible covariance across dimensions. As in Salimans et al. [2015], we are also able to improve upon the base model by adding our time-inhomogeneous Hamiltonian dynamics on top, but in a simplified regime without referring to learned reverse kernels. Rezende and Mohamed [2015] report only lower bounds on the log-likelihood for NFs, which are indeed lower than our log-likelihood estimates, although they use a much larger number of variational parameters.

fixed across layers varied across layers
Free Fixed None Free Fixed None
N/A 83.32 83.17 N/A N/A N/A
83.09 83.26 83.68 83.01 82.94 83.35
82.97 83.26 83.40 82.62 82.87 83.25
82.78 83.56 83.82 82.62 83.09 82.94
82.93 83.18 83.33 82.83 82.85 82.93
Table 1: Estimated NLL values for HVAE on MNIST. The base VAE achieves an NLL of 83.20. A more detailed version of this table is included in the supplementary material.

5 Conclusion and Discussion

We have proposed a principled way to exploit Hamiltonian dynamics within stochastic variational inference. Contrary to previous methods Salimans et al. [2015], Wolf et al. [2016], our algorithm does not rely on learned reverse Markov kernels and benefits from the use of tempering ideas. Additionally, we can use the reparameterization trick to obtain unbiased estimators of gradients of the ELBO. The resulting HVAE can be interpreted as a target-driven normalizing flow which requires the evaluation of a few gradients of the log-likelihood associated to a single data point at each stochastic gradient step. However, the Jacobian computations required for the ELBO are trivial. In our experiments, the robustness brought about by the use of target-informed dynamics can reduce the number of parameters that must be trained and improve generalizability.

We note that, although we have fewer parameters to optimize, the memory cost of using HVAE and target-informed dynamics could become prohibitively large if the memory required to store evaluations of

is already extremely large. Evaluating these gradients is not a requirement of VAEs or standard normalizing flows. However, we have shown that in the case of a fairly large generative network we are still able to evaluate gradients and backpropagate through the layers of the flow. Further tests explicitly comparing HVAE with VAEs and normalizing flows in various memory regimes are required to determine in what cases one method should be used over the other.

There are numerous possible extensions of this work. Hamiltonian dynamics preserves the Hamiltonian and hence also the corresponding target distribution, but there exist other deterministic dynamics which leave the target distribution invariant but not the Hamiltonian. This includes the Nosé-Hoover thermostat. It is possible to directly use these dynamics instead of the Hamiltonian dynamics within the framework developed in subsection 2.3. In continuous-time, related ideas have appeared in physics Cuendet [2006], Procacci et al. [2006], Schöll-Paschinger and Dellago [2006]. This comes at the cost of more complicated Jacobian calculations. The ideas presented here could also be coupled with the methodology proposed in Heng et al. [2017] – we conjecture that this could reduce the variance of the estimator (3) by an order of magnitude.

Acknowledgments

Anthony L. Caterini is a Commonwealth Scholar, funded by the UK government.

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Appendix A Full List of Tests on MNIST

Table 2 and Table 3 display the list of test runs of HVAE on MNIST. The number of flow steps is denoted by . The total number of epochs varies in training because of early stopping. The ELBO and NLL estimates are generated using 1000 importance samples from the HVAE approximate posterior; this procedure is run 3 times for each seed. The average and standard deviation of these estimates over the three runs is displayed. Table 2 refers to HVAE tests in which was fixed across flow layers, whereas Table 3 refers to HVAE tests in which was allowed to vary across flow layers.

K Tempering Seed Total Epochs ELBO Estimate NLL Estimate [head to column names]mnist_fixed_eps.csv
   ()    ()
Table 2: List of tests of HVAE for fixed across flow layers
K Tempering Seed Total Epochs ELBO Estimate NLL Estimate [head to column names]mnist_varied_eps.csv
   ()    ()
Table 3: List of tests of HVAE for fixed across flow layers

Table 4 displays the list of test runs of the base VAE on MNIST. ELBO and NLL estimates are again generated by importance sampling, but this time from the learned VAE approximate posterior.

Seed Total Epochs ELBO Estimate NLL Estimate [head to column names]mnist_cnn.csv
   ()    ()
Table 4: List of tests of VAE