Energy-based models (EBMs) have a long history in statistics and machine learning(Dayan et al., 1995; Zhu et al., 1998; LeCun et al., 2006). EBMs score configurations of variables with an energy function, which induces a distribution on the variables in the form of a Gibbs distribution. Different choices of energy function recover well-known probabilistic models including Markov random fields (Kinderman and Snell, 1980)1986; Freund and Haussler, 1992; Hinton, 2002), and conditional random fields (Lafferty et al., 2001). However, this flexibility comes at the cost of challenging inference and learning: both sampling and density evaluation of EBMs are generally intractable, which hinders the applications of EBMs in practice.
Because of the intractability of general EBMs, practical implementations rely on approximate sampling procedures (e.g.
, Markov chain Monte Carlo (MCMC)) for inference. This creates a mismatch between the model and the approximate inference procedure, and can lead to suboptimal performance and unstable training when approximate samples are used in the training procedure.
Currently, most attempts to fix the mismatch lie in designing better sampling algorithms (e.g., Hamiltonian Monte Carlo (Neal, 2005), annealed importance sampling (Neal, 2001)) or exploiting variational techniques (Kim and Bengio, 2016; Dai et al., 2017, 2018) to reduce the inference approximation error.
Instead, we bridge the gap between the model and inference by directly treating the sampling procedure as the model of interest and optimizing the log-likelihood of the the sampling procedure. We call these models energy-inspired models (EIMs) because they incorporate a learned energy function while providing tractable, exact samples. This shift in perspective aligns the training and sampling procedure, leading to principled and consistent training and inference.
To accomplish this, we cast the sampling procedure as a latent variable model. This allows us to maximize variational lower bounds (Jordan et al., 1999; Blei et al., 2017) on the log-likelihood (c.f., Kingma and Welling (2013); Rezende et al. (2014)). To illustrate this, we develop and evaluate energy-inspired models based on truncated rejection sampling (Algorithm 1), self-normalized importance sampling (Algorithm 2), and Hamiltonian importance sampling (Algorithm 3). Interestingly, the model based on self-normalized importance sampling is closely related to ranking NCE (Jozefowicz et al., 2016; Ma and Collins, 2018), suggesting a principled objective for training the “noise” distribution.
Our second contribution is to show that EIMs provide a unifying conceptual framework to explain many advances in constructing tighter variational lower bounds for latent variable models (e.g., (Burda et al., 2015; Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2017; Yin and Zhou, 2018; Molchanov et al., 2018; Sobolev and Vetrov, 2018)). Previously, each bound required a separate derivation and evaluation, and their relationship was unclear. We show that these bounds can be viewed as specific instances of auxiliary variable variational inference (Agakov and Barber, 2004; Salimans et al., 2015; Ranganath et al., 2016; Maaløe et al., 2016) with different EIMs as the variational family. Based on general results for auxiliary latent variables, this immediately gives rise to a variational lower bound with a characterization of the tightness of the bound. Furthermore, this unified view highlights the implicit (potentially suboptimal) choices made and exposes the reusable components that can be combined to form novel variational lower bounds. Concurrently, Domke and Sheldon (2019) note a similar connection, however, their focus is on the use of the variational distribution for posterior inference.
In summary, our contributions are:
The construction of a tractable class of energy-inspired models (EIMs), which lead to consistent learning and inference. To illustrate this, we build models with truncated rejection sampling, self-normalized importance sampling, and Hamiltonian importance sampling and evaluate them on synthetic and real-world tasks. These models can be fit by maximizing a tractable lower bound on their log-likelihood.
We show that EIMs with auxiliary variable variational inference provide a unifying framework for understanding recent tighter variational lower bounds, simplifying their analysis and exposing potentially sub-optimal design choices.
In this work, we consider learned probabilistic models of data . Energy-based models (LeCun et al., 2006) define in terms of an energy function
where is a tractable “prior” distribution and is a generally intractable partition function. To fit the model, many approximate methods have been developed (e.g., pseudo log-likelihood (Besag, 1975)2002; Tieleman, 2008), score matching estimator (Hyvärinen, 2005)
, minimum probability flow(Sohl-Dickstein et al., 2011), noise contrastive estimation (Gutmann and Hyvärinen, 2010)) to bypass the calculation of the partition function. Empirically, previous work has found that convolutional architectures that score images (i.e., map to a real number) tend to have strong inductive biases that match natural data (Du and Mordatch, 2019). These networks are a natural fit for energy-based models. Because drawing exact samples from these models is intractable, samples are typically approximated by Monte Carlo schemes, for example, Hamiltonian Monte Carlo (Neal and others, 2011).
Alternatively, latent variables allow us to construct complex distributions by defining the likelihood in terms of tractable components and . While marginalizing is generally intractable, we can instead optimize a tractable lower bound on using the identity
where is a variational distribution and the positive term can be omitted to form a lower bound commonly referred to as the evidence lower bound (ELBO) (Jordan et al., 1999; Blei et al., 2017). The tightness of the bound is controlled by how accurately models , so limited expressivity in the variational family can negatively impact the learned model.
3 Energy-Inspired Models
Instead of viewing the sampling procedure as drawing approximate samples from the energy-based models, we treat the sampling procedure as the model of interest. We represent the randomness in the sampler as latent variables, and we obtain a tractable lower bound on the marginal likelihood using the ELBO. Explicitly, if represents the randomness in the sampler and is the generative process, then
where is a variational distribution that can be optimized to tighten the bound. In this section, we explore concrete instantiations of models in this paradigm: one based on truncated rejection sampling (TRS), one based on self-normalized importance sampling (SNIS), and another based on Hamiltonian importance sampling (HIS) Neal (2005).
3.1 Truncated Rejection Sampling (TRS)
Consider the truncated rejection sampling process (Algorithm 1) used in (Bauer and Mnih, 2018), where we sequentially draw a sample from and accept it with probability . To ensure that the process ends, if we have not accepted a sample after steps, then we return .
In this case, , so we need to construct a variational distribution . The optimal is , which motivates choosing a similarly structured variational distribution. It is straightforward to see that where is generally intractable. So, we choose , where is a learnable variational parameter. Then, we sample and as in the generative process. This results in a simple variational bound
The TRS generative process is the same process as the Learned Accept/Reject Sampling (LARS) model (Bauer and Mnih, 2018). The key difference is the training procedure. LARS tries to directly estimate the gradient of the log likelihood. Without truncation, such a process is attractive because unbiased gradients of its log likelihood can easily be computed without knowing the normalizing constant. Unfortunately, after truncating the process, we require estimating a normalizing constant. In practice, Bauer and Mnih (2018) estimate the normalizing constant using samples during training and samples during evaluation. Even so, LARS requires additional implementation tricks (e.g., evaluating the target density, using an exponential moving average to estimate the normalizing constant) to ensure successful training, which complicate the implementation and analysis of the algorithm. On the other hand, we optimize a tractable log likelihood lower bound. As a result, no implementation tricks are necessary.
3.2 Self-Normalized Importance Sampling (SNIS)
Consider the sampling process defined by self-normalized importance sampling. That is, first sampling a set of candidate s from a proposal distribution , and then sampling from the empirical distribution composed of atoms located at each and weighted proportionally to (Algorithm 2). In this case, the latent variables are the locations of the proposal samples , …, (abbreviated ) and the index of the selected sample, .
Explicitly, the model is defined by
with . We denote the density of the process by . Choosing in Eq. 2, yields
To summarize, can be sampled from exactly and has a tractable lower bound on its log-likelihood. For the same , we expect to outperform because it considers all candidate samples simultaneously instead of sequentially.
As , becomes proportional to . For finite , interpolates between the tractable proposal and the energy model . Furthermore, Equation 3 is closely connected with the ranking NCE loss (Jozefowicz et al., 2016; Ma and Collins, 2018), a popular objective for training energy-based models. In fact, if we consider as our noise distribution and set , then up to a constant (in ), we recover the ranking NCE loss using the notation from (Ma and Collins, 2018). The ranking NCE loss is motivated by the fact that it is a consistent objective for any when the true data distribution is in our model family. As a result, it is straightforward to adapt the consistency proof from (Ma and Collins, 2018) to our setting. Furthermore, our perspective gives a coherent objective for jointly learning the noise distribution and the energy function and shows that the ranking NCE loss can be viewed as a lower bound on the log likelihood of a well-specified model regardless of whether the true data distribution is in our model family. In addition, we can recover the recently proposed InfoNCE (Oord et al., 2018) bound on mutual information by using SNIS as the variational distribution in the classic variational bound by Barber and Agakov (2003) (see Appendix C for details).
To train the SNIS model, we perform stochastic gradient ascent on Eq. 3 with respect to the parameters of the proposal distribution and the energy function . When the data are continuous, reparameterization gradients can be used to estimate the gradients to the proposal distribution (Rezende et al., 2014; Kingma and Welling, 2013). When the data are discrete, score function gradient estimators such as REINFORCE (Williams, 1992) or relaxed gradient estimators such as the Gumbel-Softmax (Maddison et al., 2016; Jang et al., 2016) can be used.
3.3 Hamiltonian importance sampling (HIS)
Simple importance sampling scales poorly with dimensionality, so it is natural to consider more complex samplers with better scaling properties. We evaluated models based on Hamiltonian importance sampling (HIS) Neal (2005), which evolve an initial sample under deterministic, discretized Hamiltonian dynamics with a learned energy function. In particular, we sample initial location and momentum variables, and then transition the candidate sample and momentum with leap frog integration steps, changing the temperature at each step (Algorithm 3). While the quality of samples from SNIS are limited by the samples initially produced by the proposal, a model based on HIS updates the positions of the samples directly, potentially allowing for more expressive power. Intuitively, the proposal provides a coarse starting sample which is further refined by gradient optimization on the energy function. When the proposal is already quite strong, drawing additional samples as in SNIS may be advantageous.
In practice, we parameterize the temperature schedule such that . This ensures that the deterministic invertible transform from to has a Jacobian determinant of (i.e., ). Applying Eq. 2 yields a tractable variational objective
We jointly optimize and the variational parameters with stochastic gradient ascent. Goyal et al. (2017) propose a similar approach that generates a multi-step trajectory via a learned transition operator.
We evaluated the proposed models on a set of synthetic datasets, binarized MNIST(LeCun, 1998) and Fashion MNIST (Xiao et al., 2017), and continuous MINST, Fashion MNIST, and CelebA (Liu et al., 2015). See Appendix D for details on the datasets, network architectures, and other implementation details. To provide a competitive baseline, we use the recently developed Learned Accept/Reject Sampling (LARS) model (Bauer and Mnih, 2018).
4.1 Synthetic data
As a preliminary experiment, we evaluated the methods on modeling synthetic densities: a mixture of 9 equally-weighted Gaussian densities, a checkerboard density with uniform mass distributed in 8 squares, and two concentric rings (Fig. 1 and Appendix Fig. 2 for visualizations). For all methods, we used a unimodal standard Gaussian as the proposal distribution (see Appendix D for further details).
TRS, SNIS, and LARS perform comparably on the Nine Gaussians and Checkerboard datasets. On the Two Rings datasets, despite tuning hyperparameters, we were unable to make LARS learn the density.
On these simple problems, the target density lies in the high probability region of the proposal density, so TRS, SNIS, and LARS only have to reweight the proposal samples appropriately. In high-dimensional problems when the proposal density is mismatched from the target density, however, we expect HIS to outperform TRS, SNIS, and LARS. To test this we ran each algorithm on the Nine Gaussians problem with a Gaussian proposal of mean 0 and variance 0.1 so that there was a significant mismatch in support between the target and proposal densities. The results in the rightmost panel ofFig. 1 show that HIS was almost unaffected by the change in proposal while the other algorithms suffered considerably.
4.2 Binarized MNIST and Fashion MNIST
|Method||Static MNIST||Dynamic MNIST||Fashion MNIST|
|VAE w/ Gaussian prior|
|VAE w/ TRS prior|
|VAE w/ SNIS prior|
|VAE w/ HIS prior|
|VAE w/ LARS prior|
ConvHVAE w/ Gaussian prior
|ConvHVAE w/ TRS prior|
|ConvHVAE w/ SNIS prior|
|ConvHVAE w/ HIS prior|
|ConvHVAE w/LARS prior|
|SNIS w/ VAE proposal|
|SNIS w/ ConvHVAE proposal|
|LARS w/ VAE proposal||—||—|
|LARS w/ small VAE proposal|
|SNIS w/ small VAE proposal|
|HIS w/ small VAE proposal|
|LARS w/ VAE proposal|
|SNIS w/ VAE proposal|
|HIS w/ VAE proposal|
Next, we evaluated the models on binarized MNIST and Fashion MNIST. MNIST digits can be either statically or dynamically binarized — for the statically binarized dataset we used the binarization from (Salakhutdinov and Murray, 2008)
, and for the dynamically binarized dataset we sampled images from Bernoulli distributions with probabilities equal to the continuous values of the images in the original MNIST dataset. We dynamically binarize the Fashion MNIST dataset in a similar manner.
First, we used the models as the prior distribution in a Bernoulli observation likelihood VAE. We summarize log-likelihood lower bounds on the test set in Table 1 (referred to as VAE w/ method prior). SNIS outperformed LARS on static MNIST and dynamic MNIST even though it used only 1024 samples for training and evaluation, whereas LARS used 1024 samples during training and samples for evaluation. As expected due to the similarity between methods, TRS performed comparably to LARS. On all datasets, HIS either outperformed or performed comparably to SNIS. We increased and for SNIS and HIS, respectively, and find that performance improves at the cost of additional computation (Appendix Fig. 3). We also used the models as the prior distribution of a convolutional heiarachical VAE (ConvHVAE, following the architecture in (Bauer and Mnih, 2018)). In this case, SNIS outperformed all methods.
Then, we used a VAE as the proposal distribution to SNIS. A limitation of the HIS model is that it requires continuous data, so it cannot be used in this way on the binarized datasets. Initially, we thought that an unbiased, low-variance estimator could be constructed similarly to VIMCO (Mnih and Rezende, 2016), however, this estimator still had high variance. Next, we used the Gumbel Straight-Through estimator (Jang et al., 2016) to estimate gradients through the discrete samples proposed by the VAE, but found that method performed worse than ignoring those gradients altogether. We suspect that this may be due to bias in the gradients. Thus, for the SNIS model with VAE proposal, we report results on training runs which ignore those gradients. Future work will investigate low-variance, unbiased gradient estimators. In this case, SNIS again outperforms LARS, however, the performance is worse than using SNIS as a prior distribution. Finally, we used a ConvHVAE as the proposal for SNIS and saw performance improvements over both the vanilla ConvHVAE and SNIS with a VAE proposal, demonstrating that our modeling improvements are complementary to improving the proposal distribution.
4.3 Continuous MNIST, Fashion MNIST, and CelebA
Finally, we evaluated SNIS and HIS on continuous versions of MNIST, Fashion MNIST, and CelebA (64x64). We use the same preprocessing as in (Dinh et al., 2016). Briefly, we dequantize pixel values by adding uniform noise, rescale them to
, and then transform the rescaled pixel values into logit space by, where . When we calculate log-likelihoods, we take into account this change of variables.
We speculated that when the proposal is already strong, drawing additional samples as in SNIS may be better than HIS. To test this, we experimented with a smaller VAE as the proposal distribution. As we expected, HIS outperformed SNIS when the proposal was weaker, especially on the more complex datasets, as shown in Table 2.
5 Variational Inference with EIMs
To provide a tractable lower bound on the log-likelihood of EIMs, we used the ELBO (Eq. 1). More generally, this variational lower bound has been used to optimize deep generative models with latent variables following the influential work by Kingma and Welling (2013); Rezende et al. (2014), and models optimized with this bound have been successfully used to model data such as natural images (Rezende and Mohamed, 2015; Kingma et al., 2016; Chen et al., 2016; Gulrajani et al., 2016), speech and music time-series (Chung et al., 2015; Fraccaro et al., 2016; Krishnan et al., 2015), and video (Babaeizadeh et al., 2017; Ha and Schmidhuber, 2018; Denton and Fergus, 2018). Due to the usefulness of such a bound, there has been an intense effort to provide improved bounds (Burda et al., 2015; Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2017; Yin and Zhou, 2018; Molchanov et al., 2018; Sobolev and Vetrov, 2018). The tightness of the ELBO is determined by the expressiveness of the variational family (Zellner, 1988), so it is natural to consider using flexible EIMs as the variational family. As we explain, EIMs provide a conceptual framework to understand many of the recent improvements in variational lower bounds.
In particular, suppose we use a conditional EIM as the variational family (i.e., is the marginalized sampling process). Then, we can use the ELBO lower bound on (Eq. 1), however, the density of the EIM is intractable. Agakov and Barber (2004); Salimans et al. (2015); Ranganath et al. (2016); Maaløe et al. (2016) develop an auxiliary variable variational bound
where is a variational distribution meant to model , and the identity follows from the fact that Similar to Eq. 1, Eq. 4 shows the gap introduced by using to deal with the intractability of . We can form a lower bound on the original ELBO and thus a lower bound on the log marginal by omitting the positive term. This provides a tractable lower bound on the log-likelihood using flexible EIMs as the variational family and precisely characterizes the bound gap as the sum of terms in Eq. 1 and Eq. 4. For different choices of EIM, this bound recovers many of the recently proposed variational lower bounds.
Furthermore, the bound in Eq. 4 is closely related to partition function estimation because
is an unbiased estimator ofwhen . To first order, the bound gap is related to the variance of this partition function estimator (e.g., (Maddison et al., 2017)), which motivates sampling algorithms used in lower variance partition function estimators such as SMC (Doucet et al., 2001) and AIS (Neal, 2001).
5.1 Importance Weighted Auto-encoders (IWAE)
To tighten the ELBO without explicitly expanding the variational family, Burda et al. (2015)
introduced the importance weighted autoencoder (IWAE) bound,
The IWAE bound reduces to the ELBO when , is non-decreasing as increases, and converges to as under mild conditions (Burda et al., 2015). Bachman and Precup (2015) introduced the idea of viewing IWAE as auxiliary variable variational inference and Naesseth et al. (2018); Cremer et al. (2017); Domke and Sheldon (2018) formalized the notion.
Consider the variational family defined by the EIM based on SNIS (Algorithm 2). We use a learned, tractable distribution as the proposal and set motivated by the fact that is the optimal variational distribution. Similar to the variational distribution used in Section 3.2, setting
Compared to the optimal choice, Eq. 6 makes the approximation which ignores the influence of on and the fact that are not independent given . A simple extension could be to learn a factored variational distribution conditional on : . Learning such an could improve the tightness of the bound, and we leave exploring this to future work.
5.2 Semi-implicit variational inference
As a way of increasing the flexibility of the variational family, Yin and Zhou (2018) introduce the idea of semi-implicit variational families. That is they define an implicit distribution
by transforming a random variablewith a differentiable deterministic transformation (i.e., ). However, Sobolev and Vetrov (2018) keenly note that can be equivalently written as with two explicit distributions. As a result, semi-implicit variational inference is simply auxiliary variable variational inference by another name.
Additionally, Yin and Zhou (2018) provide a multi-sample lower bound on the log likelihood which is generally applicable to auxiliary variable variational inference.
We can interpret this bound as using an EIM for in Eq. 4. Generally, if we introduce additional auxiliary random variables into , we can tractably bound the objective
where is a variational distribution. Analogously to the previous section, we set as an EIM based on the self-normalized importance sampling process with proposal and . If we choose
with , then Eq. 8 recovers the bound in (Yin and Zhou, 2018) (see Appendix B for details). In a similar manner, we can continue to recursively augment the variational distribution (i.e., add auxiliary latent variables to ).
This view reveals that the multi-sample bound from (Yin and Zhou, 2018) is simply one approach to choosing a flexible variational . Alternatively, Ranganath et al. (2016) use a learned variational . It is unclear when drawing additional samples is preferable to learning a more complex variational distribution. Furthermore, the two approaches can be combined by using a learned proposal instead of , which results in a bound described in (Sobolev and Vetrov, 2018).
5.3 Additional Bounds
Finally, we can also use the self-normalized importance sampling procedure to extend a proposal family to a larger family (instead of solely extending ) (Sobolev and Vetrov, 2018). Self-normalized importance sampling is a particular choice of taking a proposal distribution and moving it closer to a target. Hamiltonian Monte Carlo (Neal and others, 2011) is another choice which can also be embedded in this framework as done by (Salimans et al., 2015; Caterini et al., 2018). Similarly, SMC can be used as a sampling procedure in an EIM and when used as the variational family, it succinctly derives variational SMC (Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2017) without any instance specific tricks. In this way, more elaborate variational bounds can be constructed by specific choices of EIMs without additional derivation.
We proposed a flexible, yet tractable family of distributions by treating the approximate sampling procedure of energy-based models as the model of interest, referring to them as energy-inspired models. The proposed EIMs bridge the gap between learning and inference in EBMs. We explore three instantiations of EIMs induced by truncated rejection sampling, self-normalized importance sampling, and Hamiltonian importance sampling and we demonstrate comparably or stronger performance than recently proposed generative models. The results presented in this paper use simple architectures on relatively small datasets. Future work will scale up both the architectures and size of the datasets.
Interestingly, as a by-product, exploiting the EIMs to define the variational family provides a unifying framework for recent improvements in variational bounds, which simplifies existing derivations, reveals potentially suboptimal choices, and suggests ways to form novel bounds.
We thank Ben Poole, Abhishek Kumar, and Diederick Kingma for helpful comments. We thank Matthias Bauer for answering implementation questions about LARS.
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Appendix A IWAE bound as AVVI with an EIM
Let be an EIM based on SNIS with proposal and energy function and be
which is the IWAE bound.
Appendix B Semi-implicit Variational Inference Bound
which is equivalent to the multi-sample bound from [Yin and Zhou, 2018].
Appendix C Connection with CPC
Starting from the well-known variational bound on mutual information due to Barber and Agakov 
for a variational distribution , we can use the self-normalized importance sampling distribution and choose the proposal to be (i.e., ). Applying the bound in Eq. 3, we have
This recovers the CPC bound and proves that it is indeed a lower bound on mutual information whereas the heuristic justification in the original paper relied on unnecessary approximations.
Appendix D Implementation Details
d.1 Synthetic data
All methods used a fixed 2-D proposal distribution and a learned acceptance/energy function
parameterized by a neural network with 2 hidden layers of size 20 and tanh activations. For SNIS and LARS, the number of proposal samples drawn,, was set to 1024 and for HIS . We used batch sizes of 128 and ADAM [Kingma and Ba, 2014] with a learning rate of to fit the models. For evaluation, we report the IWAE bound with 1000 samples for HIS and SNIS. For LARS there is no equivalent to the IWAE bound, so we instead estimate the normalizing constant with 1000 samples.
The nine Gaussians density is a mixture of nine equally-weighted 2D Gaussians with variance and means . The checkerboard density places equal mass on the squares and, . The two rings density is defined as
d.2 Binarized MNIST and Fashion MNIST
We chose hyperparameters to match the MNIST experiments in Bauer and Mnih . Specifically, we parameterized the energy function by a neural network with two hidden layers of size 100 and tanh activations, and parameterized the VAE observation model by neural networks with two layers of 300 units and tanh activations. The latent spaces of the VAEs were 50-dimensional, SNIS’s was set to 1024, and HIS’s was set to . We also linearly annealed the weight of the KL term in the ELBO from 0 to 1 over the first steps and dropped the learning rate from to on step . All models were trained with ADAM [Kingma and Ba, 2014].
d.3 Continuous MNIST, Fashion MNIST, and CelebA
For the small VAE, we parameterized the VAE observation model neural networks with a single layer of 20 units and tanh activations. The latent spaces of the small VAEs were 10-dimensional. In these experiments, SNIS’s was set to 128, and HIS’s was set to . We also dropped the learning rate from to on step . All models were trained with ADAM [Kingma and Ba, 2014].