Black Box Variational Inference

12/31/2013 ∙ by Rajesh Ranganath, et al. ∙ 0

Variational inference has become a widely used method to approximate posteriors in complex latent variables models. However, deriving a variational inference algorithm generally requires significant model-specific analysis, and these efforts can hinder and deter us from quickly developing and exploring a variety of models for a problem at hand. In this paper, we present a "black box" variational inference algorithm, one that can be quickly applied to many models with little additional derivation. Our method is based on a stochastic optimization of the variational objective where the noisy gradient is computed from Monte Carlo samples from the variational distribution. We develop a number of methods to reduce the variance of the gradient, always maintaining the criterion that we want to avoid difficult model-based derivations. We evaluate our method against the corresponding black box sampling based methods. We find that our method reaches better predictive likelihoods much faster than sampling methods. Finally, we demonstrate that Black Box Variational Inference lets us easily explore a wide space of models by quickly constructing and evaluating several models of longitudinal healthcare data.

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1 Introduction

Probabilistic models with latent variables have become a mainstay in modern machine learning applications. With latent variables models, we posit a rich latent structure that governs our observations, infer that structure from large data sets, and use our inferences to summarize observations, draw conclusions about current data, and make predictions about new data. Central to working with latent variable models is the problem of computing the posterior distribution of the latent structure. For many interesting models, computing the posterior exactly is intractable: practitioners must resort to approximate methods.

One of the most widely used methods for approximate posterior estimation is variational inference 

(Wainwright and Jordan, 2008, Jordan et al., 1999)

. Variational inference tries to find the member of a family of simple probability distributions that is closest (in KL divergence) to the true posterior distribution.

For a specific class of models, those where the conditional distributions have a convenient form (and where a convenient variational family exists), this optimization can be carried out with a closed-form coordinate ascent algorithm (Ghahramani and Beal, 2001). For generic models and arbitrary variational families, however, there is no closed-form solution: computing the required expectations becomes intractable. In these settings, practitioners have resorted to model-specific algorithms (Jaakkola and Jordan, 1996, Blei and Lafferty, 2007, Braun and McAuliffe, 2007) or generic algorithms that require model specific computations (Knowles and Minka, 2011, Wang and Blei, 2013, Paisley et al., 2012).

Deriving these algorithms on a model-by-model basis is tedious work. This hinders us from rapidly exploring modeling assumptions when solving applied problems, and it makes variational methods on complicated distributions impractical for many practitioners. Our goal in this paper is to develop a “black box” variational inference algorithm, a method that can be quickly applied to almost any model and with little effort. Our method allows practitioners to quickly design, apply, and revise models of their data, without painstaking derivations each time they want to adjust the model.

Variational inference methods frame a posterior estimation problem as an optimization problem, where the parameters to be optimized adjust a variational “proxy” distribution to be similar to the true posterior. Our method rewrites the gradient of that objective as the expectation of an easy-to-implement function of the latent and observed variables, where the expectation is taken with respect to the variational distribution; and we optimize that objective by sampling from the variational distribution, evaluating the function , and forming the corresponding Monte Carlo estimate of the gradient. We then use these stochastic gradients in a stochastic optimization algorithm to optimize the variational parameters.

From the practitioner’s perspective, this method requires only that he or she write functions to evaluate the model log-likelihood. The remaining calculations (sampling from the variational distribution and evaluating the Monte Carlo estimate) are easily put into a library to be shared across models, which means our method can be quickly applied to new modeling settings.

We will show that reducing the variance of the gradient estimator is essential to the fast convergence of our algorithm. We develop several strategies for controlling the variance. The first is based on Rao-Blackwellization (Casella and Robert, 1996), which exploits the factorization of the variational distribution. The second is based on control variates  (Ross, 2002, Paisley et al., 2012), using the log probability of the variational distribution. We emphasize that these variance reduction methods preserve our goal of black box inference because they do not require computations specific to the model.

Finally, we show how to use recent innovations in variational inference and stochastic optimization to scale up and speed up our algorithm. First, we use adaptive learning rates (Duchi et al., 2011) to set the step size in the stochastic optimization. Second, we develop generic stochastic variational inference (Hoffman et al., 2013), where we additionally subsample from the data to more cheaply compute noisy gradients. This innovates on the algorithm of Hoffman et al. (2013), which requires closed form coordinate updates to compute noisy natural gradients.

We demonstrate our method in two ways. First, we compare our method against Metropolis-Hastings-in-Gibbs (Bishop, 2006), a sampling based technique that requires similar effort on the part of the practitioner. We find our method reaches better predictive likelihoods much faster than sampling methods. Second, we use our method to quickly build and evaluate several models of longitudinal patient data. This demonstrates the ease with which we can now consider models generally outside the realm of variational methods.

Related work.

There have been several lines of work that use sampling methods to approximate gradients in variational inference. Wingate and Weber (2013) have independently considered a similar procedure to ours, where the gradient is construed as an expectation and the KL is optimized with stochastic optimization. They too include a term to reduce the variance, but do not describe how to set it. We further innovate on their approach with Rao-Blackwellization, specified control variates, adaptive learning rates, and data subsampling. Salimans and Knowles (2012)

provide a framework based on stochastic linear regression. Unlike our approach, their method does not generalize to arbitrary approximating families and requires the inversion of a large matrix that becomes impractical in high dimensional settings.

Kingma and Welling (2013) provide an alternative method for variational inference through a reparameterization of the variational distributions. In constrast to our approach, their algorithm is limited to only continuous latent variables. Carbonetto et al. (2009)

present a stochastic optimization scheme for moment estimation based on the specific form of the variational objective when both the model and the approximating family are in the same exponential family. This differs from our more general modeling setting where latent variables may be outside of the exponential family. Finally,

Paisley et al. (2012) use Monte Carlo gradients for difficult terms in the variational objective and also use control variates to reduce variance. However, theirs is not a black-box method. Both the objective function and control variates they propose require model-specific derivations.

2 Black Box Variational Inference

Variational inference transforms the problem of approximating a conditional distribution into an optimization problem (Jordan et al., 1999, Bishop, 2006, Wainwright and Jordan, 2008). The idea is to posit a simple family of distributions over the latent variables and find the member of the family that is closest in KL divergence to the conditional distribution.

In a probabilistic model, let be observations, be latent variables, and the free parameters of a variational distribution . Our goal is to approximate with a setting of . In variational inference we optimize the Evidence Lower BOund (ELBO),

(1)

Maximizing the ELBO is equivalent to minimizing the KL divergence (Jordan et al., 1999, Bishop, 2006). Intuitively, the first term rewards variational distributions that place high mass on configurations of the latent variables that also explain the observations; the second term rewards variational distributions that are entropic, i.e., that maximize uncertainty by spreading their mass on many configurations.

Practitioners derive variational algorithms to maximize the ELBO over the variational parameters by expanding the expectation in Eq. 1 and then computing gradients to use in an optimization procedure. Closed form coordinate-ascent updates are available for conditionally conjugate exponential family models  (Ghahramani and Beal, 2001), where the distribution of each latent variable given its Markov blanket falls in the same family as the prior, for a small set of variational families. However, these updates require analytic computation of various expectations for each new model, a problem which is exacerbated when the variational family falls outside this small set. This leads to tedious bookkeeping and overhead for developing new models.

We will instead use stochastic optimization to maximize the ELBO. In stochastic optimization, we maximize a function using noisy estimates of its gradient (Robbins and Monro, 1951, Kushner and Yin, 1997, Bottou and LeCun, 2004)

. We will form the derivative of the objective as an expectation with respect to the variational approximation and then sample from the variational approximation to get noisy but unbiased gradients, which we use to update our parameters. For each sample, our noisy gradient requires evaluating the joint distribution of the observed and sampled variables, the variational distribution, and the gradient of the log of the variational distribution. This is a black box method in that the gradient of the log of the variational distribution and sampling method can be derived once for each type of variational distribution and reused for many models and applications.

Stochastic optimization.

We first review stochastic optimization. Let be a function to be maximized and

be the realization of a random variable

whose expectation is the gradient of . Finally, let be the learning rate. Stochastic optimization updates at the th iteration with

This converges to a maximum of when the learning rate schedule follows the Robbins-Monro conditions,

Because of its simplicity, stochastic optimization is widely used in statistics and machine learning.

A noisy gradient of the ELBO.

To optimize the ELBO with stochastic optimization, we need to develop an unbiased estimator of its gradient which can be computed from samples from the variational posterior. To do this, we write the gradient of the ELBO (Eq. 

1) as an expectation with respect to the variational distribution,

(2)

The derivation of Eq. 2 can be found in the appendix. Note that in statistics the gradient of the log of a probability distribution is called the score function (Cox and Hinkley, 1979).

With this Equation in hand, we compute noisy unbiased gradients of the ELBO with Monte Carlo samples from the variational distribution,

(3)

With Eq. 3, we can use stochastic optimization to optimize the ELBO.

The basic algorithm is summarized in Algorithm 1. We emphasize that the score function and sampling algorithms depend only on the variational distribution, not the underlying model. Thus we can easily build up a collection of these functions for various variational approximations and reuse them in a package for a variety of models. Further we did not make any assumptions about the form of the model, only that the practitioner can compute the log of the joint . This algorithm significantly reduces the effort needed to implement variational inference in a wide variety of models.

  Input: data , joint distribution , mean field variational family .
  Initialize randomly, .
  repeat
     // Draw samples from
     for  todo
        
     end for
      = th value of a Robbins Monro sequence (Eq. 2)
      =
     
  until change of is less than 0.01.
Algorithm 1 Black Box Variational Inference

3 Controlling the Variance

We can use Algorithm 1 to maximize the ELBO, but the variance of the estimator of the gradient (under the Monte Carlo estimate in Eq. 3) can be too large to be useful. In practice, the high variance gradients would require very small steps which would lead to slow convergence. We now show how to reduce this variance in two ways, via Rao-Blackwellization and easy-to-implement control variates. We exploit the structure of our problem to use these methods in a way that requires no model-specific derivations, which preserves our goal of black-box variational inference.

3.1 Rao-Blackwellization

Rao-Blackwellization (Casella and Robert, 1996) reduces the variance of a random variable by replacing it with its conditional expectation with respect to a subset of the variables. Note that the conditional expectation of a random variable is a random variable with respect to the conditioning set. This generally requires analytically computing problem-specific integrals. Here we show how to Rao-Blackwellize the estimator for each component of the gradient without needing to compute model-specific integrals.

In the simplest setting, Rao-Blackwellization replaces a function of two variables with its conditional expectation. Consider two random variables, and , and a function . Our goal is to compute its expectation with respect to the joint distribution of and .

Define , and note that . This means that can be used in place of in a Monte Carlo approximation of . The variance of is

This means that is a lower variance estimator than .

We return to the problem of estimating the gradient of . Suppose there are latent variables and we are using the mean-field variational family, where each random variable is independent and governed by its own variational distribution,

(4)

where are the variational parameters characterizing the member of the variational family we seek. Consider the th component of the gradient. Let be the distribution of variables in the model that depend on the th variable, i.e., the Markov blanket of ; and let be the terms in the joint that depend on those variables. We can write the gradient with respect to as an iterated conditional expectation which simplifies to

(5)

The derivation of this expression is in the supplement. This equation says that Rao-Blackwellized estimators can be computed for each component of the gradient without needing to compute model-specific conditional expectations.

Finally, we construct a Monte Carlo estimator for the gradient of using samples from the variational distribution,

(6)

This Rao-Blackwellized estimator for each component of the gradient has lower variance. In our empirical study, Figure 2, we plot the variance of this estimator along with that of Eq. 3.

3.2 Control Variates

As we saw above, variance reduction methods work by replacing the function whose expectation is being approximated by Monte Carlo with another function that has the same expectation but smaller variance. That is, to estimate via Monte Carlo we compute the empirical average of where is chosen so and .

A control variate (Ross, 2002) is a family of functions with equivalent expectation. Consider a function , which has a finite first moment, and a scalar . Define to be

(7)

This is a family of functions, indexed by , and note that as required. Given a particular function , we can choose to minimize the variance of .

First we note that variance of can be written as

This equation implies that good control variates have high covariance with the function whose expectation is being computed.

Taking the derivative of with respect to and setting it equal to zero gives us the value of that minimizes the variance,

With Monte Carlo estimates from the distribution, which we are collecting anyway to compute , we can estimate with the ratio of the empirical covariance and variance.

We now apply this method to Black Box Variational Inference. To maintain the generic nature of the algorithm, we want to choose a control variate that only depends on the variational distribution and for which we can easily compute its expectation. Meeting these criteria, we choose to be the score function of the variational approximation, , which always has expectation zero. (See Eq. 14 in the appendix.)

With this control variate, we have a new Monte Carlo method to compute the Rao-Blackwellized noisy gradients of the ELBO. For the th component of the gradient, the function whose expectation is being estimated and its control variate, and respectively are

(8)

The estimate for the optimal choice for the scaling is given by summing over the covariance and variance for each of the dimensions of . Letting the th dimension of and be and respectively. The optimal scaling for the gradient of the ELBO is given by

(9)

This gives us the following Monte Carlo method to compute noisy gradients using samples

(10)

Again note that we define the control variates on a per-component basis. This estimator uses both Rao-Blackwellization and control variates. We show in the empirical study that this generic control variate further reduces the variance of the estimator.

3.3 Black Box Variational Inference (II)

Putting together the noisy gradient, Rao-Blackwellization, and control variates, we present Black Box Variational Inference (II). It takes samples from the variational approximation to compute noisy gradients as in Eq. 10. These noisy gradients are then used in a stochastic optimization procedure to maximize the ELBO.

We summarize the procedure in Algorithm 2. Note that for simplicity of presentation, this algorithm stores all the samples. We can remove this memory requirement with a small modification, computing the terms on small set of examples and computing the required averages online.

Algorithm 2 is easily used on many models. It only requires samples from the variational distribution, computations about the variational distribution, and easy computations about the model.

  Input: data , joint distribution , mean field variational family .
  Initialize randomly, .
  repeat
     // Draw samples from the variational approximation
     for  todo
        
     end for
     for  todo
        for  todo
           
           
        end for
        
        
     end for
      = th value of a Robbins Monro sequence
     
     
  until change of is less than 0.01.
Algorithm 2 Black Box Variational Inference (II)

4 Extensions

We extend the main algorithm in two ways. First, we address the difficulty of setting the step size schedule. Second, we address scalability by subsampling observations.

4.1 AdaGrad

One challenge with stochastic optimization techniques is setting the learning rate. Intuitively, we would like the learning rate to be small when the variance of the gradient is large and vice-versa. Additionally, in problems like ours that have different scales111Probability distributions have many parameterizations., the learning rate needs to be set small enough to handle the smallest scale. To address this issue, we use the AdaGrad algorithm  (Duchi et al., 2011). Let be a matrix containing the sum across the first iterations of the outer products of the gradient. AdaGrad defines a per component learning rate as

(11)

This is a per-component learning rate since has the same dimension as the gradient. Note that since AdaGrad only uses the diagonal of , those are the only elements we need to compute. AdaGrad captures noise and varying length scales through the square of the noisy gradient and reduces the number of parameters to our algorithm from the standard two parameter Robbins-Monro learning rate.

4.2 Stochastic Inference in Hierarchical Bayesian Models

Stochastic optimization has also been used to scale variational inference in hierarchical Bayesian models to massive data (Hoffman et al., 2013). The basic idea is to subsample observations to compute noisy gradients. We can use a similar idea to scale our method.

In a hierarchical Bayesian model, we have a hyperparameter

, global latent variables , local latent variables , and observations having the log joint distribution

(12)

This is the same definition as in Hoffman et al. (2013), but they place further restrictions on the forms of the distributions and the complete conditionals. Under the mean field approximating family, applying Eq. 10 to construct noisy gradients of the ELBO would require iterating over every datapoint. Instead we can compute noisy gradients using a sampled observation and samples from the variational distribution. The derivation along with variance reductions can be found in the supplement.

5 Empirical Study

We use Black Box Variational Inference to quickly construct and evaluate several models on longitudinal medical data. We demonstrate the effectiveness of our variance reduction methods and compare the speed and predictive likelihood of our algorithm to sampling based methods. We evaluate the various models using predictive likelihood and demonstrate the ease at which several models can be explored.

5.1 Longitudinal Medical Data

Our data consist of longitudinal data from 976 patients (803 train + 173 test) from a clinic at New York Presbyterian hospital who have been diagnosed with chronic kidney disease. These patients visited the clinic a total of 33K times. During each visit, a subset of 17 measurements (labs) were measured.

The data are observational and consist of measurements (lab values) taken at the doctor’s discretion when the patient is at a checkup. This means both that the labs at each time step are sparse and that the time between patient visits are highly irregular. The labs values are all positive as the labs measure the amount of a particular quantity such as sodium concentration in the blood.

Our modeling goal is to come up with a low dimensional summarization of patients’ labs at each of their visits. From this, we aim to to find latent factors that summarize each visit as positive random variables. As in medical data applications, we want our factors to be latent indicators of patient health.

We evaluate our model using predictive likelihood. To compute predictive likelihoods, we need an approximate posterior on both the global parameters and the per visit parameters. We use the approximate posterior on the global parameters and calculate the approximate posterior on the local parameters on 75% of the data in the test set. We then calculate the predictive likelihood on the other 25% of the data in the validation set using Monte Carlo samples from the approximate posterior.

We initialize randomly and choose the variational families to be fully-factorized with gamma distributions for positive variables and normals for real valued variables. We use both the AdaGrad and doubly stochastic extensions on top of our base algorithm. We use 1,000 samples from the variational distribution and set the batch size at 25 in all our experiments.

5.2 Model

Figure 1: Comparison between Metropolis-Hastings within Gibbs and Black Box Variational Inference. In the x axis is time and in the y axis is the predictive likelihood of the test set. Black Box Variational Inference reaches better predictive likelihoods faster than Gibbs sampling. The Gibbs sampler’s progress slows considerably after 5 hours.

To meet our goals, we construct a Gamma-Normal time series model. We model our data using weights drawn from a Normal distribution and observations drawn from a Normal, allowing each factor to both positively and negative affect each lab while letting factors represent lab measurements. The generative process for this model with hyperparameters denoted with

is D raw Normal, an matrix For each patient : to Draw Normal

, a vector of

Define F or each visit : 1 to Draw GammaE Draw Normal, a vector of .

We set , , to be 1 and to be .01. In our model, GammaE is the expectation/variance parameterization of the (L-dimensional) gamma distribution. (The mapping between this parameterization and the more standard one can be found in the supplement.)

Black Box Variational Inference allows us to make use of non-standard parameterizations for distributions that are easier to reason about. This is an important observation, as the standard set of families used in variational inference tend to be fairly limited. In this case, the expectation parameterization of the gamma distribution allows the previous visit factors to define the expectation of the current visit factors. Finally, we emphasize that coordinate ascent variational inference and Gibbs sampling are not available for this algorithm because the required conditional distributions do not have closed form.

5.3 Sampling Methods

We compare Black Box Variational Inference to a standard sampling based technique, Metropolis-Hastings (Bishop, 2006), that also only needs the joint distribution.222Methods that involve a bit more work such as Hamiltonian Monte Carlo could work in this settings, but as our technique only requires the joint distribution and could benefit from added analysis used in more complex methods, we compare against a similar methods.

Metropolis-Hastings works by sampling from a proposal distribution and accepting or rejecting the samples based on the likelihood. Standard Metropolis-Hastings can work poorly in high dimensional models. We find that it fails for the Gamma-Normal-TS model. Instead, we compare to a Gibbs sampling method that uses Metropolis-Hastings to sample from the complete conditionals. For our proposal distribution we use the same distributions as found in the previous iteration, with the mean equal the value of the previous parameter.

We compute predictive likelihoods using the posterior samples generated by the MCMC methods on held out data in the test set.

On this model, we compared Black Box Variational Inference to Metropolis-Hastings inside Gibbs. We used a fixed computational budget of 20 hours. Figure 1 plots time versus predictive likelihood on the held out set for both methods. We found similar results with different random initializations of both models. Black Box Variational Inference gives better predictive likelihoods and gets them faster than Metropolis-Hastings within Gibbs.333Black Box Variational Inference also has better predictive mean-squared error on the labs than Gibbs style Metropolis-Hastings..

5.4 Variance Reductions

We next studied how much variance is reduced with our variance reduction methods. In Figure 2, we plot the variance of various estimators of the gradient of the variational approximation for a factor in the patient time-series versus iteration number. We compare the variance of the Monte Carlo gradient in  Eq. 3 to that of the Rao-Blackwellized gradient (Eq. 6) and that of the gradient using both Rao-Blackwellization and control variates (Eq. 10). We found that Rao-Blackwellization reduces the variance by several orders of magnitude. Applying control variates reduces the variance further. This reduction in variance drastically improves the speed at which Black Box Variational Inference converges. In fact, in the time allotted, Algorithm 1—the algorithm without variance reductions—failed to make noticeable progress.

Figure 2: Variance comparison for the first component of a random patient on the following estimators: Eq. 3, the Rao-Blackwellized estimator Eq. 6, and the Rao-Blackwellized control variate estimator Eq. 10. We find that Rao-Blackwellizing the naive estimator reduces the variance by several orders of magnitude from the naive estimator. Adding control variates reduces the variance even further.

5.5 Exploring Models

We developed Black Box Variational Inference to make it easier to quickly explore and fit many new models to a data set. We demonstrate this by considering a sequence of three other factor and time-series models of the health data. We name these models Gamma, Gamma-TS, and Gamma-Normal.

Gamma.

We model the latent factors that summarize each visit in our models as positive random variables; as noted above, we expect these to be indicative of patient health. The Gamma model is a positive-value factor model where all of the factors, weights, and observations have positive values. The generative process for this model is D raw Gamma, an matrix For each patient : to Draw Gamma, a vector of F or each visit : 1 to Draw Gamma Draw GammaE, a vector of . We set all hyperparameters save to be 1. As in the previous model, is set to .01.

Gamma-TS.

We can link the factors through time using the expectation parameterization of the Gamma distribution. (Note this is harder with the usual natural parameterization of the Gamma.) This changes to be distributed as GammaE. We draw as above. In this model, the expected values of the factors at the next visit is the same as the value at the current visit. This allows us to propagate patient states through time.

Gamma-Normal.

Similar to the above, we can change the time-series Gamma-Normal-TS (studied in the previous section) to a simpler factor model. This is similar to the Gamma model, but with Normal priors.

These combinations lead to a set of four models that are all nonconjugate and for which standard variational techniques are difficult to apply. Our variational inference method allows us to compute approximate posteriors for these models to determine which provides the best low dimensional latent representations.

We set the AdaGrad scaling parameter to for both the Gamma-Normal models and to for the Gamma models.

Model Comparisons.

Table 1 details our models along with their predictive likelihoods. From this we see that time helps in modelling our longitudinal healthcare data. We also see that the Gamma-Gamma models perform poorly. This is likely because they cannot capture the negative correlations that exist between different medical labs. More importantly, by using Black Box Variational Inference we were able to quickly fit and explore a set of complicated non-conjugate models. Without a generic algorithm, approximating the posterior of any of these models is a project in itself.

Model Predictive Likelihood
Gamma-Normal -33.9
Gamma-Normal-TS -32.7
Gamma-Gamma -175
Gamma-Gamma-TS -174
Table 1: A comparison between several models for our patient health dataset. We find that taking into account the longitudinal nature of the data in the model leads to a better fit. The Gamma weight models perform relatively poorly. This is likely due to the fact that some labs have are negatively correlated. This model cannot capture such relationships.

6 Conclusion

We developed and studied Black Box Variational Inference, a new algorithm for variational inference that drastically reduces the analytic burden. Our main approach is a stochastic optimization of the ELBO by sampling from the variational posterior to compute a noisy gradient. Essential to its success are model-free variance reductions to reduce the variance of the noisy gradient. Black Box Variational Inference works well for new models, while requiring minimal analytic work by the practitioner.

There are several natural directions for future improvements to this work. First, the software libraries that we provide can be augmented with score functions for a wider variety of variational families (each score function is simply the log gradient of the variational distribution with respect to the variational parameters). Second, we believe that number of samples could be set dynamically. Finally, carefully-selected samples from the variational distribution (e.g., with quasi-Monte Carlo methods) are likely to significantly decrease sampling variance.

7 Appendix: The Gradient of the ELBO

The key idea behind our algorithm is that the gradient of the ELBO can be written as an expectation with respect to the variational distribution. We start by differentiating Eq. 1,

where we have exchanged derivatives with integrals via the dominated convergence theorem 444The score function exists. The score and likelihoods are bounded. (Cinlar, 2011) and used .

The first term in Eq. 7 is zero. To see this, note

(14)

To simplify the second term, first observe that . This fact gives us the gradient as an expectation,

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Supplement

Derivation of the Rao-Blackwellized Gradient

To compute the Rao-Blackwellized estimators, we need to compute conditional expectations. Due to the mean field-assumption, the conditional expectation simplifies due to the factorization

(A.15)

Therefore, to construct a lower variance estimator when the joint distribution factorizes, all we need to do is integrate out some variables. In our problem this means for each component of the gradient, we should compute expectations with respect to the other factors. We present the estimator in the full mean field family of variational distributions, but note it applies to any variational approximation with some factorization like structured mean-field.

Thus, under the mean field assumption the Rao-Blackwellized estimator for the gradient becomes

(A.16)

Recall the definitions from Section 3 where we defined as the gradient of the ELBO with respect to , as the components of the log joint that include terms form the th factor, and as the expectation with respect to the set of latent variables that appear in the complete conditional for . Let bet the components of the joint that does not include terms from the th factor respectively. We can write the gradient with respect to the th factor’s variational parameters as

(A.17)

where we have leveraged the mean field assumption and made use of the identity for the expected score Eq. 14. This means we can Rao-Blackwellize the gradient of the variational parameter with respect to the the latent variables outside of the Markov blanket of without needing model specific computations.

Derivation of Stochastic Inference in Hierarchical Bayesian Models

Recall the definition of a hierarchical Bayesian model with observations given in Eq. 12

Let the variational approximation for the posterior distribution be from the mean field family. Let be the global variational parameter and let be the local variational parameters. The variational family is

(A.18)

Using the Rao Blackwellized estimator to compute noisy gradients in this family for this model gives

Unfortunately, this estimator requires iterating over every data point to compute noisy realizations of the gradient. We can mitigate this by subsampling observations. If we let , then we can write down a noisy gradient for the ELBO that does not need to iterate over every observation; this noisy gradient is

The expected value of this estimator with respect to the samples from the variational distribution and the sampled data point is the gradient of the ELBO. This means we can use it define a stochastic optimization procedure to maximize the ELBO. We can lower the variance of the estimator by introducing control variates. Let

(A.19)

We can compute the optimal scalings for the control variates, and , using the Monte Carlo by substituting Eq. A.19 into Eq. 9. This gives the following lower variance noisy gradient that does not need to iterate over all of the observations at each update

(A.20)

Gamma parameterization equivalence

The shape and rate parameterization can be written in terms of the mean and variance of the gamma as

(A.21)