Continuous latent time series models are popular tools in Bayesian machine learning. One thereby combines multiple copies of a likelihood model through a time series prior, thus enabling the latent variables to drift over time while still sharing statistical strength accross all times (Blei & Lafferty, 2006; Wang et al., 2008; Sahoo et al., 2012; Gultekin & Paisley, 2014; Charlin et al., 2015; Ranganath et al., 2015; Jerfel et al., 2017; Bamler & Mandt, 2017).
Variational Inference (VI) enables approximate inference for complex models by solving an optimization problem (Jordan et al., 1999). One chooses a variational distribution and fits it to the posterior, where the fully-factorized mean field class is the most widely used. However, the standard mean field class is not a good choice when it comes to time series models. Instead, one often uses structured variational distributions in time (Wainwright et al., 2008; Blei & Lafferty, 2006). The perhaps simplest such choice is a multivariate Gaussian with a tridiagonal precision matrix. This can be seen as using the probabilistic model of the Kalman filter as a variational distribution (variational Kalman filter) (Blei & Lafferty, 2006), and reflects the fact that the prior is a first-order Markov process.
In this paper, we introduce an efficient VI algorithm for fitting such tridiagonal Gaussian variational models to a large class of complex latent time series models. We draw on Black Box Variational Inference (BBVI) with reparameterization gradients (Salimans & Knowles, 2013; Kingma & Welling, 2014; Rezende et al., 2014; Ruiz et al., 2016), where one forms a Monte-Carlo estimate of the variational lower bound’s gradient. The problem with structured variational distributions is that computing this reparameterization gradient is expensive; a naive implementation involves a dense matrix multiplication and scales as , where is the number of time steps. In this paper, we lay out a general algorithm which gives the reparameterization gradient in . This algorithm can be thought of a variant of the forward-backward algorithm (Murphy, 2012) in the context of BBVI, and relies on a reparameterization procedure that never involves the inversion or multiplication of a dense matrix. We illustrate our approach on the dynamic word embedding model (Bamler & Mandt, 2017).
2 Background and Problem Setting
We start by specifying our generative and variational models and give an overview of the proposed algorithm.
We consider models with time-dependent observations , where is the number of time steps. At each time step , the generative process is described by an arbitrary likelihood function , and
is a latent variable. We furthermore assume that the prior is Markovian. The joint probability thus factorizes as follows,
where denotes an arbitrary prior on the first latent variable. Many models fall into this class. Our goal is an efficient posterior approximation for this model class, using a structured variational approximation and black box inference techniques.
Kalman smoothing revisited.
Before we describe our approach, we revisit the Kalman smoother (Rauch et al., 1965) as an efficient algorithm for a particularly simple realization of Eq. 1 where all conditional distributions are Gaussian. This is often called a Wiener process (more precisely, it is a Gauss-Markov process). The prior is Gaussian with tridiagonal precision matrix , and the likelihood is a Gaussian centered around with precision . Thus, the posterior is also Gaussian, , and can be obtained analytically. One finds and . Obtaining the posterior modes involves solving the linear system of equations . For an arbitrary matrix , this involves operations. However, for the Wiener process, is tridiagonal and one can solve for in linear time using a specialized algorithm.
The forward-backward algorithm (Murphy, 2012) implicitly decomposes into a product of two bidiagonal matrices and , where is lower triangular and is upper triangular. One starts with a forward pass through the data, in which one solves
for the auxiliary vectorusing forward substitution, followed by a backward pass in which one solves for using back substitution. As detailed in section 3.3, we use a similar philosophy.
For a general likelihood model , the exact posterior of Eq. 1 is intractable. To circumvent this problem, we use BBVI to minimize the KL divergence between a variational distribution and the true posterior. Here, summarizes all variational parameters. We use a structured variational distribution as our variational model that is motivated by the exact posterior of the analytically solvable model discussed above. We thus consider a Gaussian with a tridiagonal precision matrix ,
This can be understood as varying the parameters of a (fictitious) Wiener process so that its posterior approximates the posterior of the true model (Blei & Lafferty, 2006).
As we discuss below, the tridiagonal structure of allows us to fit the variational distribution efficiently. Note that the covariance matrix is dense, thus encoding long-range correlations between any two time steps. Modelling correlations is important for many time series applications, in particular when the prior is strong e.g. due to little evidence per time step.
Black box variational inference.
In BBVI, one fits to the posterior , maximizing the evidence lower bound (ELBO) using stochastic gradients. The reparameterization trick amounts to representing the ELBO as
The entropy is often an analytic function or can be estimated using other tricks. Here, is a vector ofdenotes functions that are defined such that the random variable has probability density (see Section 3.2 below).
In order to implement an efficient BBVI algorithm, one needs to be able to estimate the gradient of efficiently, i.e., in time. This involves the following three tasks:
All three of the above tasks can easily be solved efficiently if one chooses a mean field variational distribution, i.e., if factorizes over all time steps. However, the mean field approximation ignores correlations between time steps, which are important in many time series models as discussed above. In Section 3 we address each one of the above tasks individually.
In this section, we give the details of our new black box variational inference algorithm.
3.1 Evaluating the Entropy
The entropy of a multivariate Gaussian with precision matrix is , up to an additive constant. Evaluating the determinant of a general matrix takes operations in practice. To avoid this expensive operation, we parameterize the precision matrix via its Cholesky decomposition,
Here, is an upper triangular matrix. Since we restrict to have a tridiagonal structure, is bidiagonal (Kılıç & Stanica, 2013), i.e., it has the structure
with . As the mapping from to is unique, it suffices to optimize the non-zero components of in order to find the optimal entries of . It turns out that we never have to construct the matrix explicitly. The variational parameters are thus the marginal means (see Eq. 2) and the non-zero components of . Using the relation , we can evaluate the entropy in linear time,
3.2 Evaluating the reparameterization functions
In contrast to the entropy, the expected log-joint in Eq. 3 cannot be evaluated analytically for a general model. We obtain an unbiased gradient estimator of the expected log-joint by drawing independent samples for .
For what follows, let denote any of the
variational parameters. Using the chain rule, the estimate of the gradient of the expected log-joint with respect tois
where we defined
Here, is a dense (upper triangular) matrix. Instead of computing the inverse, we evaluate the term on the right-hand side of Eq. 9 by solving the linear system for using back substitution. This takes only operations due to the bidiagonal structure of , see Eq. 5. We can therefore evaluate Eq. 8 in time for each sample .
3.3 Estimating the Reparameterization Gradient
Last, we show how to efficiently estimate the reparameterization gradient in time. In this subsection we describe an efficient method to evaluate the Jacobian that appears on the right-hand side of Eq. 7.
A naive evaluation of all gradient estimates would require operations: the sums on the right-hand side of Eq. 7 run over terms, and the number of variational parameters for which the gradient estimate has to be evaluated grows at least linearly in for a reasonably flexible variational distribution.
We use the following trick to evaluate all gradient estimates in linear time in
. For any invertible matrixthat depends on some parameter , we have
Solving for , we obtain
Eq. 11 expresses the derivative of the dense (upper triangular) matrix in terms of the derivative of the bidiagonal matrix . In fact, both and are sparse matrices with only a single non-zero entry, see Eq. 5. The dense matrix still appears on the right-hand side of Eq. 11 but we avoid evaluating it explicitly. Instead, we again solve a bidiagonal linear system of equations. Combining Eqs. 7, 9 and 11, we obtain the formal expressions
where bold face denotes a column vector of derivatives of the log-joint as defined in Eq. 8. Instead of computing the inverse in Eqs. 13–14, we obtain the vectors and by solving the linear systems and , respectively, in time using the bidiagonal structure of . Since is lower-diagonal and is upper-diagonal these operations carry-out a forward and a backward pass through the time steps, respectively. This is similar to the forward-backward algorithm for the Wiener process with Gaussian likelihood discussed in Section 2.
To demonstrate the benefits of our approach, we summarize selective experimental results from work on dynamic word embeddings, which used our proposed BBVI algorithm.
Dynamic Skip-Gram Model.
Bamler & Mandt (2017) proposed the dynamic skip-gram model as a generalization of the skip-gram model to time-stamped text corpora (Mikolov et al., 2013). The model, thus, represents words as latent trajectories in an embedding space, and is able to track how words change their meaning over time.
To train the model, one first summarizes the text corpus in terms of certain sufficient statistics, using a finite vocabulary. For each time stamp, one builds co-occurrence matrices of words and their surrounding words. In addition, the model requires so-called nagative examples, expressing the likelihood of co-occurrance by mere chance. The model then models these statistics in terms of the geometrical arrangement of certain latent word and context embedding vectors. The prior over these embedding vectors is a latent Ornstein-Uhlenbeck process (a Wiener process in the presence of a restoring force), enforcing a continuous drift over time. For more details, we refer to (Bamler & Mandt, 2017).
Algorithms and Baselines.
SGI denotes the non-Bayesian skip-gram model with independent random initializations of word vectors (Mikolov et al., 2013; Hamilton et al., 2016). SGP denotes the same approach as above, but with word and context vectors being pre-initialized with the values from the previous year, as in (Kim et al., 2014). Both approaches have no dynamical priors and hence no overhead and just scale linearly with . DSG-F denotes the dynamic skip-gram filtering algorithm, proposed in (Bamler & Mandt, 2017), which also runs in , but uses the mean field approximation and only uses information form the past. Finally, DSG-S denotes the dynamic skip-gram smoothing algorithm. This is the algorithm proposed in this paper, applied to the dynamic skip-gram model.
We used data from the Google books corpus (Michel et al., 2011) from the last two centuries (). We also used the “State of the Union” (SoU) addresses of U.S. presidents, which spans more than two centuries. Finally, we used a Twitter corpus of news tweets for randomly drawn dates from to
. Details on hyperparameters and preprocessing are given in(Bamler & Mandt, 2017).
We focus on the quantitative results of (Bamler & Mandt, 2017), showing show that the dynamic skip-gram smoothing algorithm (described and generalized in this paper) generalizes better to unseen data than all baselines at similar run time. We thereby analyze held-out predictive likelihoods on word-context pairs at a given time , where is excluded from the training set.
The predictive likelihoods as a function of time are shown in Figure 1. On all three corpora, differences between the two implementations of the static model (SGI and SGP) are small, which suggests that pre-initializing the embeddings with the previous result seems to have little impact on the predictive power. Log-likelihoods for the skip-gram filter (DSG-F) grow over the first few time steps as the filter sees more data, and then saturate. Skip-gram smoothing (this paper) further outperforms skip-gram filtering.
To conclude, we stress that a structured BBVI algorithm with quadratic instead of linear run time in would be impractical. We therefore hope that our structured reparameterization trick will spur further research on complex latent time series models.
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