Introduction
Despite the recent success of deep generative models such as Variational Autoencoders (VAEs) BIBREF0 and Generative Adversarial Networks (GANs) BIBREF1 in different areas of Machine Learning, they have failed to produce similar generative quality in NLP. In this paper we focus on VAEs and their mathematical underpinning to explain their behaviors in the context of text generation.
The vanilla VAE applied to text BIBREF2 consists of an encoder (inference) and decoder (generative) networks: Given an input $x$, the encoder network parameterizes $q_\phi (z|x)$ and infers about latent continuous representations of $x$, while the decoder network parameterizes $p_\theta (x|z)$ and generates $x$ from the continuous code $z$. The two models are jointly trained by maximizing the Evidence Lower Bound (ELBO), $\mathcal {L}(\theta , \phi ; x,z)$:
where the first term is the reconstruction term, and the second term is the Kullback-Leibler (KL) divergence between the posterior distribution of latent variable $z$ and its prior $p({z})$ (i.e., $\mathcal {N}(0,I)$). The KL term can be interpreted as a regularizer which prevents the inference network from copying ${x}$ into ${z}$, and for the case of a Gaussian prior and posterior has a closed-form solution.
With powerful autoregressive decoders, such as LSTMs, the internal decoder's cells are likely to suffice for representing the sentence, leading to a sub-optimal solution where the decoder ignores the inferred latent code ${z}$. This allows the encoder to become independent of $x$, an issue known as posterior collapse ($q_\phi ({z}|{x})\approx p({z})$) where the inference network produces uninformative latent variables. Several solutions have been proposed to address the posterior collapse issue: (i) Modifying the architecture of the model by weakening decoders BIBREF2, BIBREF3, BIBREF4, BIBREF5, or introducing additional connections between the encoder and decoder to enforce the dependence between $x$ and $z$ BIBREF6, BIBREF7, BIBREF8; (ii) Using more flexible or multimodal priors BIBREF9, BIBREF10; (iii) Alternating the training by focusing on the inference network in the earlier stages BIBREF11, or augmenting amortized optimization of VAEs with instance-based optimization of stochastic variational inference BIBREF12, BIBREF13.
All of the aforementioned approaches impose one or more of the following limitations: restraining the choice of decoder, modifying the training algorithm, or requiring a substantial alternation of the objective function. As exceptions to these, $\delta $-VAE BIBREF14 and $\beta $-VAE BIBREF15 aim to avoid the posterior collapse by explicitly controlling the regularizer term in eqn. DISPLAY_FORM2. While $\delta $-VAE aims to impose a lower bound on the divergence term, $\beta $-VAE (betavae) controls the impact of regularization via an additional hyperparameter (i.e., $\beta D_{KL}\big (q_\phi ({z}|{x}) || p({z})\big )$). A special case of $\beta $-VAE is annealing BIBREF2, where $\beta $ increases from 0 to 1 during training.
In this study, we propose to use an extension of $\beta $-VAE BIBREF16 which permits us to explicitly control the magnitude of the KL term while avoiding the posterior collapse issue even in the existence of a powerful decoder. We use this framework to examine different properties of the estimated posterior and the generative behaviour of VAEs and discuss them in the context of text generation via various qualitative and quantitative experiments.
Kullback-Leibler Divergence in VAE
We take the encoder-decoder of VAEs as the sender-receiver in a communication network. Given an input message $x$, a sender generates a compressed encoding of $x$ denoted by $z$, while the receiver aims to fully decode $z$ back into $x$. The quality of this communication can be explained in terms of rate (R) which measures the compression level of $z$ as compared to the original message $x$, and distortion (D) which quantities the overall performance of the communication in encoding a message at sender and successfully decoding it at the receiver. Additionally, the capacity of the encoder channel can be measured in terms of the amount of mutual information between $x$ and $z$, denoted by $\text{I}({x};{z})$ BIBREF17.
Kullback-Leibler Divergence in VAE ::: Reconstruction vs. KL
The reconstruction loss can naturally measure distortion ($D := - \big \langle \log p_\theta ({x}|{z}) \big \rangle $), while the KL term quantifies the amount of compression (rate; $R := D_{KL}[q_\phi ({z}|{x})|| p({z})]$) by measuring the divergence between a channel that transmits zero bit of information about $x$, denoted by $p(z)$, and the encoder channel of VAEs, $q_\phi (z|x)$.
BIBREF18 introduced the $H-D \le \text{I}({x};{z}) \le R$ bounds, where $H$ is the empirical data entropy (a constant). These bounds on mutual information allow us to analyze the trade-off between the reconstruction and KL terms in eqn. (DISPLAY_FORM2). For instance, since $\text{I}({x};{z})$ is non-negative (using Jensen's inequality), the posterior collapse can be explained as the situation where $\text{I}({x};{z})=0$, where encoder transmits no information about $x$, causing $R=0, D=H$. Increasing $\text{I}({x};{z})$ can be encouraged by increasing both bounds: increasing the upper-bound (KL term) can be seen as the mean to control the maximum capacity of the encoder channel, while reducing the distortion (reconstruction loss) will tighten the bound by pushing the lower bound to its limits ($H-D\rightarrow H$). A similar effect on the lower-bound can be encouraged by using stronger decoders which could potentially decrease the reconstruction loss. Hence, having a framework that permits the use of strong decoders while avoiding the posterior collapse is desirable. Similarly, channel capacity can be decreased.
Kullback-Leibler Divergence in VAE ::: Explicit KL Control via @!START@$\beta $@!END@-VAE
Given the above interpretation, we now turn to a slightly different formulation of ELBO based on $\beta $-VAE BIBREF15. This allows control of the trade-off between the reconstruction and KL terms, as well as to set explicit KL value. While $\beta $-VAE offers regularizing the ELBO via an additional coefficient $\beta \in {\rm I\!R}^+$, a simple extension BIBREF16 of its objective function incorporates an additional hyperparameter $C$ to explicitly control the magnitude of the KL term,
where $C\!\! \in \!\! {\rm I\!R}^+$ and $| . |$ denotes the absolute value. While we could apply constraint optimization to impose the explicit constraint of $\text{KL}\!\!=\!\!C$, we found that the above objective function satisfies the constraint (experiment). Alternatively, it has been shown BIBREF21 the similar effect could be reached by replacing the second term in eqn. DISPLAY_FORM6 with $\max \big (C,D_{KL}\big (q_\phi ({z}|{x}) || p({z})\big )\big )$ at the risk of breaking the ELBO when $\text{KL}\!\!<\!\!C$ BIBREF22.
Experiments
We conduct various experiments to illustrate the properties that are encouraged via different KL magnitudes. In particular, we revisit the interdependence between rate and distortion, and shed light on the impact of KL on the sharpness of the approximated posteriors. Then, through a set of qualitative and quantitative experiments for text generation, we demonstrate how certain generative behaviours could be imposed on VAEs via a range of maximum channel capacities. Finally, we run some experiments to find if any form of syntactic information is encoded in the latent space. For all experiments, we use the objective function of eqn. DISPLAY_FORM6 with $\beta =1$. We do not use larger $\beta $s because the constraint $\text{KL}=C$ is always satisfied.
Experiments ::: Corpora
We use 5 different corpora covering different domains and size through this section: Yelp and Yahoo BIBREF4 both have ($100k$,$10k$,$10k$) sentences in (train, dev, test) sets and $20k$ words in vocabulary, Children's Book Test (CBT; BIBREF23) has ($192k$,$10k$,$12k$) sentences and $12k$ vocab, Wikipedia (WIKI; BIBREF24) has ($2m$,$270k$,$270k$) sentences and $20k$ vocab, and WebText BIBREF25 has ($1m$,$23k$,$24k$) sentences and $22k$ vocab.
Experiments ::: Models
We examine three VAE architectures, covering a range of decoding strengths to examine if the objective function in eqn. DISPLAY_FORM6 is immune to posterior collapse regardless of the choice of encoder-decoder architectures: $\beta _C$-VAELSTM with (LSTM encoder, LSTM decoder), $\beta _C$-VAEGRU with (GRU encoder, GRU decoder) BIBREF26, and $\beta _C$-VAECNN with (LSTM encoder, CNN decoder) BIBREF27. The dimension of word embeddings is 256 and the dimension of the latent variable is 64. The encoder and the decoder, for both VAELSTM and VAEGRU, have hidden size of 512 dimensions. VAECNN has exactly the same encoder as VAELSTM, while the decoder follows similar architecture to GLU with a bottleneck structure (with two blocks) BIBREF27 and has 512 channels externally and 128 internally for the convolutions with the filter size of 20. All models were trained for 10 epochs and optimised the objective function (eqn. DISPLAY_FORM6) with Adam BIBREF28 with following learning rates: $10^{-5}\times 85$ for VAEGRU and VAELSTM, and $10^{-4}$ for VAECNN. To couple the encoder with the decoder we concatenate the latent variable to word embeddings at each time step without initialisation of hidden state.
Experiments ::: Rate and Distortion
To analyse the dependence between the values of explicit rate ($C$) and distortion, we trained our models with different values of $C$, ranging from 10 to 100. Figure FIGREF8 reports the results for $\beta _C$-VAEGRU, $\beta _C$-VAELSTM, and $\beta _C$-VAECNN models on Yahoo and Yelp corpora. In all our experiments we found that $C\!-\!1\!\le KL\!\le \! C\!+\!1$, demonstrating that the objective function effectively imposed the desired constraint on KL term. Hence, setting any $C>0$ can in practice avoid the collapse issue.
The general trend is that by increasing the value of $C$ one can get a better reconstruction (lower distortion) while the amount of gain varies depending on the VAE's architecture and corpus. Additionally, we measured rate and distortion on CBT, WIKI, and WebText corpora using $\beta _C$-VAELSTM and observed the same trend with the increase of $C$, see Table TABREF12. This observation is consistent with the bound on $\text{I}({x};{z})$ we discussed earlier (expl) such that with an increase of KL we increase an upper bound on $\text{I}({x};{z})$ which in turn allows to have smaller values of reconstruction loss. Additionally, as reported in Table TABREF12, encouraging higher rates (via larger $C$) encourages more active units (AU; BIBREF29) in the latent code $z$.
As an additional verification, we also group the test sentences into buckets based on their length and report BLEU-2/4 and ROUGE-2/4 metrics to measure the quality of reconstruction step in Table TABREF12. As expected, we observe that increasing rate has a consistently positive impact on improving BLEU and ROUGE scores.
Experiments ::: Aggregated Posterior
To understand how the approximated posteriors are being affected by the magnitude of the KL, we adopted an approach from BIBREF6 and looked at the divergence between the aggregated posterior, $q_\phi (z)=\sum _{x\sim q(x)} q_\phi (z|x)$, and prior $p(z$). Since during generation we generate samples from the prior, ideally we would like the aggregated posterior to be as close as possible to the prior.
We obtained unbiased samples of ${z}$ first by sampling an ${x}$ from data and then ${z} \sim q_\phi ({z}|{x})$, and measured the log determinant of covariance of the samples ($\log \det (\mathrm {Cov}[q_\phi ({z})])$). As reported in Figure FIGREF8, we observed that $\log \det (\mathrm {Cov}[q_\phi ({z})])$ degrades as $C$ grows, indicating sharper approximate posteriors. We then consider the difference of $p(z)$ and $q(z)$ in their means and variances, by computing the KL divergence from the moment-matching Gaussian fit of $q(z)$ to $p(z)$: This returns smaller values for $\beta _{C=5}$-VAEGRU (Yelp: 0, Yahoo: 0), and larger values for $\beta _{C=100}$-VAEGRU (Yelp: 8, Yahoo: 5), which illustrates that the overlap between $q_\phi ({z})$ and $p(z)$ shrinks further as $C$ grows.
The above observation is better pronounced in Table TABREF12, where we also report the mean ($||\mu ||^2_2$) of unbiased samples of $z$, highlighting the divergence from the mean of the prior distribution as rate increases. Therefore, for the case of lower $C$, the latent variables observed during training are closer to the generated sample from the prior which makes the decoder more suitable for generation purpose. We will examine this hypothesis in the following section.
Experiments ::: Text Generation
To empirically examine how channel capacity translates into generative capacity of the model, we experimented with the $\beta _C$-VAELSTM models from Table TABREF12. To generate a novel sentence, after a model was trained, a latent variable $z$ is sampled from the prior distribution and then transformed into a sequence of words by the decoder $p(x|z)$.
During decoding for generation we try three decoding schemes: (i) Greedy: which selects the most probable word at each step, (ii) Top-k BIBREF30: which at each step samples from the K most probable words, and (iii) Nucleus Sampling (NS) BIBREF31: which at each step samples from a flexible subset of most probable words chosen based on their cumulative mass (set by a threshold $p$, where $p = 1$ means sampling from the full distribution). While similar to Top-k, the benefit of NS scheme is that the vocabulary size at each time step of decoding varies, a property that encourages diversity and avoids degenerate text patterns of greedy or beam search decoding BIBREF31. We experiment with NS $(p=\lbrace 0.5, 0.9\rbrace )$ and Top-k $(k=\lbrace 5, 15\rbrace )$.
Experiments ::: Text Generation ::: Qualitative Analysis
We follow the settings of homotopy experiment BIBREF2 where first a set of latent variables was obtained by performing a linear interpolation between $z_1 \sim p(z)$ and $z_2 \sim p(z)$. Then each $z$ in the set was converted into a sequence of words by the decoder $p(x|z)$. Besides the initial motivation of BIBREF2 to examine how neighbouring latent codes look like, our additional incentive is to analyse how sensitive the decoder is to small variations in the latent variable when trained with different channel capacities, $C=\lbrace 3,15,100\rbrace $.
Table TABREF17 shows the generated sentences via different decoding schemes for each channel capacity. For space reason, we only report the generated sentences for greedy, Top-$k=15$, and NS $p=0.9$. To make the generated sequences comparable across different decoding schemes or C values, we use the same samples of $z$ for decoding.
Experiments ::: Text Generation ::: Qualitative Analysis ::: Sensitivity of Decoder
To examine the sensitivity of the decoder to variations of the latent variable, we consider the sentences generate with the greedy decoding scheme (the first column in Table TABREF17). The other two schemes are not suitable for this analysis as they include sampling procedure. This means that if we decode the same latent variable twice we will get two different sentences. We observed that with lower channel capacity ($C=3$) the decoder tends to generate identical sentences for the interpolated latent variables (we highlight these sentences in gray), exhibiting decoder's lower sensitivity to $z$'s variations. However, with the increase of channel capacity ($C=15,100$) the decoder becomes more sensitive. This observation is further supported by the increasing pattern of active units in Table TABREF12: Given that AU increases with increase of $C$ one would expect that activation pattern of a latent variable becomes more complex as it comprises more information. Therefore small change in the pattern would have a greater effect on the decoder.
Experiments ::: Text Generation ::: Qualitative Analysis ::: Coherence of Sequences
We observe that the model trained with large values of $C$ compromises sequences' coherence during the sampling. This is especially evident when we compare $C=3$ with $C=100$. Analysis of Top-15 and NS (p=0.9) generated samples reveals that the lack of coherence is not due to the greedy decoding scheme per se, and can be attributed to the model in general. To understand this behavior further, we need two additional results from Table TABREF12: LogDetCov and $||\mu ||^2_2$. One can notice that as $C$ increases LogDetCov decreases and $||\mu ||^2_2$ increases. This indicates that the aggregated posterior becomes further apart from the prior, hence the latent codes seen during the training diverge more from the codes sampled from the prior during generation. We speculate this contributes to the coherence of the generated samples, as the decoder is not equipped to decode prior samples properly at higher $C$s.
Experiments ::: Text Generation ::: Quantitative Analysis
Quantitative analysis of generated text without gold reference sequences (e.g. in Machine Translation or Summarization) has been a long-standing challenge. Recently, there have been efforts towards this direction, with proposal such as self-BLEU BIBREF32, forward cross entropy BIBREF33 and Fréchet InferSent Distance BIBREF33. We opted for FCE as a complementary metric to our qualitative analysis. To calculate FCE, first a collection of synthetic sentences are generated by sampling $z\sim p(z)$ and decoding the samples into sentences. The synthetic sequences are then used to train a language model (an LSTM with the parametrisation of our decoder). The FCE score is estimated by reporting the negative log likelihood (NLL) of the trained LM on the set of human generated sentences.
We generated synthetic corpora using trained models from Table TABREF12 with different C and decoding schemes and using the same exact $z$ samples for all corpora. Since the generated corpora using different C values would have different coverage of words in the test set (i.e., Out-of-Vocabulary ratios), we used a fixed vocabulary to minimize the effect of different vocabularies in our analysis. Our dictionary contains words that are common in all of the three corpora, while the rest of the words that don't exist in this dictionary are replaced with 〈unk〉 symbol. Similarly, we used this fixed dictionary to preprocess the test sets. Also, to reduce bias to a particular set of sampled $z$'s we measure the FCE score three times, each time we sampled a new training corpus from a $\beta _C$-VAELSTM decoder and trained an LM from scratch. In Table TABREF20 we report the average FCE (NLL) for the generated corpora.
In the qualitative analysis we observed that the text generated by the $\beta _C$-VAELSTM trained with large values of $C=100$ exhibits lower quality (i.e., in terms of coherence). This observation is supported by the FCE score of NS(p=0.9) decoding scheme (TABREF20), since the performance drops when the LM is trained on the corpus generated with $C=100$. The generated corpora with $C=3$ and $C=15$ achieve similar FCE score. However, these patterns are reversed for Greedy decoding scheme, where the general tendency of FCE scores suggests that for larger values of $C$ the $\beta _C$-VAELSTM seems to generate text which better approximates the natural sentences in the test set. To understand this further, we report additional statistics in Table TABREF20: percentage of 〈unk〉 symbols, self-BLEU and average sentence length in the corpus.
The average sentence length, in the generated corpora is very similar for both decoding schemes, removing the possibility that the pathological pattern on FCE scores was caused by difference in sentence length. However, we observe that for Greedy decoding more than $30\%$ of the test set consists of 〈unk〉. Intuitively, seeing more evidence of this symbol during training would improve our estimate for the 〈unk〉. As reported in the table, the $\%$unk increases on almost all corpora as $C$ grows, which is then translated into getting a better FCE score at test. Therefore, we believe that FCE at high $\%$unk is not a reliable quantitative metric to assess the quality of the generated syntactic corpora. Furthermore, for Greedy decoding, self-BLEU decreases when $C$ increases. This suggests that generated sentences for higher value of $C$ are more diverse. Hence, the LM trained on more diverse corpora can generalise better, which in turn affects the FCE.
In contrast, the effect the 〈unk〉 symbol has on the corpora generated with the NS(p=0.9) decoding scheme is minimal for two reasons: First, the vocabulary size for the generated corpora, for all values of $C$ is close to the original corpus (the corpus we used to train the $\beta _C$-VAELSTM). Second, the vocabularies of the corpora generated with three values of $C$ is very close to each other. As a result, minimum replacement of the words with the 〈unk〉 symbol is required, making the experiment to be more reflective of the quality of the generated text. Similarly, self-BLEU for the NS(p=0.9) is the same for all values of $C$. This suggests that the diversity of sentences has minimal, if any, effect on the FCE.
Experiments ::: Syntactic Test
In this section, we explore if any form of syntactic information is captured by the encoder and represented in the latent codes despite the lack of any explicit syntactic signal during the training of the $\beta _C$-VAELSTM. To train the models we used the same WIKI data set as in BIBREF24, but we filtered out all the sentences that are longer than 50 space-separated tokens. We use the data set of BIBREF24 which consists of pairs of grammatical and ungrammatical sentences to test various syntactic phenomenon. For example, a pair in subject-verb agreement category would be: (The author laughs, The author laugh). We encode both the grammatical and ungrammatical sentences into the latent codes $z^+$ and $z^-$, respectively. Then we condition the decoder on the $z^+$ and try to determine whether the decoder assigns higher probability to the grammatical sentence (denoted by $x^+$): $p(x^-|z^+) < p(x^+|z^+)$ (denoted by p1 in Table TABREF28). We repeat the same experiment but this time try to determine whether the decoder, when conditioned on the ungrammatical code ($z^-$), still prefers to assign higher probability to the grammatical sentence: $p(x^-|z^-) < p(x^+|z^-)$ (denoted by p2 in Table TABREF28). Table TABREF28 shows the p1 and p2 for the $\beta _C$-VAELSTM model trained with $C=\lbrace 3,100\rbrace $. Both the p1 and p2 are similar to the accuracy and correspond to how many times a grammatical sentence was assigned a higher probability.
As reported for C=3, p1 and p2 match in almost all cases. This is to some degree expected since lower channel capacity encourages a more dominating decoder which in our case was trained on grammatical sentences from the WIKI. On the other hand, this illustrates that despite avoiding the KL-collapse issue, the dependence of the decoder on the latent code is so negligible that the decoder hardly distinguishes the grammatical and ungrammatical inputs. This changes for $C=100$, as in almost all the cases the decoder becomes strongly dependent on the latent code and can differentiate between what it has seen as input and the closely similar sentence it hasn't received as the input: The decoder assigns larger probability to the ungrammatical sentence when conditioned on the $z^-$ and, similarly, larger probability to the grammatical sentence when conditioned on the $z^+$.
However, the above observations neither confirm nor reject existence of grammar signal in the latent codes. We run a second set of experiments where we aim to discard sentence specific information from the latent codes by averaging the codes inside each syntactic category. The averaged codes are denoted by $\bar{z}^+$ and $\bar{z}^-$, and the corresponding accuracies are reported by p̄1 and p̄2 in Table TABREF28. Our hypothesis is that the only invariant factor during averaging the codes inside a category is the grammatical property of its corresponding sentences.
As expected, due to the weak dependence of decoder on latent code, the performance of the model under $C=3$ is almost identical (not included for space limits) when comparing p1 vs. p̄1, and p2 vs. p̄2. However, for $C=100$ the performance of the model deteriorates. While we leave further exploration of this behavior to our future work, we speculate this could be an indication of two things: the increase of complexity in the latent code which encourages a higher variance around the mean, or the absence of syntactic signal in the latent codes.
Discussion and Conclusion
In this paper we analysed the interdependence of the KL term in Evidence Lower Bound (ELBO) and the properties of the approximated posterior for text generation. To perform the analysis we used an information theoretic framework based on a variant of $\beta $-VAE objective, which permits explicit control of the KL term, and treats KL as a mechanism to control the amount of information transmitted between the encoder and decoder.
The immediate impact of the explicit constraint is avoiding the collapse issue ($D_{KL}=0$) by setting a non-zero positive constraint ($C\ge 0$) on the KL term ($|D_{KL}\big (q_\phi ({z}|{x}) || p({z})\big )-C|$). We experimented with a range of constraints ($C$) on the KL term and various powerful and weak decoder architectures (LSTM, GRU, and CNN), and empirically confirmed that in all cases the constraint was satisfied.
We showed that the higher value of KL encourages not only divergence from the prior distribution, but also a sharper and more concentrated approximated posteriors. It encourages the decoder to be more sensitive to the variations on the latent code, and makes the model with higher KL less suitable for generation as the latent variables observed during training are farther away from the prior samples used during generation. To analyse its impact on generation we conducted a set of qualitative and quantitative experiments.
In the qualitative analysis we showed that small and large values of KL term impose different properties on the generated text: the decoder trained under smaller KL term tends to generate repetitive but mainly plausible sentences, while for larger KL the generated sentences were diverse but incoherent. This behaviour was observed across three different decoding schemes and complemented by a quantitative analysis where we measured the performance of an LSTM LM trained on different VAE-generated synthetic corpora via different KL magnitudes, and tested on human generated sentences.
Finally, in an attempt to understand the ability of the latent code in VAEs to represent some form of syntactic information, we tested the ability of the model to distinguish between grammatical and ungrammatical sentences. We verified that at lower (and still non-zero) KL the decoder tends to pay less attention to the latent code, but our findings regarding the presence of a syntactic signal in the latent code were inconclusive. We leave it as a possible avenue to explore in our future work. Also, we plan to develop practical algorithms for the automatic selection of the $C$'s value, and verify our findings under multi-modal priors and complex posteriors.
Acknowledgments
The authors would like to thank the anonymous reviewers for their helpful suggestions. This research was supported by an EPSRC Experienced Researcher Fellowship (N. Collier: EP/M005089/1), an MRC grant (M.T. Pilehvar: MR/M025160/1) and E. Shareghi is supported by the ERC Consolidator Grant LEXICAL (648909). We gratefully acknowledge the donation of a GPU from the NVIDIA.

Question: What different properties of the posterior distribution are explored in the paper?
Answer: interdependence between rate and distortion, impact of KL on the sharpness of the approximated posteriors, demonstrate how certain generative behaviours could be imposed on VAEs via a range of maximum channel capacities, some experiments to find if any form of syntactic information is encoded in the latent space

Question: Why does proposed term help to avoid posterior collapse?
Answer: by setting a non-zero positive constraint ($C\ge 0$) on the KL term ($|D_{KL}\big (q_\phi ({z}|{x}) || p({z})\big )-C|$)

Question: How does explicit constraint on the KL divergence term that authors propose looks like?
Answer: Answer with content missing: (Formula 2) Formula 2 is an answer: 
\big \langle\! \log p_\theta({x}|{z}) \big \rangle_{q_\phi({z}|{x})}  -  \beta |D_{KL}\big(q_\phi({z}|{x}) || p({z})\big)-C|