This document proposes applying stochastic gradient variational Bayes (SGVB) to latent Dirichlet allocation (LDA) topic modeling to obtain an efficient posterior estimation. SGVB introduces randomness into variational inference for LDA by estimating expectations with Monte Carlo integration and using reparameterization to sample from approximate posterior distributions. Evaluation on several text corpora shows perplexities comparable to existing LDA inference methods, with the potential for faster parallelization using techniques like GPU processing. Future work will explore applying SGVB to other probabilistic document models like correlated topic models.

1. A Simple Stochastic Gradient
Variational Bayes
for Latent Dirichlet Allocation
Tomonari MASADA (正田备也)
Nagasaki University (长崎大学)
masada@nagasaki-u.ac.jp

2. Aim
•Obtain an informative summary of a large
set of documents
•by extracting word lists, each relating to a
specific topic
 Topic modeling
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4. Contribution
•We propose a new posterior estimation for
latent Dirichlet allocation (LDA) [Blei+ 03]
•by applying stochastic gradient variational Bayes
(SGVB) [Kingma+ 14] to LDA
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6. LDA [Blei+ 03]
• Achieve a clustering of word tokens by assigning each word
token to one among the 𝐾 topics.
• 𝑧 𝑑𝑖: the topic to which the 𝑖-th word token in document 𝑑 is
assigned.
• 𝜃 𝑑𝑘: How often the topic 𝑘 is talked about in document 𝑑?
• Topic probability distribution in each document
• 𝜙 𝑘𝑣: How often the word 𝑣 is used to talk about the topic 𝑘?
• Word probability distribution for each topic
discrete variables
continuous variables
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7. Variational Bayesian (VB) inference
= maximization of evidence lower bound (ELBO)
•VB tries to approximate the true posterior.
•An approximate posterior is introduced when ELBO is
obtained by applying Jensen's inequality:
• 𝒛: discrete hidden variables (topic assignments)
• 𝚯: continuous hidden variables (multinomial parameters)
evidence approximate posterior 𝑞(𝒛, 𝚯)
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8. Factorization assumption
•We assume the approximate posterior 𝑞 𝒛, 𝚯
factorizes as 𝑞 𝒛 𝑞 𝚯 to make the inference
tractable.
•Then ELBO can be written as
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9. Stochastic gradient variational Bayes
(SGVB) [Kingma+ 14]
•A general framework for estimating evidence
lower bound (ELBO) in variational Bayes (VB)
•Only applicable to continuous distributions
𝑞 𝚯
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10. (SGVB) Monte Carlo integration
•By using Monte Carlo integration, ELBO can be
estimated with 𝐿 random samples as
• The discrete part 𝑞 𝒛 is estimated in a similar manner
to the original VB for LDA [Blei+ 03].
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11. (SGVB) Reparameterization
• SGVB can be applied "under certain mild conditions."
• We use the logistic normal distributions for approximating
the true posterior of
𝜃 𝑑𝑘: per-doc topic probability distributions, and
𝜙 𝑘𝑣: per-topic word probability distributions.
• We can efficiently sample from the logistic normal with
reparameterization.
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12. Maximize ELBO using gradient ascent
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14. "Stochastic" gradient VB
•The expectation integrations in ELBO are estimated
by Monte Carlo method.
•The derivatives of ELBO depend on random samples.
•Randomness is incorporated into maximization.
• SGVB = VB where gradients are stochastic.
• (Observation) It seems easier to avoid poor local minima.
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15. without randomness
= with zero standard deviation
•A special case of the proposed method is quite
similar to CVB0 [Asuncion+ 09].
•Our method has a context.
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16. Data sets for evaluation
# docs
# vocabulary
words
NYT 99,932 46,263
MOVIE 27,859 62,408
NSF 128,818 21,471
MED 125,490 42,830
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21. Not that efficient in time…
•500 iters for NYT data set when 𝐾 = 200
•LNV: 43 hours
•CGS: 14 hours
•VB: 23 hours
•However, parallelization with GPU works.
•(preparing an implementation with TensorFlow)
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22. Conclusion
•We incorporate randomness into variational
inference for LDA by applying SGVB to LDA.
•The proposed method gives perplexities
comparable to the existing inferences for LDA.
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23. Future work
•SGVB is a general framework for devising a
posterior inference for probabilistic models.
•We've already applied SGVB to CTM [Blei+ 05].
• This will be poster-presented at APWeb'16.
•SGVB is also applicable to other document models.
• NVDM [Miao+ 16]: document modeling with MLP
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