ICCSA 2016 @ Beijing

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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