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A Simple Stochastic Gradient Variational
Bayesfor Latent Dirichlet
Allocation
Tomonari MASADA ( 正田备也 )Nagasaki University (长崎大学 )
masada@nagasaki-u.ac.jp
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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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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LDA [Blei+ 03]• Achieve a clustering of word tokens by assigning each word
token to one among the topics.
• : the topic to which the -th word token in document is
assigned.
• : How often the topic is talked about in document ?
• Topic probability distribution in each document
• : How often the word is used to talk about the topic ?
•Word probability distribution for each topic
discrete variablescontinuous variables
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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:
• : discrete hidden variables (topic assignments)• : continuous hidden variables (multinomial parameters)
evidence approximate posterior
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Factorization assumption•We assume the approximate posterior factorizes as
to make the inference tractable.
•Then ELBO can be written as
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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
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(SGVB) Monte Carlo integration•By using Monte Carlo integration, ELBO can be
estimated with random samples as
• The discrete part is estimated in a similar manner to
the original VB for LDA [Blei+ 03].10
(SGVB) Reparameterization• SGVB can be applied "under certain mild conditions."•We use the logistic normal distributions for approximating
the true posterior of: per-doc topic probability distributions, and: per-topic word probability distributions.
•We can efficiently sample from the logistic normal with reparameterization.
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Maximize ELBO using gradient ascent
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"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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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.15
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 42,83016
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Not that efficient in time…
•500 iters for NYT data set when
•LNV: 43 hours
•CGS: 14 hours
•VB: 23 hours
•However, parallelization with GPU works.
• (preparing an implementation with TensorFlow)
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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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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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