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
• 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
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)
•We assume the approximate posterior 𝑞 𝒛, 𝚯
factorizes as 𝑞 𝒛 𝑞 𝚯 to make the inference
•Then ELBO can be written as
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
(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].
• 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
"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.
= with zero standard deviation
•A special case of the proposed method is quite
similar to CVB0 [Asuncion+ 09].
•Our method has a context.
Data sets for evaluation
NYT 99,932 46,263
MOVIE 27,859 62,408
NSF 128,818 21,471
MED 125,490 42,830
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)
•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.
•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