A Simple Stochastic Gradient Variational Bayes for the Correlated Topic Model

A Simple SGVB(Stochastic Gradient Variational Bayes)

for the CTM(Correlated Topic Model)

Tomonari MASADA ( 正田备也 )Nagasaki University (长崎大学 )

[email protected]

APWeb 2016 @ Suzhou

Aim•Make an informative summary of

large document sets by

•extracting word lists, each relating to

a different and particular topic.

Topic modeling2

Contribution•We propose a new posterior estimation for the correlated topic model (CTM) [Blei+ 07],•an extension of LDA [Blei+ 03] for modeling topic correlations,

•with stochastic gradient variational Bayes (SGVB) [Kingma+ 14].

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LDA [Blei+ 03]

•Clustering word tokens by assigning each word token to

one among the topics.• : To which topic is the -th word token in document is assigned?

• : How often is the topic talked about in document ?

•Multinomial distribution for each

• : How often is the word used to talk about the topic ?

•Multinomial distribution for each

discrete variables

continuous variables

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CTM [Blei+ 05]

•Clustering word tokens by assigning each word token to

one among the topics.• : To which topic is the -th word token in document is assigned?

• : How often is the topic talked about in document ?

• where (logistic normal distribution)

• : How often is the word used to talk about the topic ?

•Multinomial distribution for each

discrete variables

continuous variables

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Variational BayesMaximization of ELBO (evidence lower bound)

•VB (variational Bayes) approximates the true posterior.•An approximate posterior is introduced when ELBO is

obtained by Jensen's inequality:

• : discrete hidden variables (topic assignments)• : continuous hidden variables (multinomial parameters)

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log evidence approximate posterior

Factorization assumption

•We assume the approximate posterior factorizes

as .

•Then ELBO can be written as

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×discrete continuous

SGVB[Kingma+ 14]

•SGVB (stochastic gradient variational Bayes) is a general framework for estimating ELBO in VB.

•SGVB is only applicable to continuous distributions .•Monte Carlo integration for expectation

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Reparameterization

•We use the diagonal logistic normal for approximating the true posterior of .•We can efficiently sample from the logistic normal with reparameterization.

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Monte Carlo integration

•ELBO is estimated with a sample from the approximate posterior.

• The discrete part is estimated as in the original VB. 11

Parameter updatesNo explicit inversion (only Cholesky factorization)

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"Stochastic" gradient•The expectation integrations are estimated by Monte Carlo method.•The derivatives of ELBO depend on samples.•Randomness is incorporated into the maximization of ELBO.•Does this make it easier to avoid local minima?

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Data sets# docs # word types

NYT 149,890 46,650MOVIE 27,859 62,408

NSF 128,818 21,471

MED 125,490 42,83014

Conclusion•We incorporate randomness into the posterior inference for the CTM by using SGVB.

•The proposed method gives perplexities comparable to those achieved by LDA.

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Pro/Con•No explicit inversion of covariance matrix is required.

•Careful tuning of gradient descent seems required.

•Only Adam was tested.20

Future work•Online learning for topic models with NN•NN may achieve a better approximate posterior.

•SGVB can be used to estimate ELBO in a similar

manner.

•Document batches can be fed to VB indefinitely.•Topic word lists are then updated indefinitely.

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