Sparse Stochastic Inference for Latent Dirichlet Allocation

Transcription

1 Sparse Stochastic Inference for Latent Dirichlet Allocation David Mimno 1, Matthew D. Hoffman 2, David M. Blei 1 1 Dept. of Computer Science, Princeton U. 2 Dept. of Statistics, Columbia U. Presentation led by Miao Liu March 7, 2013

2 1 Introduction 2 Hybrid Stochastic-MCMC Inference 3 Related Work 4 Empirical Results

3 A hybrid algorithm for Bayesian topic models sparse Gibbs sampling to estimate variational expectation of local variables (efficiency) online VB to update the variational distribution of global variables (scalability) Applications: find large numbers of topics in massive collections of documents 1.2 million books 33 billion words

4 Latent Dirichlet Allocation (Blei et al., 2003) Notations α: the parameter of Dirichlet prior on the per-document topic distribution β: the parameter of Dirichlet prior on the per-topic word distribution θ i : the topic distribution for document i φ i : the word distribution for topic k z ij : the topic for the jth word in document i w ij : the specific word The generative process θ i Dir(α), where i {1,, M} (1) φ k Dir(β), where k {1,, K } (2) For each of the words w ij, where j {1,, N i } (3) z ij Multinomial(θ i ), w ij Multinomial(φ zij ) (4)

5 Online LDA (Hoffman et al. (2010)) pros less memory faster convergence does not scale to large number of topics

6 Hybrid Stochastic-MCMC Inference Sampling: a second source of stochasticity for the gradient Marginalize out the topic proportions θ d Corpus-level global variables: K topic-word distributions β 1,, β K over the V-dimensional vocabulary. Document-level local variables: For a document d of length N d θ d z d = (z d1,, z dnd ) variational distribution q(z 1,, z D, β 1, β K ) = d q(z d) k q(β k). q(z d ) is a single distribution over the K N d possible topic configurations The VB lower bound log p(w α, η) d E q log [ p(z d α) i ] β zdi w di + E q log p(β k η)+h(q) k (5)

7 The optimal variational distribution over topic configurations for a document, holding all other variational distribution fixed q (z d ) exp { E q( zd )[log p(z d α)p(w d z d, β)] } (6) Γ(K α) Γ(α + i = I z di =k) (7) Γ(K α + N d ) Γ(α) The hyper parameter of the optimal variational distribution over topic-word distribution, holding the other distributions fixed λ kw = η + E q [I zdi =ki wdi =w] (8) d i k

8 Online Stochastic Inference for λ kw Recast the variational LB as a summation over per-document terms l d l d = ) w (E q [N dkw ] + 1D (η λ) E q [log β kw ]+ ( 1 log Γ( λ kw ) ) log Γ(λ kw ), (9) D w w [ ] D d λ k l d = E B d B λ k l d E q [N dkw ] = i E q[i zdi =ki wdi =w] The natural gradient (Hoffman et al., 2010) E q [N dkw ] + 1 D (η λ kw) (10) faster convergence cheaper computation Issue: the natural gradient cannot be directly evaluated (with a combinatorial number of topic configurations z d ).

9 MCMC within Stochastic Inference Solution: use MCMC to sample z d from q (z d ) and the empirical average to estimate E q [N dkw ] q (z di = k z\i) (α + j i I z j =k) exp{e q [log β kwdi ]} E q [N dkw ] N kw = 1 S s d B i [I zdi s =k I wdi =w] Impacts from MCMC 1 add stochasticity to the gradient 2 allows using collapsed objective function that does not represent θ d 3 provides a sparse estimate of the gradient (for many words and topics, the estimate of E q [N dkw ] will be zero)

10 Previous variational methods lead to dense update to KV topic parameters, which is expensive when K and V are large. Algorithm 1 exploit the sparsity exhibited by samples from q.

11 Two sources of zero-mean noise in constructing an approximate gradient for VB Subsampling the data (SS) Monte Carlo inference (MC) SAEM (Delyon et al., 1999): EM + MC Kuhn & Lavielle (2004): extends SAEM to MCMC estimates online EM (Cappé & Moulines, 2009): EM + SS Collapsed VB (Teh et al. 2006) also analytically marginalizes over θ d, but still maintains a fully factorized distribution over z d The prosed method does not restrict to such factored distributions, hence reduces bias Parallelization

12 Evaluation Held-out probability: calculates the marginal likelihood for held out documents using the left-to-right sequential sampling (Wallach et al. 2009; Buntine, 2009) Topic coherence measures the semantic quality of a topic by approximating the experience of a user viewing the W most probable words for the topic (Mimno et al., 2011) D(w): the number of document containing one or more token of type w D(w i, w j ): the number of documents containing at least one token of w i and w j C(W ) = W i=2 i 1 j=1 log D(w i,w j )+ɛ D(w j ) C(W ) is related to point-wise mutual information (Newman et al., 2010) Wallclock time

13 Dataset Science/Nature/PNAS articles 350, 00 research articles Vocabulary size: 19, % articles for training, 10% for testing Pre-1922 books 1.2 million books 33 billion words

14 Comparison to Online VB (Hoffman et al. 2010) Each iteration consists of a mini-batch of 100 documents the number of coordinate ascent steps in VB is equal to the number of Gibbs sweeps both methods use the same learning schedule Standard online VB takes time linear in K

15 Comparison to Online VB (Hoffman et al. 2010) The entropy of the topic distributions: the proposed method: 6.8 ± 0.46 online VB: 6.0 ± 0.58 This result could indicate that coordinate ascent over the local variables for online LDA is not converging?

16 Comparison to Sequential Monte Carlo (Ahmed et al 2012) The SMC sampler sweeps through each document the same number of times as the sampled online algorithm The learning rate schedule allows sampled online inference to forget its initial topics SMC weights all documents equally

17 Effect of parameter settings Number of samples: B + S Topic-word smoothing: η Forgetting factors learning rate Size of corpus D

18 scalability