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Online Inference of Topics with Latent Dirichlet Allocation Kevin R. Canini Lei Shi Thomas L. Griﬃths Computer Science Division Helen Wills Neuroscience Institute Department of Psychology University of California University of California University of California Berkeley, CA 94720 Berkeley, CA 94720 Berkeley, CA 94720 kevin@cs.berkeley.edu lshi@berkeley.edu tom griffiths@berkeley.edu Abstract arrive in a continuous stream, and decisions must be made on a regular basis, without waiting for future documents to arrive. In these settings, repeatedly run- Inference algorithms for topic models are typ- ning a batch algorithm can be infeasible or wasteful. ically designed to be run over an entire col- lection of documents after they have been In this paper, we explore the possibility of using online observed. However, in many applications of inference algorithms for topic models, whereby the rep- these models, the collection grows over time, resentation of the topics in a collection of documents making it infeasible to run batch algorithms is incrementally updated as each document is added. repeatedly. This problem can be addressed In addition to providing a solution to the problem of by using online algorithms, which update es- growing document collections, online algorithms also timates of the topics as each document is open up diﬀerent routes for parallelization of infer- observed. We introduce two related Rao- ence from batch algorithms, providing ways to draw Blackwellized online inference algorithms for on the enhanced computing power of multiprocessor the latent Dirichlet allocation (LDA) model – systems, and diﬀerent tradeoﬀs in runtime and perfor- incremental Gibbs samplers and particle ﬁl- mance from other algorithms. ters – and compare their runtime and perfor- We discuss algorithms for a particular topic model: la- mance to that of existing algorithms. tent Dirichlet allocation (LDA) (Blei et al., 2003). The state space from which these algorithms draw samples is deﬁned at time i to be all possible topic assignments1 INTRODUCTION to each of the words in the documents observed up to time i. The result is a Rao-Blackwellized samplingProbabilistic topic models are often used to analyze scheme (Doucet et al., 2000), analytically integratingcollections of documents, each of which is represented out the distributions over words associated with topicsas a mixture of topics, where each topic is a proba- and the per-document weights of those topics.bility distribution over words. Applying these mod-els to a document collection involves estimating the The plan of the paper is as follows. Section 2 intro-topic distributions and the weight each topic receives duces the LDA model in more detail. Section 3 dis-in each document. A number of algorithms exist for cusses one batch and three online algorithms for sam-solving this problem (e.g., Hofmann, 1999; Blei et al., pling from LDA. Section 4 discusses the eﬃcient im-2003; Minka and Laﬀerty, 2002; Griﬃths and Steyvers, plementation of one of the online algorithms – particle2004), most of which are intended to be run in “batch” ﬁlters. Section 5 describes a comparative evaluationmode, being applied to all the documents once they are of the algorithms, and Section 6 concludes the paper.collected. However, many applications of topic mod-els are in contexts where the collection of documentsis growing. For example, when inferring the topics 2 INFERRING TOPICSof news articles or communications logs, documents Latent Dirichlet allocation (Blei et al., 2003) is widelyAppearing in Proceedings of the 12th International Confe- used for identifying the topics in a set of documents,rence on Artiﬁcial Intelligence and Statistics (AISTATS) building on previous work by Hofmann (1999). In this2009, Clearwater Beach, Florida, USA. Volume 5 of JMLR: model, each document is represented as a mixture ofW&CP 5. Copyright 2009 by the authors. a ﬁxed number of topics, with topic z receiving weight 65
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Online Inference of Topics with Latent Dirichlet Allocation (d)θz in document d, and each topic is a probability dis- Algorithm 1 batch Gibbs sampler for LDAtribution over a ﬁnite vocabulary of words, with word 1: initialize zN randomly from {1, . . . , T }N (z)w having probability φw in topic z. The generative 2: loopmodel assumes that documents are produced by inde- 3: choose j from {1, . . . , N }pendently sampling a topic z for each word from θ(d) 4: sample zj from P (zj |zN j , wN )and then independently sampling the word from φ(z) .The independence assumptions mean that the docu-ment is treated as a bag of words, so word ordering 3.1 BATCH GIBBS SAMPLERis irrelevant to the model. Symmetric Dirichlet priorsare placed on θ(d) and φ(z) , with θ(d) ∼ Dirichlet(α) Griﬃths and Steyvers (2004) presented a collapsedand φ(z) ∼ Dirichlet(β), where α and β are hyper- Gibbs sampler for LDA, where the state space is the setparameters that aﬀect the relative sparsity of these of all possible topic assignments to the words in everydistributions. The complete probability model is thus document. The Gibbs sampler is “collapsed” because the variables θ and φ are analytically integrated out, wi |zi , φ(zi ) ∼ Discrete(φ(zi ) ), i = 1, . . . , N, and only the latent topic variables zN are sampled. φ(z) ∼ Dirichlet(β), z = 1, . . . , T, The topic assignment of word j is sampled according zi |θ(di ) ∼ Discrete(θ(di ) ), i = 1, . . . , N, to its conditional distribution θ(d) ∼ Dirichlet(α), d = 1, . . . , D, (w ) (d )where N is the total number of words in the collection, nzj ,N j + β nzj j j + α j ,N P (zj |zN j , wN ) ∝ , (1)T is the number of topics, D is the number of docu- (·) (d ) j nzj ,N j + W β n·,N j + T αments, and di and zi are, respectively, the documentand topic of the ith word, wi . The goal of inference in where zN j indicates (z1 , . . . , zj−1 , zj+1 , . . . , zN ), W isthis model is to identify the values of φ and θ, given a j (w ) the size of the vocabulary, nzj ,N j is the number ofdocument collection represented by the sequence of N (·)words wN = (w1 , . . . , wN ). Estimation is complicated times word wj is assigned to topic zj , nzj ,N j is theby the latent variables zN = (z1 , . . . , zN ), the topic total number of words assigned to topic zj , nzj j j is (d ) ,Nassignments of the words. Various algorithms have the number of times a word in document dj is assignedbeen proposed for solving this problem, including a (dj )variational Expectation-Maximization algorithm (Blei to topic zj , and n·,N j is the total number of words inet al., 2003) and Expectation-Propagation (Minka and document dj , and all the counts are taken over wordsLaﬀerty, 2002). In the collapsed Gibbs sampling algo- 1 through N , excluding the word at position j itselfrithm of Griﬃths and Steyvers (2004), φ and θ are ana- (hence the N j subscripts).lytically integrated out of the model to collect samples The Gibbs sampling procedure, outlined in Algo-from P (zN |wN ). The use of conjugate Dirichlet priors rithm 1, converges to the desired posterior distribu-on φ and θ makes this analytic integration straightfor- tion P (zN |wN ). This batch Gibbs sampler can beward, and also makes it easy to recover the posterior extended in several ways, leading to eﬃcient onlinedistribution on φ and θ given zN and wN , meaning sampling algorithms for LDA.that a set of samples from P (zN |wN ) is suﬃcient toestimate φ and θ. 3.2 O-LDAExisting inference algorithms provide users with sev-eral options in trading oﬀ bias and runtime. However, A simple modiﬁcation of the batch Gibbs samplermost of these algorithms are designed to be run over an yields an online algorithm presented by Song et al.entire document collection, requiring multiple sweeps (2005) and called “o-LDA” by Banerjee and Basuto produce good estimates of φ and θ. While some ap- (2007). This procedure, outlined in Algorithm 2, ﬁrstplications of these models involve the analysis of static applies the batch Gibbs sampler to a preﬁx of the fulldatabases, more typically, users work with document dataset, then samples the topic of each new word i bycollections that grow over time. In the remainder of conditioning on the words observed so far1 :the paper, we outline three related algorithms that can (w ) i (d ) ibe used for inference in such a setting. nzi ,ii + β nzi ,ii + α P (zi |zi−1 , wi ) ∝ (·) (d ) . (2) i nzi ,ii + W β n·,ii + T α3 ALGORITHMS 1 The o-LDA algorithm as presented by Banerjee and Basu (2007) samples the next topic by conditioning onlyIn this section, we describe a batch sampling algorithm on the topics of the words up to the end of the previousfor LDA. We then discuss ways in which this algorithm document, rather than all previous words. The algorithmcan be extended to yield three online algorithms. presented here is slightly slower, but more accurate. 66
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Canini, Shi, GriﬃthsAlgorithm 2 o-LDA (initialized with ﬁrst σ words) Algorithm 4 particle ﬁlter for LDA 1: sample zσ using batch Gibbs sampler (p) 1: initialize weights ω0 = P −1 for p = 1, . . . , P 2: for i = σ + 1, . . . , N do 2: for i = 1, . . . , N do 3: sample zi from P (zi |zi−1 , wi ) 3: for p = 1, . . . , P do (p) (p) (p) 4: set ωi = ωi−1 P (wi |zi−1 , wi−1 )Algorithm 3 incremental Gibbs sampler for LDA (p) (p) (p) 5: sample zi from P (zi |zi−1 , wi ) 1: for i = 1, . . . , N do 6: normalize weights ω i to sum to 1 2: sample zi from P (zi |zi−1 , wi ) 7: if ω i −2 ≤ ESS threshold then 3: for j in R(i) do 8: resample particles 4: sample zj from P (zj |zij , wi ) 9: for j in R(i) do 10: for p = 1, . . . , P do (p) (p) (p) 11: sample zj from P (zj |zij , wi ) After its batch initialization phase, o-LDA applies (p) 12: set ωi = P −1 for p = 1, . . . , PEquation (2) incrementally for each new word wi ,never resampling old topic variables. For this reason,its performance depends critically on the accuracy ofthe topics inferred during the batch phase. If the doc- If |R(i)| is bounded as a function of i, then the overalluments used to initialize o-LDA are not representative runtime is linear. However, if |R(i)| grows logarith-of the full dataset, it could be led to make poor infer- mically or linearly with i, then the overall runtime isences. Also, because each topic variable is sampled by log-linear or quadratic, respectively. R(i) can also beconditioning only on previous words and topics, sam- chosen to be nonempty only at certain intervals, lead-ples drawn with o-LDA are not distributed according ing to an incremental Gibbs sampler that only rejuve-to the true posterior distribution P (zN |wN ). To rem- nates itself periodically (for example, whenever thereedy these issues, we consider online algorithms that re- is time to spare between observing documents).vise their decisions about previous topic assignments. An alternative approach to frequently resampling pre- vious topic assignments is to concurrently maintain3.3 INCREMENTAL GIBBS SAMPLER multiple samples of zi , rejuvenating them less fre- quently. This option is desirable because it allows theExtending o-LDA to occasionally resample topic vari- algorithm to simultaneously explore several regions ofables, we introduce the incremental Gibbs sampler, an the state space. It is also useful in a multi-processoralgorithm that rejuvenates old topic assignments in environment, since it is simpler to parallelize multiplelight of new data. The incremental Gibbs sampler, samples – dedicating each sample to a single machineoutlined in Algorithm 3, does not have a batch initial- – than it is to parallelize operations on one sample. Anization phase like o-LDA, but it does use Equation (2) ensemble of independent samples from the incremen-to sample topic variables of new words. After each step tal Gibbs sampler could be used to approximate thei, the incremental Gibbs sampler resamples the topics posterior distribution P (zN |wN ); however, if the sam-of some of the previous words. The topic assignment ples are not rejuvenated often enough, they will notzj of each index j in the “rejuvenation sequence” R(i) have the desired distribution. With this motivation,is drawn from its conditional distribution we turn to particle ﬁlters, which perform importance (w ) (d ) nzj ,ij + β nzj j + α j weighting on a set of sequentially-generated samples. ,ij P (zj |zij , wi ) ∝ (·) (d ) . (3) j nzj ,ij + W β n·,ij + T α 3.4 PARTICLE FILTERIf these rejuvenation steps are performed often enough Particle ﬁlters are a sequential Monte Carlo method(depending on the mixing time of the induced Markov commonly used for approximating a probability dis-chain), the incremental Gibbs sampler closely approxi- tribution over a latent variable as observations are ac-mates the posterior distribution P (zi |wi ) at every step quired (Doucet et al., 2001). We can extend the incre-i. Indeed, convergence is guaranteed as the number of mental Gibbs sampler to obtain a Rao-Blackwellizedtimes each zj is resampled goes to inﬁnity, since the al- particle ﬁlter (Doucet et al., 2000), again analyticallygorithm becomes a batch Gibbs sampler for P (zi |wi ) integrating out φ and θ to sample from P (zi |wi ). Thisin the limit. More generally, the incremental Gibbs use of particle ﬁlters is slightly nonstandard, since thesampler is an instance of the decayed MCMC frame- state space grows with each observation.work introduced by Marthi et al. (2002). The choiceof the number of rejuvenation steps to perform deter- The particle ﬁlter for LDA, outlined in Algorithm 4,mines the runtime of the incremental Gibbs sampler. updates samples from P (zi−1 |wi−1 ) to generate sam- 67
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Online Inference of Topics with Latent Dirichlet Allocationples from the target distribution P (zi |wi ) after each is used after particle resampling to restore diversityword wi is observed. It does this by ﬁrst generat- to the particle set in the same way that the incremen- (p)ing a value of zi for each particle p from a proposal tal Gibbs sampler rejuvenates its samples, by choosing (p) (p)distribution Q(zi |zi−1 , wi ). The prior distribution, a rejuvenation sequence R(i) of topic variables to re- (p) (p) sample. The length of R(i) can be chosen to trade oﬀP (zi |zi−1 , wi−1 ), is typically used for the proposal runtime against performance, and the variables to bebecause it is often infeasible to sample from the pos- (p) (p) resampled can be randomly selected either uniformlyterior, P (zi |zi−1 , wi ). However, since zi is drawn or using a decayed distribution that favors more recentfrom a constant, ﬁnite set of values, we can use the history, as in Marthi et al. (2002). While a uniformposterior, which is given by Equation (2) and min- schedule visits earlier sites more overall, using a distri-imizes the variance of the resulting particle weights bution that approaches zero quickly enough for sites(Doucet et al., 2000). Next, the unnormalized impor- in the past ensures that in expectation, each site istance weights of the particles are calculated using the sampled the same number of times.standard iterative equation (p) ωi (p) (p) P (wi |zi , wi−1 )P (zi |zi−1 ) (p) 4 EFFICIENT IMPLEMENTATION (p) ∝ (p) (p) (4) ωi−1 Q(zi |zi−1 , wi ) In order to be feasible as an online algorithm, the par- (p) = P (wi |zi−1 , wi−1 ). (5) ticle ﬁlter must be implemented with an eﬃcient data representation. In particular, the amount of time itThe weights are then normalized to sum to 1. Equa- takes to incrementally process a document must nottion (5) is designed so that after the weight normaliza- grow with the amount of data previously seen. In ini-tion step, the particle ﬁlter approximates the posterior tial implementations of the algorithm, individual par-distribution over topic assignments as follows: ticles were represented as linear arrays of topic assign- P ment values, consistent with the interpretation of the P (zi |wi ) ≈ (p) (p) ωi 1zi (zi ), (6) state space zi = (z1 , . . . , zi ) as a sequence of variables p=1 stored in an array. It was found that nearly all of the computing time was spent resampling the parti-where 1zi (·) is the indicator function for zi . As P → cles (line 8 of Algorithm 4). This is due to the fact∞, the right side converges to the left side, since zi that when a particle is resampled more than once,can assume only a ﬁnite number of values. the na¨ implementation makes copies of the zi ar- ıve ray for each child particle. Since these structures growOver time, the weights assigned to particles diverge linearly with the observed data and resampling is per-signiﬁcantly, as a few particles come to provide a sig- formed at a roughly constant rate, the total time spentniﬁcantly better account of the observed data than the resampling particles grows quadratically.others. Resampling addresses this issue by producing anew set of particles that are more highly concentrated This problem can be alleviated by using a shared rep-on states with high weight whenever the variance of the resentation of the particles, exploiting the high degreeweights becomes large. A standard measure of weight of redundancy among particles with common lineages.variance is an approximation to the eﬀective sample When a particle is resampled multiple times, the re-size, ESS ≈ ω −2 , and a threshold can be expressed sulting copies all share the same parent particle andas some proportion of the number of particles, P . implicitly inherit its zi vector as their history of topic assignments. Each particle maintains a hash table thatThe simplest form of resampling is to draw from is used to store the diﬀerences between its topic as-the multinomial distribution deﬁned by the normal- signments and its parent’s; consequently, the compu-ized weights. However, more sophisticated resampling tational complexity of the resampling step is reducedmethods also exist, such as stratiﬁed sampling (Kita- from quadratic to linear. As illustrated in Figure 1,gawa, 1996), quasi-deterministic methods (Fearnhead, the particles are thus stored as a directed tree2 , with2004), and residual resampling (Liu and Chen, 1998), parent-child relationships indicating the hierarchy ofwhich produce more diverse sets of particles. Residual inheritance for topic variables.resampling was used in our evaluations. Whenever the (p)particles are resampled, their weights are all reset to To look up the value zi of topic assignment i in par-P −1 , since each is now a draw from the same distri- ticle p, the particle’s hash table is consulted ﬁrst. Ifbution and the previous weights are reﬂected in their the value is missing, the particle’s parent’s hash tablerelative resampling frequencies. is checked, recursing up the tree towards the root andAs in the resample-move algorithm of Gilks and 2 More precisely, a forest of directed trees, since it isBerzuini (2001), Markov chain Monte Carlo (MCMC) possible that not all particles share a common ancestor. 68
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Canini, Shi, Griﬃths !"#$%&()* idence conﬁrms that the time overhead of maintaining !"#$% &(# )*!+ , -.()- / a directed tree of hash tables is negligible compared to 0 -1!2$- / the increase in speed and decrease in memory usage it / -3(.1!)4- 5 5 -!"616$7- 0 aﬀords. Speciﬁcally, by reducing the resampling step 8 -+97!":- 0 ; -&9$($- , to have linear runtime, this implementation detail is < -)- , the key to making the particle ﬁlter feasible to run. = -#(.&- / > -)9$- , ? -1!"$- / 5 EVALUATION In online topic modeling settings, such as news article !"#$%&()+ ,-.)!"#$%&(/ clustering, we care about two aspects of performance: !"#$% &(# )*!+ !"#$% &(# )*!+ 8 -+97!":- 5 , -.()- 0 the quality of the solutions recovered and runtime. As ; -&9$($- 0 8 -+97!":- 5 documents arrive and are incrementally processed, we = -#(.&- 0 would like online algorithms to maintain high-quality inferences and to produce topic labels quickly for new documents. Since there is a tradeoﬀ between runtime !"#$%&()0 !"#$%&()1 and inference quality, the algorithms were evaluated !"#$% &(# )*!+ !"#$% &(# )*!+ by comparing the quality of their inferences while con- , -.()- / ; -&9$($- / ? -1!"$- 5 straining the amount of time spent per document. We compared the performance of the three online algo-Figure 1: An example of the “directed tree of hashta- rithms presented in Section 3: o-LDA, the incrementalbles” implementation of the particle ﬁlter. Particle 2 Gibbs sampler, and the particle ﬁlter. Our evaluationis the root, so all other particles descended from it. is a variation of the comparison of o-LDA to other on-Particle 0 directly depends on particle 2, altering the line algorithms by Banerjee and Basu (2007), using thetopics of the words “choosing” and “where”. The other same datasets and performance metric.child of particle 2 is not itself an active sample, but aninactive remnant of an old particle that was not resam- 5.1 DATASETSpled. It is retained because multiple active particles The datasets used to test the algorithms are each adepend on its hashed topic values. If either particle 1 collection of categorized documents. They consist ofor particle 3 is not resampled in the future, the remain- four subsets derived from the 20 Newsgroups corpus3 :ing one will be merged with its parent, maintaining the (1) diff-3 (2995 documents, 7670 word types, 3 cate-bound on the tree depth. gories), (2) rel-3 (2996 documents, 10091 word types, 3 categories), (3) sim-3 (2980 documents, 5950 word types, 3 categories), and (4) subset-20 (1997 docu-eventually terminating when the value is found in an ments, 13341 word types, 20 categories), represent- (p)ancestor’s hash table. To change the value zi , the ing diﬀerent levels of size and diﬃculty, as well asnew value is inserted into particle p’s hash table, and news articles harvested from the Slashdot website: (5)the old value is inserted into each of particle p’s chil- slash-7 (6714 documents, 5769 word types, 7 cate-dren’s hash tables (if they don’t already have an entry) gories) and (6) slash-6 (5182 documents, 4498 wordto ensure consistency. types, 6 categories).In order to ensure that variable lookup is a constant-time operation, it is necessary that the depth of the 5.2 METHODOLOGYtree does not grow with the amount of data. This can Our testing methodology is designed to approximate abe ensured by selectively pruning the tree just after real-world online inference task. For each dataset, thethe resampling step. After resampling is performed, algorithms were given the ﬁrst 10% of the documentsa node is called active if it has been sampled one or to use for initialization. A single sample drawn usingmore times, and inactive otherwise. If an entire sub- the batch Gibbs sampler on this initial set was used totree of nodes is inactive, it is deleted. If an inactive initialize all of the online algorithms. This constitutednode has only one active descendant depending on its the explicit batch initialization phase of o-LDA, andhistory, that descendant’s hash table is merged with the other two online algorithms used the same startingits own. When this operation is performed after each conﬁguration.resampling step, it can be shown that the depth of thetree is never greater than P . Furthermore, merging the 3 Available online at http://people.csail.mit.edu/hash tables takes linear amortized time. Empirical ev- jrennie/20Newsgroups/ 69
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Online Inference of Topics with Latent Dirichlet AllocationAfter the batch initialization set is chosen, the o-LDA clustered a randomly-chosen held-out set consisting ofalgorithm has no remaining parameters and is the 10% of the documents, as a function of the amount offastest of the three online algorithms, since it does the training set that had been observed so far. That is,not rejuvenate its topic assignments. The incremental at regular intervals throughout the training set, eachGibbs sampler has one parameter: the choice of re- algorithm was run on the held-out documents as if theyjuvenation sequence R(i). The particle ﬁlter has two were the next ones to be observed, the nMI score wasparameters: the eﬀective sample size (ESS) threshold, calculated for the held-out documents, and the algo-which controls how often the particles are resampled, rithm was returned to its original state and positionand the choice of rejuvenation sequence. Since the run- in the training set.time and performance of the incremental Gibbs sam-pler and the particle ﬁlter depend on these parameters, 5.3 RESULTSthere is a compromise to be made. We set the param-eters so that these algorithms ran within roughly 6 The results of the training set evaluation are showntimes the amount of time taken by o-LDA on each in Figure 2. The particle ﬁlter and incremental Gibbsdataset. For the incremental Gibbs sampler, R(i) was sampler perform about equally well, with the parti-chosen to be a set of 4 indices from 1 to i chosen uni- cle ﬁlter performing better for some datasets. As ex-formly at random. For the particle ﬁlter, the ESS pected, o-LDA consistently has the lowest score ofthreshold was set at 20 for the diff-3, rel-3, and the three algorithms. The dashed horizontal line insim-3 datasets and at 10 for the subset-20, slash-6, each ﬁgure represents the performance of the batchand slash-7 datasets, |R(i)| was set at 30 for the Gibbs sampler on the entire dataset, which is approx-diff-3, rel-3, and sim-3 datasets and at 10 for the imately the best possible performance an online algo-subset-20, slash-6, and slash-7 datasets, and the rithm could achieve using the LDA model.particular values of R(i) were chosen uniformly at ran- The results of the evaluation on the held-out set aredom from 1 to i. The runtime of these algorithms can shown in Figure 3. For each held-out set, the meanbe chosen to ﬁt any constraints, but we selected one performance of the particle ﬁlter is consistently betterpoint that we felt was reasonable. Indeed, an impor- than that of the incremental Gibbs sampler, which istant strength of these two online algorithms is that consistently better than that of o-LDA. With the ex-they can take full advantage of any amount of comput- ception of the sim-3 and subset-20 datasets, the al-ing power by appropriate choice of their parameters. gorithms’ performances are separated by at least twoThe LDA hyperparameters α and β were both set to standard deviations. Interestingly, the performance ofbe 0.1. The particle ﬁlter was run with 100 particles; all the algorithms on all the held-out document setsto allow the other algorithms the same advantage of does not change signiﬁcantly as more training data ismultiple samples, they were each run 100 times inde- observed. This seems to indicate that a majority ofpendently, with the sample having highest posterior the information about the topics comes from the ﬁrstprobability at each step being used for evaluation. 10% of the documents. In rel-3, performance on the held-out set seems to decrease as more of the trainingSince the datasets are collections of documents with set is observed. This could be because the held-outknown category memberships, we evaluated how well documents are more closely related to those at the be-the clustering implied by the inferred topics matched ginning of the training set than those at the end.the true categories. That is, for each dataset, the num-ber of topics T was set equal to the number of cate- As mentioned earlier, the algorithms’ performancegories, and the documents were clustered according to strongly depends on their parameters; allowing moretheir most frequent topic. Normalized mutual infor- time for rejuvenation of old topic assignments wouldmation (nMI) was used to measure the similarity of improve the performance of the particle ﬁlter and thethis implied partition to the true document categories incremental Gibbs sampler. Table 1 summarizes the(Banerjee and Basu, 2007). Scores are between 0 and total runtimes of each algorithm on each dataset. Al-1, with a perfect match receiving a score of 1. though there is some variation, the incremental Gibbs sampler and particle ﬁlter each take about 6 timesTwo diﬀerent evaluations were made for each algo- longer than o-LDA.rithm on each dataset. First, we evaluated how wellthe algorithms clustered the documents on which they The top ten words from 5 of the 20 topics found by thewere trained. That is, at regular intervals through- particle ﬁltering algorithm on the subset-20 datasetout each dataset, the sample with maximum posterior are listed in Table 2. Although the normalized mutualprobability was drawn, and the quality of the induced information is not as high as that of the batch Gibbsclustering of the documents observed so far was mea- sampler, the recovered topics seem intelligible.sured. Second, we evaluated how well the algorithms We also noticed that the particle ﬁlter used signiﬁ- 70
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Canini, Shi, GriﬃthsFigure 2: nMI traces for each algorithm on each dataset. The algorithms were initialized with the same conﬁg-uration on the ﬁrst 10% of the documents. Each dashed horizontal line represents the nMI score for the batchGibbs sampler on an entire dataset. Solid lines show mean performance over 30 runs, and shading indicates plusand minus one sample standard deviation.Figure 3: nMI traces for each algorithm on held-out test sets, as a function of the amount of the training setobserved. The algorithms were initialized with the same conﬁguration on the ﬁrst 10% of the documents. Eachdashed horizontal line represents the nMI score on the held-out set for the batch Gibbs sampler given the entiretraining set. Solid lines show mean performance over 30 runs, and shading indicates plus and minus one samplestandard deviation. 71
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Online Inference of Topics with Latent Dirichlet Allocation AcknowledgementsTable 1: Runtimes of algorithms in seconds. Numbersin parentheses give multiples of o-LDA runtime. This work was supported by the DARPA CALO project o-LDA inc. Gibbs particle ﬁlter and NSF grant BCS-0631518. The authors thank Jason diff-3 34 185 (5.4) 183 (5.4) Wolfe for helpful discussions and Sugato Basu for providing the datasets and nMI code used for evaluation. rel-3 57 338 (5.9) 251 (4.4) sim-3 32 176 (5.5) 148 (4.6) References subset-20 150 1221 (8.1) 1029 (6.9) slash-6 44 255 (5.8) 256 (5.8) Banerjee, A. and Basu, S. 2007. Topic models over text slash-7 69 420 (6.1) 521(7.6) streams: a study of batch and online unsupervised learn- ing. In Proc. 7th SIAM Int’l. Conf. on Data Mining. Blei, D. M., Griﬃths, T. L., Jordan, M. I., and Tenen- baum, J. B. 2004. Hierarchical topic models and theTable 2: The top 10 words from 5 topics found by the nested Chinese restaurant process. In Advances in Neu- ral Information Processing Systems 16.particle ﬁlter for the subset-20 dataset. Blei, D. M. and Laﬀerty, J. D. 2005. Correlated topic mod- els. In Advances in Neural Information Processing Sys- gun list space god wire tems 18. police mail cost bible ground Blei, D. M., Ng, A. Y., and Jordan, M. I. 2003. Latent guns users nasa moral wiring Dirichlet allocation. Journal of Machine Learning Re- semi email mass choose cable search, 3:993–1022. Doucet, A., de Freitas, N., and Gordon, N., eds. 2001. ﬁre internet rockets christian power Sequential Monte Carlo Methods in Practice. Springer. cops access station jesus neutral Doucet, A., de Freitas, N., Murphy, K. P., and Russell, S. J. revolver unix orbit church nec 2000. Rao-Blackwellised particle ﬁltering for dynamic carry address launch christianity circuit Bayesian networks. In Proc. 16th Conf. in Uncertainty auto security dod absolute box in Artiﬁcial Intelligence. Fearnhead, P. 2004. Particle ﬁlters for mixture models koresh system drink life current with an unknown number of components. Statistics and Computing, 14(1):11–21. Gilks, W. R. and Berzuini, C. 2001. Following a mov- ing target–Monte Carlo inference for dynamic Bayesiancantly less memory than the other online algorithms. models. Journal of the Royal Statistical Society. SeriesThis is due to the shared representation of the par- B (Statistical Methodology), 63(1):127–146.ticles, as discussed in Section 4, which allows more Griﬃths, T. L. and Steyvers, M. 2004. Finding scientiﬁcaccurate inferences to be computed with less memory topics. Proc. National Academy of Sciences of the USA, 101(Suppl. 1):5228–5235.than algorithms that maintain independent samples. Griﬃths, T. L., Steyvers, M., Blei, D. M., and Tenenbaum, J. B. 2004. Integrating topics and syntax. In Advances in Neural Information Processing Systems 17.6 DISCUSSION Hofmann, T. 1999. Probabilistic latent semantic indexing. In Proc. 22nd Annual Int’l. ACM SIGIR Conf. on Re-We have discussed a number of algorithms for the search and Development in Information Retrieval.problem of inferring topics using LDA. We have shown Kitagawa, G. 1996. Monte Carlo ﬁlter and smoother for non-Gaussian nonlinear state space models. Journal ofhow to extend a batch algorithm into a series of Computational and Graphical Statistics, 5(1):1–25.online algorithms, each more ﬂexible than the last. Liu, J. S. and Chen, R. 1998. Sequential Monte CarloOur results demonstrate that these algorithms per- methods for dynamic systems. Journal of the Americanform eﬀectively in recovering the topics used in multi- Statistical Association, 93(443):1032–1044.ple datasets. Latent Dirichlet allocation has been ex- Marthi, B., Pasula, H., Russell, S. J., and Peres, Y. 2002. Decayed MCMC ﬁltering. In Proc. 18th Conf. in Uncer-tended in a number of directions, including incorpora- tainty in Artiﬁcial Intelligence.tion of hierarchical representations (Blei et al., 2004), Minka, T. P. and Laﬀerty, J. D. 2002. Expectation-minimal syntax (Griﬃths et al., 2004), the interests of propagation for the generative aspect model. In Proc.authors (Rosen-Zvi et al., 2004), and correlations be- 18th Conf. in Uncertainty in Artiﬁcial Intelligence.tween topics (Blei and Laﬀerty, 2005). We anticipate Rosen-Zvi, M., Griﬃths, T. L., Steyvers, M., and Smyth, P. 2004. The author-topic model for authors and docu-that the algorithms we have outlined here will natu- ments. In Proc. 20th Conf. in Uncertainty in Artiﬁcialrally generalize to many of these models. In particu- Intelligence.lar, applications of the hierarchical Dirichlet process to Song, X., Lin, C.-Y., Tseng, B. L., and Sun, M.-T. 2005.text (Teh et al., 2006) can be viewed as an analogue to Modeling and predicting personal information dissem-LDA in which the number of topics is allowed to vary. ination behavior. In Proc. 11th ACM SIGKDD Int’l. Conf. on Knowledge Discovery and Data Mining.Our particle ﬁltering framework requires little modiﬁ- Teh, Y. W., Jordan, M. I., Beal, M. J., and Blei, D. M.cation to handle this case, providing an online alter- 2006. Hierarchical Dirichlet processes. Journal of thenative to existing inference algorithms for this model. American Statistical Association, 101(476):1566–1581. 72
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