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Proceedings of the Seventh Italian Conference on Computational Linguistics CLiC-it 2020

 | 
Felice Dell'Orletta
, 
Johanna Monti
, 
Fabio Tamburini

Contributed Papers

Topic Modelling Games

Rocco Tripodi

Résumé

This paper presents a new topic modelling framework inspired by game theoretic principles. It is formulated as a normal form game in which words are represented as players and topics as strategies that the players select. The strategies of each player are modelled with a probability distribution guided by a utility function that the players try to maximize. This function induces players to select strategies similar to those selected by similar players and to choice strategies not shared with those selected by dissimilar players. The proposed framework is compared with state-of-the-art models demonstrating good performances on standard benchmarks.

Texte intégral

1. Introduction

1Topic modeling is a technique that discovers the underlying topics contained in a collection of documents (Blei 2012; Griffiths and Steyvers 2004). It can be used in different tasks of text classification, document retrieval, and sentiment analysis, providing together vector representations of words and documents. State-of-the-art systems are based on probabilistic (Blei, Ng, and Jordan 2003; Mcauliffe and Blei 2008; Chong, Blei, and Li 2009) and neural networks models (Bengio et al. 2003; Hinton and Salakhutdinov 2009; Larochelle and Lauly 2012; Cao et al. 2015). A different perspective based on game theory is proposed in this article.

2The use of game-theoretic principles in machine learning (Goodfellow et al. 2014), pattern recognition (Pavan and Pelillo 2007) and natural language processing (Tripodi, Vascon, and Pelillo 2016; Tripodi and Navigli 2019) problems is developing a promising field of research with the development of original models. The main difference between computational models based on optimization techniques and game-theoretic models is that the former tries to maximize (minimize) a function (that in many cases is non-convex) and the latter tries to find the equilibrium state of a dynamical system. The equilibrium concept is useful because it represents a state in which all the constraints of a given system are satisfied and no object of the system has an incentive to deviate from it, because a different configuration will immediately lead to a worse situation in terms of payoff and fitness, at object and system level. Furthermore, it is guaranteed that the system converges to a mixed strategy Nash equilibrium (Nash 1951). So far, game-theoretic models have been used in classification and clustering tasks (Pavan and Pelillo 2007; Tripodi and Pelillo 2017). In this work, it is proposed a game-theoretic model for inferring a low dimensional representation of words that can capture their latent semantic representation.

3In this work, topic modeling is interpreted as a symmetric non-cooperative game (Weibull 1997) in which, the words are the players and the topics are the strategies that the players can select. Two players are matched to play the games together according to the co-occurrence patterns found in the corpus under study. The players use a probability distribution over their strategies to play the games and obtain a payoff for each strategy. This reward helps them to adjust their strategy selection in future games, considering what strategy has been effective in previous games. It allows concentrating more mass on the strategies that get high reward. The underlying idea to model the payoff function is to create two influence dynamics, the first one forces similar players (words that appear in similar contexts) to select similar strategies; the second one forces dissimilar players (words that do not share any context) to select different strategies. The games are played repeatedly until the system converges, that is, the difference among the strategy distributions of the players at time t and at time t-1 is under a small threshold. The convergence of the system corresponds to an equilibrium, a situation in which there is an optimal association of words and topics.

2. Related Work

4Hofmann (1999) proposed one of the earliest topic models, probabilistic Latent Semantic Indexing (pLSI). It represents each word in a document as a sample from a mixture model, where topics are represented as multinomial random variables and documents as a mixture of topics. Latent Dirichlet Allocation (LDA) (Blei, Ng, and Jordan 2003), the most widely used topic model, is a generalization of pLSI that introduces Dirichlet priors for both the word multinomial distributions over topics and topic multinomial distributions over documents. This line of research has been developed building on top of LDA different features to infer correlations among topics (Lafferty and Blei 2006) or to model jointly words and labels in a supervised way (Mcauliffe and Blei 2008).

5Topic models based on neural network principles have been introduced with the neural network language model proposed in (Bengio et al. 2003). This paradigm is very popular in NLP and many topic models are based on it because with these techniques it is possible to obtain a low-dimensional representation of the data. In particular, auto-encoders (Ranzato and Szummer 2008), Boltzmann machines (Hinton and Salakhutdinov 2009) and autoregressive distributions (Larochelle and Lauly 2012) have been used to model documents with layer-wise neural network tools. Neural Topic Model (NTM; (Cao et al. 2015)) tries to overcome some limitations of classical topic models, such as the initialization problem and the generalization to n-grams. It exploits word embedding to represent n-grams and uses backpropagation to adjust the weights of the network between the embedding and the word-topic and document-topic layers. A general framework for topic modeling based also on neural networks is Sparse Contextual Hidden and Observed Language AutoencodeR (SCHOLAR; (Card, Tan, and Smith 2018)). It allows using covariates to influence the topic distributions and labels to include supervision. As Sparse Additive GEnerative models (SAGE; (Eisenstein, Ahmed, and Xing 2011)) it can produce sparse topic representations but differently from it and Structural Topic Model (STM; (Roberts et al. 2014)) it can easily consider a larger set of metadata. A graphical topic model was proposed by Gerlach et al. (2018). In this framework, the task of finding topical structures is interpreted as the task of finding communities in complex networks. It is particularly interesting because it shows analogies with traditional topic models and overcomes some of their limitations such as the bound with a Bayesian prior and the need to specify the number of topics in advance.

3. Topic Modelling Games

Normal-form games consist of a finite set of players N=(1,..,n), a finite set of pure strategies, Si={1,...,mi} for each player Image 100000000000002D0000000E18124F372A361E3C.jpgand a payoff (utility) function Image 1000000000000054000000109A0D311188FE9EA9.jpg, that associates a payoff to each combination of strategies S = S1 x S2  x ... x Sn. The payoff function does not depend only on the strategy chosen by a single player but by the combination of strategies played at the same time by the players. Each player tries to maximize the value of ui. Furthermore, in non-cooperative games the players choose their strategies independently, considering what other players can play and trying to find the best response to the strategy of the co-players. Nash equilibria (Nash 1951) represent the key concept of game theory and can be defined as those strategy combinations in which each strategy is a best response to the strategy of the co-player and no player has the incentive to unilaterally deviate from them because there is no way to do better. In addition to play pure strategies, that correspond to selecting just one strategy from those available in Si , a player i can also use mixed strategies, which are probability distributions over pure strategies. A mixed strategy over Si is defined as a vector Image 100000000000008A0000001333BCB50E1D12D25B.jpg, such that xj≥0 and Image 100000000000004E0000001AA6F91E52603FE541.jpg. In a two-player game, a strategy profile can be defined as a pair (xixj). The expected payoff for this strategy profile is computed as:

Image 10000000000000A600000016CCA7B0796E04EC52.jpg

where Aij is the mi x mj payoff matrix between player i and j.

6Evolutionary game theory (Weibull 1997) has introduced two important modifications: 1. the games are played repeatedly, and 2. the players update their mixed strategy over time until it is not possible to improve the payoff. The players, with these two modifications, can develop an inductive learning process, that allows them to learn their strategy distribution according to what other players are selecting. The payoff corresponding to the h-th pure strategy is computed as:

Image 10000000000000BD00000035BBBDB1C1980B32D0.jpg     (1)

7The average payoff of player i is calculated as:

Image 100000000000008700000033765F70A63D49F55C.jpg     (2)

8To find the Nash equilibrium of the game, it is common to use the replicator dynamics equation (Weibull 1997). It allows better than average strategies to grow at each iteration. It can be considered as an inductive learning process, in which the players learn from past experiences how to play their best strategy. It is important to notice that each player optimizes its individual strategy space, but this operation is done according to what other players simultaneously are doing so the local optimization is the result of a global process.

Data Preparation

9The players of the topic modelling games are the words v = (1,…,n) in the vocabulary V of the corpus under analysis and the strategies S = (1,…,m) are the topics to extract from the same corpus. The strategy space xi of each player i is represented as a probability distribution that can be interpreted as the mixture of topics typically used in topic modeling. The interactions among the players are modeled using the n x n adjacency matrix (W) of an undirected weighted graph. Each entry wij encodes the similarity between two words. The strategy space of the games can be represented as a n x n matrix X, where each row represents the probability distribution of a player over its m strategies (topics that have to be extracted from the corpus).

Payoff Function and System Dynamics

10The payoff function of the game is constructed exploiting the information stored in W. This matrix gives us the structural information of the corpus. It allows us to select the players with whom each player is playing the games, indicated with the presence of an edge between two nodes (players), and to quantify the level of influence that each player has on the other, indicated with the weight on each edge. The absence of an edge in this graph indicates that two words are distributional dissimilar. Using these three sources of information we model a payoff function that forces similar players to choose similar strategies (topics) and dissimilar players to choose different ones. The payoff of a player is calculated as,

Image 100000000000011D000000369A7A699839514115.jpg     (3)

where the first summation is over all the ni direct neighbors of player i that are the players with whom i share some similarity and the second summation is over the negi negative players of player i, that are players with whom player i does not share any similarity. With the first summation player i will negotiate with its neighbors a correlated strategy (topic), with the second he will deviate from the strategies chosen by negative players, this is done by subtracting the payoff that i would have gained if these negative players would have been his neighbors. The negative players are sampled from V according to frequency, in the same way, negative samples are selected in word embeddings models (Mikolov et al. 2013; Tripodi and Pira 2017). The equation that gives us the probability of selecting a word as negative is:

Image 10000000000000BF00000032A869BCA5A31A7E29.jpg     (4)

where f(wi) is the frequency of word wi. Since the similarity with negative players is 0 we introduced the parameter Image 100000000000000700000008A70B892EA5FEF5A5.jpgto weight their influence and set it to (A>0). The number of negative players, negi, is set to ni (number of neighbours of player i ).

Once the players have played all the games with their neighbors and negative players, the average payoff of each player can be calculated with Equation (2). The payoff is higher when two words are highly correlated and have a similar mixed strategy. For this reason the replicator dynamics equation (Weibull 1997) is used to compute the dynamics of the system. It pushes the players to be influenced by the mixed strategy of the co-players. This influence is proportional to the similarity between two players (Aij). Once the influence dynamics do not affect the players the Nash equilibrium of the system is reached. The stopping criteria of the dynamics and are: 1. the maximum number of iterations (105); and 2. the minimum difference between two different iterations (10-3) that is calculated as Image 10000000000000C30000001EBF87EE7BCFD1A9C0.jpg.

4. Experimental Results

11In this section, we evaluate TMG and compare it with state-of-the-art systems.

4.1 Data and Setting

12The datasets used to evaluate TMG are 20 Newsgroups1 (20NG) and NIPS2. 20NG is a collection of about 20,000 documents organized into 20 different classes. NIPS is composed of about 1,700 NIPS conference papers published between 1987 and 1999 with no class information. Each text was tokenized and lowercased. The stop-words were removed and the vocabulary was constructed considering the 1000 and 2000 most frequent words in 20NG and NIPS, respectively. This choice is in line with previous work (Card, Tan, and Smith 2018). To keep the model as simple as possible, the tf-idf weighting was used to construct the feature vectors of the words and the cosine similarity was employed to create the adjacency matrix A. It is important to notice here that other sources of information can be easily included at this stage, derived from pre-trained word embeddings, syntactic structures or document metadata. Then A is sparsified taking only the r nearest neighbours of each node. r is calculated as r = log(n) this operation reduces the computational cost of the algorithm and guarantees that the graph remains connected (Von Luxburg 2007).

  • 3 Experimentally it was also observed that using a Dirichlet distribution to initialize the strategy (...)

The strategy space of the players was initialized using a normal distribution to reduce the parameters of the framework3. The last two parameters of the systems concern the stopping criteria of the dynamics and are: 1. the maximum number of iterations (105); and 2. the minimum difference between two different iterations (10^{-3}) that is calculated as Image 10000000000000C00000001E5D7D63E9BBE8C1BE.jpg.

13TMG has been compared with SCHOLAR4, LDA5 and NVDM6. We configured the NVDM network with two encoder layers (500-dimensional) and ReLu non-linearities. SCHOLAR has been configured using a more complex setting that consists in a single layer encoder and a 4-layer generator. LDA has been run with the following parameters: α=50, iterations=1000 and topicthreshold=0.

4.2 Evaluation

14In this section, we compared the generalization performances of TMG and compared them with the models presented in the previous section. For the evaluation we used perplexity (PPL), even if it is has been shown to not correlate with human interpretation of topics (Chang et al. 2009). We computed perplexity on unobserved documents (C), as.

Image 1000000000000126000000325CB730AA416ED9C6.jpg     (5)

where N is the number of documents in the collection C. Low perplexity suggests less uncertainties about the documents. Held out documents represent the 15% of each dataset. Perplexity is computed for 10 topics for the NIPS dataset and 20 topics for the 20 Newsgroups dataset. These numbers correspond to the real number of classes of each dataset.

Table 1: Comparison of the models as perplexity

Dataset

TMG

SCHOLAR

NVDM

LDA

20NG

824

819

927

791

NIPS

1311

1370

1564

1017

15Table 1 shows the comparison of perplexity. As reported in previous work (Card, Tan, and Smith 2018), it is difficult to achieve a lower perplexity than LDA. The results in these experiments follow the same pattern, with LDA that has the lowest perplexity, TMG, and SCHOLAR that have similar results, and NVDM that performs slightly worse on both datasets.

4.3 Topic Coherence and Interpretability

16It has been shown that perplexity does not necessarily correlate well with topic coherence (Chang et al. 2009; Srivastava and Sutton 2017). For this reason, we evaluated the performances of our system also on coherence (Chang et al. 2009; Das, Zaheer, and Dyer 2015). The coherence is calculated by computing the relatedness between topic words using the pointwise mutual information (PMI). We used Wikipedia (2018.05.01 dump) as corpus to compute co-occurrence statistics using a sliding window of 5 words on the left and on the right of each target word. For each topic, we selected the 10 words with the highest mass. Then we calculated the PMI among all the words pair and finally compute the coherence as the arithmetic mean of all these values. This metric has been shown to correlate well with human judgments (Lau, Baldwin, and Cohn 2017). We used two different sources of information for the computation of the PMI: one is internal and corresponds to the dataset under analysis; the other one is external and is represented by the English Wikipedia corpus.

Internal PMI

17Figure 1 presents the PMI values of the different models computed on the two corpora. As it is possible to see from figure 1a, TMG has a low PMI compared to all other systems on the 20 Newsgroups dataset when there are few topics to extract (i.e.: 2 and 5). The situation changes drastically when the number of topics increases. In fact, it has the highest performances on this dataset when extracts 10, 20, 50, 100 topics. The performances of NDVM and SCHOLAR are similar and follow a decreasing pattern, with very high values at the beginning. On the contrary, the performances of LDA follow an opposite pattern this model seems to work better when the number of topics to extract is high. On NIPS (Figure 1b) the performances of the systems are similar to those on 20 Newsgroups. The only exception is that TMG has always the highest PMI and seems to behave better also when the number of topics to extract is high. This probably because the number of words in NIPS is higher and for this, it is reasonable to have also a higher number of topics. This is also confirmed from a qualitative analysis of the topics in Section 4.4, where it is demonstrated that with low values of k it is possible to extract general topics and increasing its value it is possible to extract more specific ones.

18In general, we can find three different patterns in these experiments: 1. NDVM and SCHOLAR work well on extracting a low number of topics; 2. LDA works well when it has to extract a large number of topics; 3. TMG works well on extracting a number of topics that is close to the real number of classes in the datasets.

19Another aspect to take into account is the fact that even if TMG has the highest performances, its results have also a high standard deviation. This is due to the stochastic nature of negative sampling.

Sparsity

We compared the sparsity of the word-topics matrices, X, in Figure 3a and 3b, computed as Image 10000000000000780000002E504A3B1F3BFF66C1.jpg. From both figures, we can see that TMG can produce highly sparse representations especially when the number of topics to extract is low. This is a nice feature since it provides more interpretable results. Only SCHOLAR produces more sparse representations when the number of topics to extract is high. Experimentally we also noticed that we can control the sparsity of X, in TMG, increasing the number of iterations of the game dynamics.

4.4 Qualitative Evaluation

  • 7 for space limitation we presented only 15 topics for 20NG

20Examples of topics extracted from 20NG and NIPS are presented in Table 2 and 3, respectively7. The first difference that emerges from these results are the external PMI values. This is due to the fact that the texts in NIPS have a very specific language and for this reason the PMI values are very high. We can also see that TMG groups highly coherent set of words in each topic. We can easily identify in Table 2 the topics in which the dataset is organized and especially: talk.politics.midleast, alt.atheism, comp.graphics, soc.religion.christian, talk.politics.misc, rec.motorcycles, sci.crypt, talk.politics.guns, rec.sport.hockey, sci.space, talk.politics.misc.

Table 2

turks

schneider

drive

vms

god

intellect

bike

providing

fbi

gun

team

space

male

tim

amateur

soviet

allan

ide

disclaimer

jesus

banks

ride

encryption

compound

firearms

game

orbit

gay

israel

georgia

turkish

morality

scsi

vnews

christians

gordon

riding

clipper

batf

guns

play

shuttle

men

israeli

intelligence

armenian

keith

controller

vax

christ

surrender

dod

key

fire

criminals

season

launch

sexual

arab

ai

armenia

atheists

drives

necessarily

christianity

univ

bikes

escrow

waco

crime

hockey

earth

percentage

jews

programs

passes

moral

mb

represents

bible

pittsburgh

motorcycle

crypto

children

weapons

league

mission

study

arabs

michael

roads

political

disk

views

christian

significant

bmw

keys

koresh

criminal

nhl

flight

sex

policy

radio

armenians

pasadena

isa

expressed

faith

hospital

honda

chip

gas

violent

players

nasa

apparent

war

adams

argic

objective

bus

news

church

level

road

secure

branch

weapon

cup

moon

showing

land

ignore

proceeded

animals

floppy

poster

belief

blood

advice

wiretap

started

armed

stanley

solar

women

north

occur

29.71

15.27

12.7

11.72

10.79

10.18

8.94

8.93

8.55

7.52

7.45

7.14

6.92

6.21

6.13

Best topics (each topic is represented on the columns) extracted from 20 Newsgroup using TMG (setting k = 20) ordered using external PMI (bottom row).

Table 3

ocular

dendrites

oscillatory

crowdsourcing

kaiming

retina

auditory

graph

disturbances

lifted

eye

dendritic

oscillations

crowds

shaoqing

photoreceptor

sound

edges

plant

propositional

fovea

soma

oscillators

workers

xiangyu

retinal

sounds

graphs

controllers

predicate

dominance

dendrite

oscillator

worker

jian

vertebrate

cochlear

optimisation

controller

grounding

saccades

axonal

oscillation

labelers

yangqing

schulten

ear

edge

disturbance

predicates

saccadic

axons

synchronization

crowd

karen

photoreceptors

hearing

vertices

plants

domingos

fixation

nmda

decoding

turk

sergey

ganglion

ears

optimise

activate

clauses

foveal

pyramidal

locking

wisdom

trevor

kohonen

acoust

optimising

activated

compilation

eyes

somatic

synchronize

expertise

sergio

bipolar

tone

optimised

activating

formulas

saccade

axon

synchronized

dawid

jitendra

visualizing

cochlea

vertex

activates

logical

304.85

283.66

276.39

230.5

218.51

196.86

176.75

146.3

146.25

145.84

Topics extracted from NIPS using TMG (setting k = 10) ordered using external PMI (bottom row).

21We can also easily identify from Table 3 highly coherent topics, related to optic, signal analysis, optimization, crowdsourcing, audio, graph theory and logics. We noticed from these topics that they are general and that it is possible to discover more specific topics increasing the number of topics to extract. For example, we discovered topics related to topic modelling and generative adversarial networks.

5. Conclusion and Future Work

22In this paper, it is presented a new topic modeling framework based on game-theoretic principles. The results of its evaluation show that the model performs well compared to state-of-the-art systems and that it can extract topically and semantically related groups of words. In this work, the model was left as simple as possible to assess if a game-theoretic framework itself is suited for topic modeling. In future work, it will be interesting to introduce the topic-document distribution and to test it on classification tasks and covariates to extract topics using different dimensions, such as time, authorship, or opinion. The framework is open and flexible and in future work, it will be tested with different initializations of the strategy space, graph structures, and payoff functions. It will be particularly interesting to test it using word embedding and syntactic information.

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Notes

1 http://qwone.com/~jason/20Newsgroups/

2 https://cs.nyu.edu/~roweis/data.html

3 Experimentally it was also observed that using a Dirichlet distribution to initialize the strategy space with different α parameters did not affect much the performances of the model.

4 https://github.com/dallascard/scholar

5 http://mallet.cs.umass.edu

6 https://github.com/ysmiao/nvdm

7 for space limitation we presented only 15 topics for 20NG

Auteur

Sapienza NLP Group, Department of Computer Science, Sapienza University of Rome – tripodi@di.uniroma1.it

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