Diversity-Promoting Bayesian Learning of Latent Variable Models

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1 Diversity-Promoting Bayesian Learning of Latent Variable Models Pengtao Xie 1, Jun Zhu 1,2 and Eric Xing 1 1 Machine Learning Department, Carnegie Mellon University 2 Department of Computer Science and Technology, Tsinghua University 1

2 Latent Variable Models (LVMs) Patterns Hidden Markov Model, Topic Models, Restricted Boltzmann Machine, Deep Belief Network, Factor Analysis, Neural Network, Sparse Coding, Matrix Factorization, Distance Metric learning, Principal Component Analysis, etc. 2

3 Patterns and Components Latent Patterns Behind Data Themes in Documents Components in LVMs Topic Models Politics Obama Constitution Government Economics GDP Bank Marketing Education University Knowledge Student Groups in Images Gaussian Mixture Model Tiger Car Food 3

4 Precision Training Time Num. of Docs. Challenges How to capture infrequent patterns? Category popularity (measured by #docs labeled with each category) in Wikipedia (Partalas, 2015) Category ID How to reduce model size (computational complexity) without compromising modeling power? Num. of Components How to prevent overfitting? Num. of Components 4

5 Diversify LVMs Encourage components to be diverse Goals Infrequent-Patterns Extraction Model Size Reduction without Losing Modeling Power Overfitting Reduction [Zou and Adams, 2012] [Xie, et. al, 2015] Non-diversified LVMs Diversified LVMs 5

6 Diversify LVMs in Point Estimation Frequentist-style point estimation of LVMs Component vectors (parameters) are deterministic Single best estimate of parameters Formulated as an optimization problem Diversity-promoting regularization Define a regularizer that encourages components to be diverse Use the regularizer to control LVMs max A LD ( ; A) ( A) Examples: determinantal point process, mutual angular regularizer, inverse covariance [Zou and Adams, 2012] [Xie, et. al, 2015] 6

7 Bayesian Learning of LVMs Component vectors are random variables Infer a posterior distribution over the components p( A D) Advantages over point estimation p( D A) p( A) pd ( ) Alleviates overfitting via model averaging Quantify uncertainty 7

8 Diversify LVMs in Bayesian Learning Prior control: define diversity-biased priors and propagate the diversity to posterior via Bayes rule Mutual Angular Prior p( A D) p( D A) p( A) pd ( ) Angle-based notion of diversity: component vectors are deemed to be more diverse if their pairwise angles are larger. Bayesian network and von Mises-Fisher distribution Facilitate efficient (approximate) posterior inference 8

9 Mutual Angular Prior Decompose a component vector a i into its magnitude g = a i 2 and direction a i = a i /g K Develop a mutual angular prior over A ai i 1 i Bayesian network: the parents of a i are a 1 j j 1 K Joint distribution p A = p(a 1 ) i=2 p(a i pa(a i )) Local probability distribution Encourage larger mutual angles von Mises-Fisher (vmf) distribution p x = C p κ exp (κμ T x) μ 2 = 1 x 2 = 1 p a i pa a i = C p κ exp (κ( i 1 j=1 a j i 1 j=1 a j 2 ) T a i ) a 4 a 3 a 1 a 2 9

10 Mutual Angular Prior (cont d) Prior over magnitude K i i 1 G g pg ( i ) is gamma distribution Put pa ( ) and pg ( ) together Generative process K p( G) p( gi ) i 1 Joint distribution 10

11 Posterior Inference Posterior Inference: compute p(a D) Exact solution is intractable Variational inference: an approximate inference method Minimize the KL divergence between the true posterior and a simple variational distribution Maximize a variational lower bound 11

12 Variational Inference p a i pa a i = C p κ exp (κ( i 1 j=1 a j i 1 j=1 a j 2 ) T a i ) Not amenable for variational inference 12

13 Variational Inference (cont d) p a i pa a i = C p κ exp (κ( i 1 j=1 a j i 1 j=1 a j 2 ) T a i ) Not amenable for variational inference Reparametrize p a i pa a i 13

14 Variational Inference (cont d) p a i pa a i = C p κ exp (κ( i 1 j=1 a j i 1 j=1 a j 2 ) T a i ) Not amenable for variational inference Reparametrize p a i pa a i C p ( x) x p/2 1 I ( x) x / 2 p/2 (2 ) Ip/2 1( x) v v k 0 x 2 /4 k! ( v k 1) k 14

15 Variational Inference (cont d) Upper bound of the log of partition function Z i = 1/C p (κ i 1 j=1 a j 2 ) Lower bound of E q(a) log p(a) 15

16 Diversity-Promoting Bayesian Learning of MoE 16

17 Results Non-diversified Diversified Diversification improves classification accuracy 17

18 Results Non-diversified Diversified Diversification reduces model size without losing modeling power 18

19 Results Frequent Infrequent Diversification effectively captures infrequent patterns 19

20 Conclusions Promoting diversity in Bayesian latent variable models Capture infrequent patterns Reduce model size without sacrificing modeling power Prevent overfitting Mutual angular prior Angle-based notion of diversity Bayesian network and von Mises-Fisher distribution Facilitate approximate posterior inference Variational inference Reparametrize local probability distribution Derive upper bound of the log partition function Experiments corroborate the efficacy of our methods 20

21 Thank you! 21

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