Learning the Structure of Sum-Product Networks. Robert Gens Pedro Domingos
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1 Learning the Structure of Sum-Product Networks Robert Gens Pedro Domingos
2 w 20 10x O(n) X Y LL PLL CLL CMLL Motivation SPN Structure Experiments Review Learning
3 Graphical Models Representation Inference Learning Compact and expressive Global independence Exponential in treewidth of graph Extremely difficult because: Learning requires inference Approximate inference is unreliable Hidden variables no global optimum
4 Sum-Product Networks Representation Inference Compact and expressive Local independence Linear in number of edges Learning Much easier because exact inference Only weight learning to date
5 This Paper: SPN Structure Learning General-purpose SPN structure learning Discrete or continuous Learns layers of hidden variables Fully leverages context-specific independence Simple and intuitive 1-3 orders of magnitude faster, and more accurate at query time
6 w 20 10x O(n) X Y LL PLL CLL CMLL Motivation SPN Structure Experiments Review Learning
7 Compactly Representable Probability Distributions Graphical Models Sum-Product Networks Existing Tractable Models
8 Compactly Representable Probability Distributions Graphical Models Sum-Product Networks Existing Exact inference in time Tractable linear in network size Models
9 A Univariate Distribution Is an SPN. X
10 A Univariate Distribution Is an SPN. X... Multinomial Gaussian Poisson
11 A Univariate Distribution Is an SPN. X
12 X
13 A Product of SPNs over Disjoint Variables Is an SPN. X Y
14 A Product of SPNs over Disjoint Variables Is an SPN. X Y
15 X Y
16 A Weighted Sum of SPNs over the Same Variables Is an SPN. w 1 w 2 X Y X Y
17 A Weighted Sum of SPNs over the Same Variables Is an SPN. w 1 w 2 Sums out a mixture variable X Y X Y
18 X1 X1 X1 X1 X2 X2 X2 X2 X3 X3 X3 X3 X4 X4 X4 X4 X5 X5 X5 X5 X6 X6 X6 X6
19 All Marginals Are Computable in Linear Time
20 All Marginals Are Computable in Linear Time X Y
21 All Marginals Are Computable in Linear Time w 1 w 2 X Y X Y
22 All Marginals Are Computable in Linear Time X Y X Y
23 All Marginals Are Computable in Linear Time? X Y X Y
24 All Marginals Are Computable in Linear Time? X Y X Y Evidence Evidence
25 All Marginals Are Computable in Linear Time? X Y X Y Evidence Evidence
26 All Marginals Are Computable in Linear Time? X Y X Y Evidence Marginalize Evidence Marginalize
27 All Marginals Are Computable in Linear Time? X Y X Y Evidence Marginalize Evidence Marginalize
28 All Marginals Are Computable in Linear Time = X Y X Y Evidence Marginalize Evidence Marginalize
29 X Y X Y
30 All MAP States Are Computable in Linear Time X Y X Y
31 All MAP States Are Computable in Linear Time max X Y X Y
32 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h)? max X Y X Y
33 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h)? max X Y X Y Evidence Evidence
34 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h)? max X Y X Y Evidence Mode Evidence Mode
35 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h)? max X Y X Y Evidence Mode Evidence Mode
36 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h) = 0.12 max X Y X Y Evidence Mode Evidence Mode
37 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h) = 0.12 max X Y X Y Evidence Mode Evidence Mode
38 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h) = 0.12 max X Y X Y Evidence Mode Evidence Mode
39 All MAP States Are Computable in Linear Time max y,h P (X =0,Y = y, H = h) = 0.12 max X Y X Y Evidence Mode Evidence Mode
40 What Does an SPN Mean? Products = Features Sums = Clusterings
41 What Does an SPN Mean? Products = Features Sums = Clusterings Hay Greenery Truck Sign Flowers Gravel
42 What Does an SPN Mean? Products = Features Sums = Clusterings Hay Greenery Truck Sign Flowers Gravel Calf Cow Bull Person patting Person on bike
43 What Does an SPN Mean? Products = Features Sums = Clusterings Hay Greenery Truck Sign Flowers Gravel Calf Cow Bull Person patting Person on bike Tail Body Legs Head Arms Body Legs Head
44 What Does an SPN Mean? Products = Features Sums = Clusterings Hay Greenery Truck Sign Flowers Gravel Calf Cow Bull Person patting Person on bike Tail Body Legs Head Arms Body Legs Head
45 What Does an SPN Mean? Products = Features Sums = Clusterings Scene decomposition Hay Greenery Truck Sign Flowers Mixture Gravel Calf Cow Bull Person patting Person on bike Pooling Tail Body Legs Head Arms Body Legs Head Part decomposition
46 Special Cases Hierarchical mixture models Thin junction trees (e.g.: hidden Markov models) Non-recursive probabilistic context-free grammars Etc.
47 Weight Learning Generative (Poon & Domingos, UAI 2011) UAI 2011 Best Paper Award Discriminative (Gens & Domingos, NIPS 2012) NIPS 2012 Outstanding Student Paper Award
48 Weight Learning Generative (Poon & Domingos, UAI 2011) UAI 2011 Best Paper Award Discriminative (Gens & Domingos, NIPS 2012) NIPS 2012 Outstanding Student Paper Award Key limitation: Requires a structure
49 w 20 10x O(n) X Y LL PLL CLL CMLL Motivation SPN Structure Experiments Review Learning
50 LearnSPN Recursive algorithm Instances Variables
51 LearnSPN Recursive algorithm Instances Variables
52 LearnSPN Return: Instances Multinomial Gaussian X 1 Poisson
53 Instances Variables
54 Establish approximate independence Instances Variables
55 Establish approximate independence Instances Variables Return: Recurse Recurse Recurse
56 Establish approximate independence Instances If no independence, cluster similar instances Variables Return: Recurse Recurse Recurse
57 Establish approximate independence Instances If no independence, cluster similar instances Variables Return: Return: w 1 w 2 w 3 Recurse Recurse Recurse Recurse Recurse Recurse
58 If no detectable dependencies Instances Variables Return: Fully factorized R... R... R... R... R... distribution
59 If no detectable dependencies Instances Variables Return: Fully factorized distribution
60 Instances Variables If never finds independence Return: w 1 w 7 R... R... R... R... R... Kernel density estimate
61 Instances If never finds independence Variables Return: w 1 w 2 w 7 Kernel density estimate
62 Establish approximate independence Return: Instances Variables If no independence, cluster similar instances Return: w 1 w 2 w 3 Recurse Recurse Recurse Recurse Recurse Recurse
63 Establish approximate independence Return: Instances If no independence, cluster similar instances Return: w 1 w 2 w 3 Recurse Recurse Recurse Variables Recurse Recurse Recurse
64 Establish approximate independence Instances If no independence, cluster similar instances Variables Return: Return: w 1 w 2 w 3 Recurse Recurse Recurse Recurse Recurse Recurse Recurse Recurse
65 Establish approximate independence Instances If no independence, cluster similar instances Variables Return: Return: w 1 w 2 w 3 Recurse Recurse Recurse Recurse Recurse Recurse Recurse Recurse
66 Instances Variables
67 Instances R... R... Variables
68 Instances R... R... R... R... R... Variables
69 Instances R... R... R... R... R... R... R... R... R... Variables
70 Instances R... R... R... R... R... R... R... R... R... Variables High treewidth
71 C Instances C Variables High treewidth
72 Instances C Variables High treewidth
73 Training LearnSPN set Establish approximate independence Return: Instances Variables If no independence, cluster similar instances Return: w 1 w 2 w 3 Recurse Recurse Recurse If V =1, Return: Recurse Recurse Recurse
74 Variable Splits ( ) Pairwise independence test p-value (validation set) X 6 X 7 X 5 X 1 X 4 X 2 X 3
75 Variable Splits ( ) Pairwise independence test p-value (validation set) X 6 X 7 X 5 X 1 X 4 X 2 X 3
76 Variable Splits ( ) Pairwise independence test p-value (validation set) Recurse Recurse X 6 X 5 X 7 X 1 X 2 X 3 X 4
77 Variable Splits ( ) Pairwise independence test p-value (validation set) X 6 X 7 X 5 X 1 X 4 X 2 X 3
78 Instance Clustering (+) X 6 X 7 X 5 X 1 X 4 X 2 X 3
79 Instance Clustering (+) X 1 X 2 X 3 X 4 X 5 X 6 X 7
80 Instance Clustering (+) m 1 m 2 m 3 m 4 m 5 m 6 m 7 X 1 X 2 X 3 X 4 X 5 X 6 X 7
81 Instance Clustering (+) m 1 m 2 m 3 m 4 m 5 m 6 m 7
82 Instance Clustering (+) Online hard EM Naive Bayes mixture model Cluster penalty (validation set) P (V )= X i Learned weights are the mixture priors P (C i ) Y j P (X j C i ) m 1 m 2 m 3 m 4 m 5 m 6 m 7
83 Instance Clustering (+) Online hard EM Naive Bayes mixture model Cluster penalty (validation set) P (V )= X i Learned weights are the mixture priors P (C i ) Y j P (X j C i ) m 1 m 2 m 3 m 4 m 5 m 6 m 7 C 1 C 2 C 3
84 Instance Clustering (+) Online hard EM Naive Bayes mixture model Cluster penalty (validation set) P (V )= X i Learned weights are the mixture priors P (C i ) Y j P (X j C i ) m 1 m 2 m 3 m 4 m 5 m 6 m 7 C 1 C 2 C 3
85 Instance Clustering (+) Online hard EM Naive Bayes mixture model Cluster penalty (validation set) P (V )= X i Learned weights are the mixture priors P (C i ) Y j P (X j C i ) Recurse Recurse Recurse m 3 m 4 m 1 m 2 C 2 C 1 m 5 m 6 C 3 m 7
86 Instance Clustering (+) Online hard EM Naive Bayes mixture model Cluster penalty (validation set) P (V )= X i Learned weights are the mixture priors P (C i ) Y j P (X j C i ) Recurse Recurse Recurse 2 7 m 3 m 4 m 1 m 2 C 2 C 1 m 5 m 6 C 3 m 7
87 LearnSPN Locally Optimizes Likelihood
88 LearnSPN Locally Optimizes Likelihood X i X 1 X 2 X j No loss of likelihood if truly independent X k
89 LearnSPN Locally Optimizes Likelihood X i X 1 X 2 X j No loss of likelihood if truly independent X k Naive Bayes likelihood LearnSPN likelihood w 1 w 2 w 1 w 2 Recurse Recurse
90 w 20 10x O(n) X Y LL PLL CLL CMLL Motivation SPN Structure Experiments Review Learning
91 Experiments 20 Datasets collaborative filtering click-through logs nucleic acid sequences... Representation Learning Inference SPN LearnSPN Exact Bayesian Gibbs WinMine Network Loopy BP Instances 2k-388k Variables Markov Network Della Pietra L1 Gibbs Loopy BP Gibbs Loopy BP Total: 1680 Experiments
92 Learning Results SPN Wins (p=0.05) SPN Ties SPN Losses WinMine Della Pietra L1 Log-likelihood Pseudo Log-likelihood
93 Learning Results SPN Wins (p=0.05) SPN Ties SPN Losses WinMine Della Pietra L1 Log-likelihood Pseudo Log-likelihood
94 Inference
95 P( Query Evidence ) Inference
96 Inference P( Query Evidence ) Q E 10% 30% 20% 30% 30% 30% 40% 30% 50% 30% 30% 0% 30% 10% 30% 20% 30% 40% 30% 50% For each proportion, generate 1000 queries from test set
97 Inference P( Query Evidence ) Q E 10% 30% 20% 30% 30% 30% 40% 30% 50% 30% 30% 0% 30% 10% 30% 20% 30% 40% 30% 50% { WinMine Della Pietra L1 X Gibbs {Loopy BP { X 20 Datasets X { 10 Variable proportions = 1200 Experiments (In addition to 400 for SPNs) For each proportion, generate 1000 queries from test set
98 Inference Accuracy EachMovie, 500 variables SPN Conditional Log-Likelihood WinMine L % 20% 30% 40% 50% Della Pietra Fraction of Query Variables
99 Inference Time Gibbs ms/query WinMine Della Pietra L1 SPN ms/query
100 Inference Time Gibbs ms/query WinMine Della Pietra L SPN ms/query
101 Inference Time Gibbs ms/query WinMine Della Pietra L SPN ms/query
102 Inference Time Gibbs ms/query WinMine Della Pietra L SPN ms/query
103 Inference Time Gibbs ms/query WinMine Della Pietra L SPN ms/query
104 Inference Time Gibbs ms/query WinMine Della Pietra L1 1 SPN is 1-3 orders of magnitude faster SPN ms/query
105 Inference Time Loopy Belief Propagation 1000 Single variable marginals Gibbs ms/query { 100 WinMine Della Pietra L1 1 { X 20 Datasets X 10 WinMine Variable proportions Della Pietra L1 10 = 600 Experiments 10x 30x SPNs had higher conditional marginal log-likelihood on 78% of experiments 0.1 (95% higher CLL vs. Gibbs) SPN ms/query
106 Experiment Conclusions SPN learning accuracy is comparable SPN exact inference is 1-3 orders of magnitude faster, and more accurate Inference does not involve tuning settings or diagnosing convergence Inference takes a predictable amount of time Can now apply SPNs to many domains
107 Future Work Other SPN structure and weight learning algorithms Approximating intractable distributions with SPNs Parallelizing SPN learning and inference Code and supplemental results available at spn.cs.washington.edu/learnspn/
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