Sum-Product Networks: A New Deep Architecture
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1 Sum-Product Networks: A New Deep Architecture Pedro Domingos Dept. Computer Science & Eng. University of Washington Joint work with Hoifung Poon 1
2 Graphical Models: Challenges Bayesian Network Markov Network Restricted Boltzmann Machine (RBM) Sprinkler Rain Grass Wet Advantage: Compactly represent probability Problem: Inference is intractable Problem: Learning is difficult 2
3 Deep Learning Stack many layers E.g.: DBN [Hinton & Salakhutdinov, 2006] CDBN [Lee et al., 2009] DBM [Salakhutdinov & Hinton, 2010] Potentially much more powerful than shallow architectures [Bengio, 2009] But Inference is even harder Learning requires extensive effort 3
4 Learning: Requires approximate inference Inference: Still approximate Graphical Models 4
5 Graphical Models E.g., hierarchical mixture model, thin junction tree, etc. Problem: Too restricted Existing Tractable Models 5
6 This Talk: Sum-Product Networks Compactly represent partition function using a deep network Graphical Models Sum-Product Networks Existing Tractable Models 6
7 Graphical Models Sum-Product Networks Exact Existing inference linear time Tractable in network size Models 7
8 Can compactly represent many more distributions Graphical Models Sum-Product Networks Existing Tractable Models 8
9 Learn optimal way to reuse computation, etc. Graphical Models Sum-Product Networks Existing Tractable Models 9
10 Outline Sum-product networks (SPNs) Learning SPNs Experimental results Conclusion 10
11 Why Is Inference Hard? 1 P X X X X Z ( 1,, N) j 1,, N Bottleneck: Summing out variables E.g.: Partition function Sum of exponentially many products j Z X j j X 11
12 Alternative Representation X 1 X 2 P(X) P(X) = 0.4 I[X 1 =1] I[X 2 =1] I[X 1 =1] I[X 2 =0] I[X 1 =0] I[X 2 =1] I[X 1 =0] I[X 2 =0] Network Polynomial [Darwiche, 2003] 12
13 Alternative Representation X 1 X 2 P(X) P(X) = 0.4 I[X 1 =1] I[X 2 =1] I[X 1 =1] I[X 2 =0] I[X 1 =0] I[X 2 =1] I[X 1 =0] I[X 2 =0] Network Polynomial [Darwiche, 2003] 13
14 Shorthand for Indicators X 1 X 2 P(X) P(X) = 0.4 X 1 X X 1 X X 1 X X 1 X 2 Network Polynomial [Darwiche, 2003] 14
15 Sum Out Variables X 1 X 2 P(X) e: X 1 = 1 P(e) = 0.4 X 1 X X 1 X X 1 X X 1 X 2 Set X 1 = 1, X 1 = 0, X 2 = 1, X 2 = 1 Easy: Set both indicators to 1 15
16 Graphical Representation X 1 X 2 P(X) X 1 X 1 X 2 X 2 16
17 But Exponentially Large Example: Parity Uniform distribution over states with even number of 1 s 2 N-1 N2 N-1 X 1 X 1 X 2 X 2 X 3 X 3 X 4 X 4 X 5 X 5 17
18 But Exponentially Large Example: Parity Can we make this more compact? Uniform distribution over states of even number of 1 s X 1 X 1 X 2 X 2 X 3 X 3 X 4 X 4 X 5 X 5 18
19 Use a Deep Network Example: Parity Uniform distribution over states with even number of 1 s O(N) 19
20 Use a Deep Network Example: Parity Uniform distribution over states of even number of 1 s Induce many hidden layers Reuse partial computation 20
21 Arithmetic Circuits (ACs) Data structure for efficient inference Darwiche [2003] Compilation target of Bayesian networks Key idea: Use ACs instead to define a new class of deep probabilistic models Develop new deep learning algorithms for this class of models 21
22 Sum-Product Networks (SPNs) Rooted DAG Nodes: Sum, product, input indicator Weights on edges from sum to children X 1 X 1 X 2 X 2 22
23 Distribution Defined by SPN P(X) S(X) X 1 X 1 X 2 X 2 23
24 Can We Sum Out Variables? P(e) Xe S(X)? = S(e) e: X 1 = 1 X 1 X 1 X 2 X
25 Valid SPN SPN is valid if S(e) = Xe S(X) for all e Valid Can compute marginals efficiently Partition function Z can be computed by setting all indicators to 1 25
26 Valid SPN: General Conditions Theorem: SPN is valid if it is complete & consistent Complete: Under sum, children cover the same set of variables Consistent: Under product, no variable in one child and negation in another Incomplete Inconsistent S(e) Xe S(X) S(e) Xe S(X) 26
27 Semantics of Sums and Products Product Feature Form feature hierarchy Sum Mixture (with hidden var. summed out) i i w ij I[Y i = j] j Sum out Yi w ij j 27
28 Inference Probability: P(X) = S(X) / Z X: X 1 = 1, X 2 = 0 X 1 1 X 1 0 X 2 0 X X 1 X 1 X 2 X
29 Inference Marginal: P(e) = S(e) / Z e: X 1 = 1 X 1 1 X 1 0 X 2 1 X = X 1 X 1 X 2 X
30 Inference MPE: Replace sums with maxes e: X 1 = 1 X 1 1 X 1 0 X 2 1 X = MAX = MAX MAX MAX MAX X 1 X 1 X 2 X Darwiche [2003] 30
31 Inference MAX: Pick child with highest value e: X 1 = 1 X 1 1 X 1 0 X 2 1 X = MAX = MAX MAX MAX MAX X 1 X 1 X 2 X Darwiche [2003] 31
32 Handling Continuous Variables Sum Integral over input Simplest case: Indicator Gaussian SPN compactly defines a very large mixture of Gaussians 32
33 SPNs Everywhere Graphical models Existing tractable models & inference methods Determinism, context-specific indep., etc. Can potentially learn the optimal way 33
34 SPNs Everywhere Graphical models Methods for efficient inference E.g., arithmetic circuits, AND/OR graphs, case-factor diagrams SPNs are a class of probabilistic models SPNs have validity conditions SPNs can be learned from data 34
35 SPNs Everywhere Graphical models Models for efficient inference General, probabilistic convolutional network Sum: Average pooling Max: Max pooling 35
36 SPNs Everywhere Graphical models Models for efficient inference General, probabilistic convolutional network Grammars in vision and language E.g., object detection grammar, probabilistic context-free grammar Sum: Non-terminal Product: Production rule 36
37 Outline Sum-product networks (SPNs) Learning SPNs Experimental results Conclusion 37
38 General Approach Start with a dense SPN Find the structure by learning weights Zero weights signify absence of connections Can learn with gradient descent or EM 38
39 The Challenge Gradient diffusion: Gradient quickly dilutes Similar problem with EM Hard EM overcomes this problem 39
40 Our Learning Algorithm Online learning Hard EM Sum node maintains counts for each child For each example Find MPE instantiation with current weights Increment count for each chosen child Renormalize to set new weights Repeat until convergence 40
41 Outline Sum-product networks (SPNs) Learning SPNs Experimental results Conclusion 41
42 Task: Image Completion Methodology: Learn a model from training images Complete unseen test images Measure mean square errors Very challenging Good for evaluating deep models 42
43 Datasets Main evaluation: Caltech-101 [Fei-Fei et al., 2004] 101 categories, e.g., faces, cars, elephants Each category: images Also, Olivetti [Samaria & Harter, 1994] (400 faces) Each category: Last third for test Test images: Unseen objects 43
44 SPN Architecture Whole Image Region Pixel x y 44
45 Systems SPN DBM [Salakhutdinov & Hinton, 2010] DBN [Hinton & Salakhutdinov, 2006] PCA [Turk & Pentland, 1991] Nearest neighbor [Hays & Efros, 2007] 45
46 Caltech: Mean-Square Errors Left Bottom NN PCA DBN DBM SPN NN PCA DBN DBM SPN 46
47 SPN vs. DBM / DBN SPN is order of magnitude faster SPN DBM / DBN Learning 2-3 hours Days Inference < 1 second Minutes or hours No elaborate preprocessing, tuning Reduced errors by 30-60% Learned up to 46 layers 47
48 Example Completions Original SPN DBM DBN PCA Nearest Neighbor 48
49 Example Completions Original SPN DBM DBN PCA Nearest Neighbor 49
50 Example Completions Original SPN DBM DBN PCA Nearest Neighbor 50
51 Example Completions Original SPN DBM DBN PCA Nearest Neighbor 51
52 Example Completions Original SPN DBM DBN PCA Nearest Neighbor 52
53 Example Completions Original SPN DBM DBN PCA Nearest Neighbor 53
54 Open Questions Other learning algorithms Discriminative learning Architecture Continuous SPNs Sequential domains Other applications 54
55 End-to-End Comparison Data Approximate General Graphical Models Approximate Performance Given same computation budget, which approach has better performance? Data Approximate Sum-Product Networks Exact Performance 55
56 Conclusion Sum-product networks (SPNs) DAG of sums and products Compactly represent partition function Learn many layers of hidden variables Exact inference: Linear time in network size Deep learning: Online hard EM Substantially outperform state of the art on image completion 56
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