Inductive Principles for Restricted Boltzmann Machine Learning
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1 Inductive Principles for Restricted Boltzmann Machine Learning Benjamin Marlin Department of Computer Science University of British Columbia Joint work with Kevin Swersky, Bo Chen and Nando de Freitas Department of Computer Science, University of British Columbia 1
2 Introduction: Maximum Likelihood Maximum Likelihood Estimation is statistically consistent and efficient but is not computationally tractable for many models of interest like RBM s, MRF s, CRF s due to the partition function. Consistent Efficient Tractable Department of Computer Science, University of British Columbia 2
3 Introduction: Alternative Estimators Recent work has seen the proposal of many new estimators that trade consistency/efficiency for computational tractability including RM, SM, GSM, MPF, NCE, NLCE. Consistent Efficient Tractable Department of Computer Science, University of British Columbia 3
4 Introduction: Alternative Estimators Our main interest is uncovering the relationships between these estimators and studying their theoretical and empirical properties. Consistent Efficient Tractable Department of Computer Science, University of British Columbia 4
5 Outline: Boltzmann Machines and RBMs Inductive Principles Maximum Likelihood Contrastive Divergence Pseudo-Likelihood Ratio Matching Generalized Score Matching Minimum Probability Flow Experiments Demo Department of Computer Science, University of British Columbia 5
6 Introduction: Restricted Boltzmann Machines K Hidden Units H 1 H 2 H 3 H 4 X 1 X 2 X 3 D Visible Units A Restricted Boltzmann Machine (RBM) is a Boltzmann Machine with a bipartite graph structure. Typically one layer of nodes are fully observed variables (the visible layer), while the other consists of latent variables (the hidden layer). Department of Computer Science, University of British Columbia 6
7 Introduction: Restricted Boltzmann Machines The joint probability of the visible and hidden variables is defined through a bilinear energy function. Department of Computer Science, University of British Columbia 7
8 Introduction: Restricted Boltzmann Machines The bipartite graph structure gives the RBM a special property: the visible variables are conditionally independent given the hidden variables and vice versa. Department of Computer Science, University of British Columbia 8
9 Introduction: Restricted Boltzmann Machines The marginal probability of the visible vector is obtained by summing out over all joint states of the hidden variables. Department of Computer Science, University of British Columbia 9
10 Introduction: Restricted Boltzmann Machines This construction eliminates the latent, hidden variables, leaving a distribution defined in terms of the visible variables. However, computing the normalizing constant (partition function) still has exponential complexity in D. Department of Computer Science, University of British Columbia 10
11 Outline: Boltzmann Machines and RBMs Inductive Principles Maximum Likelihood Contrastive Divergence Pseudo-Likelihood Ratio Matching Generalized Score Matching Experiments Demo Department of Computer Science, University of British Columbia 11
12 Stochastic Maximum Likelihood Exact maximum likelihood learning is intractable in an RBM. Stochastic ML estimation can instead be applied, usually using a simple block Gibbs sampler. Department of Computer Science, University of British Columbia 12
13 Stochastic Maximum Likelihood Exact maximum likelihood learning is intractable in an RBM. Stochastic ML estimation can instead be applied, usually using a simple block Gibbs sampler. Update Weights Update Weights L. Younes. Parametric inference for imperfectly observed Gibbsian fields. Prob. Th. and Related Fields, 82(4): , T. Tieleman. Training restricted Boltzmann machines using approximations to the likelihood gradient. ICML 25, Department of Computer Science, University of British Columbia 13
14 Contrastive Divergence The contrastive divergence principle results in a gradient that looks identical to stochastic maximum likelihood. The difference is that CD samples from the T-step Gibbs distribution. Department of Computer Science, University of British Columbia 14
15 Contrastive Divergence The contrastive divergence principle results in a gradient that looks identical to stochastic maximum likelihood. The difference is that CD samples from the T-step Gibbs distribution. Update Weights & Reset Chain to Data Department of Computer Science, University of British Columbia 15
16 Pseudo-Likelihood The principle of maximum pseudo-likelihood is based on optimizing a product of one-dimensional conditional densities under a log loss. Department of Computer Science, University of British Columbia 16
17 Pseudo-Likelihood The principle of maximum pseudo-likelihood is based on optimizing a product of one-dimensional conditional densities under a log loss. Department of Computer Science, University of British Columbia 17
18 Ratio Matching The ratio matching principle is very similar to pseudolikelihood, but is based on minimizing a squared difference between one dimensional conditional distributions. Aapo Hyvarinen. Some extensions of score matching. Computational Statistics & Data Analysis, 51(5): , Department of Computer Science, University of British Columbia 18
19 Generalized Score Matching The generalized score matching principle is similar to ratio matching, except that the difference between inverse one dimensional conditional distributions is minimized. Siwei Lyu. Interpretation and generalization of score matching. In Uncertainty in Artificial Intelligence 25, Department of Computer Science, University of British Columbia 19
20 Minimum Probability Flow Minimize the flow of probability from data states to non-data states (as we ve just seen!). Jascha Sohl-Dickstein, Peter Battaglino, Michael R. DeWeese. Minimum Probability Flow Learning. Department of Computer Science, University of British Columbia 20
21 Comparison: Gradients Department of Computer Science, University of British Columbia 21
22 Comparison: Weighting Functions Department of Computer Science, University of British Columbia 22
23 Comparison: Weighting Functions What about MPF? Department of Computer Science, University of British Columbia 23
24 Comparison: A Manifold of Estimators? Dimensions: 1. Neighborhood structure around data configurations. 2. Form of loss function on the probability ratio. Smooth Monotonically decreasing Bounded below Others? Covers: PL, GLP, NLCE, RM, GSM, MPF Limitations: No good for missing data/explicit latent variables. Department of Computer Science, University of British Columbia 24
25 Outline: Boltzmann Machines and RBMs Inductive Principles Maximum Likelihood Contrastive Divergence Pseudo-Likelihood Ratio Matching Generalized Score Matching Experiments Demo Department of Computer Science, University of British Columbia 25
26 Experiments: Data Sets: MNIST handwritten digits 20 News Groups CalTech 101 Silhouettes Evaluation Criteria: Log likelihood (using AIS estimator) Classification error Reconstruction error De-noising Novelty detection Department of Computer Science, University of British Columbia 26
27 Experiments: Log Likelihood Department of Computer Science, University of British Columbia 27
28 Experiments: Classification Error Department of Computer Science, University of British Columbia 28
29 Experiments: De-noising Department of Computer Science, University of British Columbia 29
30 Experiments: Novelty Detection Department of Computer Science, University of British Columbia 30
31 Experiments: Learned Weights on MNIST Department of Computer Science, University of British Columbia 31
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