The need for consistent treatment of uncertainty has recently triggered increased interest in probabilistic deep learning methods. However, most current approaches have severe limitations when it comes to inference, since many of these models do not even permit to evaluate exact data likelihoods. Sum-product networks (SPNs), on the other hand, are an excellent architecture in that regard, as they allow to efficiently evaluate likelihoods, as well as arbitrary marginalization and conditioning tasks. Nevertheless, SPNs have not been fully explored as serious deep learning models, likely due to their special structural requirements, which complicate learning. In this paper, we make a drastic simplification and use random SPN structures which are trained in a "classical deep learning manner", i.e. employing automatic differentiation, SGD, and GPU support. The resulting models, called RAT-SPNs, yield prediction results comparable to deep neural networks, while still being interpretable as generative model and maintaining well-calibrated uncertainties. This property makes them highly robust under missing input features and enables them to naturally detect outliers and peculiar samples.
Learning the parameters of SPNs, i.e. the sum weights and the parameters of input distributions, can be addressed in various ways. By interpreting the sum nodes as discrete latent variables @cite_26 @cite_20 @cite_17 , SPNs can be trained using the classical expectation-maximization (EM) algorithm @cite_17 . @cite_23 derived a concave-convex procedure, which interestingly coincides with the EM updates for sum-weights. Moreover, SPN parameters can be treated in the Bayesian framework, as proposed in @cite_25 @cite_10 @cite_5 . @cite_35 introduced a safe semi-supervised learning scheme for discriminative and generative parameter learning, providing guarantees for the performance in the semi-supervised case. @cite_13 employed SPNs as probabilistic autoencoders and unsupervised representation learners. The structure of SPNs can be crafted by hand @cite_26 @cite_29 or learned from data. Most structure learners @cite_27 @cite_1 @cite_12 @cite_28 are variations of the divide-and-conquer scheme known as @cite_19 . This schemed recursively splits the data via clustering (determining sum nodes) and independence tests (determining product nodes).
@cite_13 @cite_35 @cite_12 @cite_26 @cite_28 @cite_29 @cite_1 @cite_19 @cite_27 @cite_23 @cite_5 @cite_10 @cite_25 @cite_20 @cite_17
2963276753 2128744540 2420777250 2159687189
One of the central themes in Sum-Product networks (SPNs) is the interpretation of sum nodes as marginalized latent variables (LVs). This interpretation yields an increased syntactic or semantic structure, allows the application of the EM algorithm and to efficiently perform MPE inference. In literature, the LV interpretation was justified by explicitly introducing the indicator variables corresponding to the LVs’ states. However, as pointed out in this paper, this approach is in conflict with the completeness condition in SPNs and does not fully specify the probabilistic model. We propose a remedy for this problem by modifying the original approach for introducing the LVs, which we call SPN augmentation. We discuss conditional independencies in augmented SPNs, formally establish the probabilistic interpretation of the sum-weights and give an interpretation of augmented SPNs as Bayesian networks. Based on these results, we find a sound derivation of the EM algorithm for SPNs. Furthermore, the Viterbi-style algorithm for MPE proposed in literature was never proven to be correct. We show that this is indeed a correct algorithm, when applied to selective SPNs, and in particular when applied to augmented SPNs. Our theoretical results are confirmed in experiments on synthetic data and 103 real-world datasets. Sum-product networks are a new deep architecture that can perform fast, exact inference on high-treewidth models. Only generative methods for training SPNs have been proposed to date. In this paper, we present the first discriminative training algorithms for SPNs, combining the high accuracy of the former with the representational power and tractability of the latter. We show that the class of tractable discriminative SPNs is broader than the class of tractable generative ones, and propose an efficient backpropagation-style algorithm for computing the gradient of the conditional log likelihood. Standard gradient descent suffers from the diffusion problem, but networks with many layers can be learned reliably using "hard" gradient descent, where marginal inference is replaced by MPE inference (i.e., inferring the most probable state of the non-evidence variables). The resulting updates have a simple and intuitive form. We test discriminative SPNs on standard image classification tasks. We obtain the best results to date on the CIFAR-10 dataset, using fewer features than prior methods with an SPN architecture that learns local image structure discriminatively. We also report the highest published test accuracy on STL-10 even though we only use the labeled portion of the dataset. Sum-Product Networks (SPNs) are probabilistic inference machines that admit exact inference in linear time in the size of the network. Existing parameter learning approaches for SPNs are largely based on the maximum likelihood principle and are subject to overfitting compared to more Bayesian approaches. Exact Bayesian posterior inference for SPNs is computationally intractable. Even approximation techniques such as standard variational inference and posterior sampling for SPNs are computationally infeasible even for networks of moderate size due to the large number of local latent variables per instance. In this work, we propose a novel deterministic collapsed variational inference algorithm for SPNs that is computationally efficient, easy to implement and at the same time allows us to incorporate prior information into the optimization formulation. Extensive experiments show a significant improvement in accuracy compared with a maximum likelihood based approach. Sum-product networks (SPNs) are a new class of deep probabilistic models. SPNs can have unbounded treewidth but inference in them is always tractable. An SPN is either a univariate distribution, a product of SPNs over disjoint variables, or a weighted sum of SPNs over the same variables. We propose the first algorithm for learning the structure of SPNs that takes full advantage of their expressiveness. At each step, the algorithm attempts to divide the current variables into approximately independent subsets. If successful, it returns the product of recursive calls on the subsets; otherwise it returns the sum of recursive calls on subsets of similar instances from the current training set. A comprehensive empirical study shows that the learned SPNs are typically comparable to graphical models in likelihood but superior in inference speed and accuracy.