Approximation Complexity of Maximum A Posteriori Inference in Sum-Product Networks

We discuss the computational complexity of approximating maximum a posteriori inference in sum-product networks. We first show NP-hardness in trees of height two by a reduction from maximum independent set; this implies non-approximability within a sublinear factor. We show that this is a tight bound, as we can find an approximation within a linear factor in networks of height two. We then show that, in trees of height three, it is NP-hard to approximate the problem within a factor $2^{f(n)}$ for any sublinear function $f$ of the size of the input $n$. Again, this bound is tight, as we prove that the usual max-product algorithm finds (in any network) approximations within factor $2^{c \cdot n}$ for some constant $c < 1$. Last, we present a simple algorithm, and show that it provably produces solutions at least as good as, and potentially much better than, the max-product algorithm. We empirically analyze the proposed algorithm against max-product using synthetic and realistic networks.Comment: 18 page

Credal Sum-Product Networks

Sum-product networks are a relatively new and increasingly popular class of (precise) probabilistic graphical models that allow for marginal inference with polynomial effort. As with other probabilistic models, sum-product networks are often learned from data and used to perform classification. Hence, their results are prone to be unreliable and overconfident. In this work, we develop credal sum-product networks, an imprecise extension of sum-product networks. We present algorithms and complexity results for common inference tasks. We apply our algorithms on realistic classification task using images of digits and show that credal sum-product networks obtained by a perturbation of the parameters of learned sum-product networks are able to distinguish between reliable and unreliable classifications with high accuracy

Cascading Sum-Product Networks using Robustness

Sum-product networks are an increasingly popular family of probabilistic graphical models for which marginal inference can be performed in polynomial time. They have been shown to achieve state-of-the-art performance in several tasks. When learning sum-product networks from scarce data, the obtained model may be prone to robustness issues. In particular, small variations of parameters could lead to different conclusions. We discuss the characteristics of sum-product networks as classifiers and study the robustness of them with respect to their parameters. Using a robustness measure to identify (possibly) unreliable decisions, we build a hierarchical approach where the classification task is deferred to another model if the outcome is deemed unreliable. We apply this approach on benchmark classification tasks and experiments show that the robustness measure can be a meaningful manner to improve classification accuracy

Cascading Sum-Product Networks using Robustness

Sum-product networks are an increasingly popular family of probabilistic graphical models for which marginal inference can be performed in polynomial time. They have been shown to achieve state-of-the-art performance in several tasks. When learning sum-product networks from scarce data, the obtained model may be prone to robustness issues. In particular, small variations of parameters could lead to different conclusions. We discuss the characteristics of sum-product networks as classifiers and study the robustness of them with respect to their parameters. Using a robustness measure to identify (possibly) unreliable decisions, we build a hierarchical approach where the classification task is deferred to another model if the outcome is deemed unreliable. We apply this approach on benchmark classification tasks and experiments show that the robustness measure can be a meaningful manner to improve classification accuracy

Cascading sum-product networks using robustness

Sum-product networks are an increasingly popular family of probabilistic graphical models for which marginal inference can be performed in polynomial time. They have been shown to achieve state-of-the-art performance in several tasks. When learning sum-product networks from scarce data, the obtained model may be prone to robustness issues. In particular, small variations of parameters could lead to different conclusions. We discuss the characteristics of sum-product networks as classifiers and study the robustness of them with respect to their parameters. Using a robustness measure to identify (possibly) unreliable decisions, we build a hierarchical approach where the classification task is deferred to another model if the outcome is deemed unreliable. We apply this approach on benchmark classification tasks and experiments show that the robustness measure can be a meaningful manner to improve classification accuracy

A hierarchy of sum-product networks using robustness

Sum-product networks are a popular family of probabilistic graphical models that have been shown to achieve state-of-the-art performance in several tasks. When learning sum-product networks from scarce data, the obtained model may be prone to robustness issues, and where small variations of parameters could lead to different conclusions. We discuss the characteristics of sum-product networks as classifiers and study the robustness of them with respect to their parameters. Using a robustness measure to identify (possibly) unreliable decisions, we build a hierarchical approach where the classification task in testing time is deferred to another model if the outcome is deemed unreliable. We apply this approach on benchmark classification tasks across 47 datasets. Experiments show that the robustness measure can be a meaningful manner to build dynamic ensemble of classifiers and that our Hierarchical Sum-Product Network guarantees an improvement in accuracy

Cascading Sum-Product Networks using Robustness

Sum-product networks are an increasingly popular family of probabilistic graphical models for which marginal inference can be performed in polynomial time. They have been shown to achieve state-of-the-art performance in several tasks. When learning sum-product networks from scarce data, the obtained model may be prone to robustness issues. In particular, small variations of parameters could lead to different conclusions. We discuss the characteristics of sum-product networks as classifiers and study the robustness of them with respect to their parameters. Using a robustness measure to identify (possibly) unreliable decisions, we build a hierarchical approach where the classification task is deferred to another model if the outcome is deemed unreliable. We apply this approach on benchmark classification tasks and experiments show that the robustness measure can be a meaningful manner to improve classification accuracy

A hierarchy of sum-product networks using robustness

Sum-product networks are a popular family of probabilistic graphical models that have been shown to achieve state-of-the-art performance in several tasks. When learning sum-product networks from scarce data, the obtained model may be prone to robustness issues, and where small variations of parameters could lead to different conclusions. We discuss the characteristics of sum-product networks as classifiers and study the robustness of them with respect to their parameters. Using a robustness measure to identify (possibly) unreliable decisions, we build a hierarchical approach where the classification task in testing time is deferred to another model if the outcome is deemed unreliable. We apply this approach on benchmark classification tasks across 47 datasets. Experiments show that the robustness measure can be a meaningful manner to build dynamic ensemble of classifiers and that our Hierarchical Sum-Product Network guarantees an improvement in accuracy

Robustifying Sum-Product Networks

Sum-product networks are a relatively new and increasingly popular family of probabilistic graphical models that allow for marginal inference with polynomial effort. They have been shown to achieve state-of-the-art performance in several tasks involving density estimation. Sum-product networks are typically learned from data; as such, inferences produced with them are prone to be unreliable and overconfident when data is scarce. In this work, we develop the credal sum-product networks, a generalization of sum-product networks that uses set-valued parameters. We present algorithms and complexity results for common inference tasks with this class of models. We also present an approach for assessing the reliability of classifications made with sum-product networks. We apply this approach on benchmark classification tasks as well as a new application in predicting the age of stars. Our experiments show that the use of credal sum-product networks allow us to distinguish between reliable and unreliable classifications with higher accuracy than standard approaches based on (precise) probability values