Bayesian Learning of Sum-Product Networks

Sum-product networks (SPNs) are flexible density estimators and have received significant attention due to their attractive inference properties. While parameter learning in SPNs is well developed, structure learning leaves something to be desired: Even though there is a plethora of SPN structure learners, most of them are somewhat ad-hoc and based on intuition rather than a clear learning principle. In this paper, we introduce a well-principled Bayesian framework for SPN structure learning. First, we decompose the problem into i) laying out a computational graph, and ii) learning the so-called scope function over the graph. The first is rather unproblematic and akin to neural network architecture validation. The second represents the effective structure of the SPN and needs to respect the usual structural constraints in SPN, i.e. completeness and decomposability. While representing and learning the scope function is somewhat involved in general, in this paper, we propose a natural parametrisation for an important and widely used special case of SPNs. These structural parameters are incorporated into a Bayesian model, such that simultaneous structure and parameter learning is cast into monolithic Bayesian posterior inference. In various experiments, our Bayesian SPNs often improve test likelihoods over greedy SPN learners. Further, since the Bayesian framework protects against overfitting, we can evaluate hyper-parameters directly on the Bayesian model score, waiving the need for a separate validation set, which is especially beneficial in low data regimes. Bayesian SPNs can be applied to heterogeneous domains and can easily be extended to nonparametric formulations. Moreover, our Bayesian approach is the first, which consistently and robustly learns SPN structures under missing data.Comment: NeurIPS 2019; See conference page for supplemen

Learning Logistic Circuits

This paper proposes a new classification model called logistic circuits. On MNIST and Fashion datasets, our learning algorithm outperforms neural networks that have an order of magnitude more parameters. Yet, logistic circuits have a distinct origin in symbolic AI, forming a discriminative counterpart to probabilistic-logical circuits such as ACs, SPNs, and PSDDs. We show that parameter learning for logistic circuits is convex optimization, and that a simple local search algorithm can induce strong model structures from data.Comment: Published in the Proceedings of the Thirty-Third AAAI Conference on Artificial Intelligence (AAAI19

Learning Tractable Probabilistic Models for Fault Localization

In recent years, several probabilistic techniques have been applied to various debugging problems. However, most existing probabilistic debugging systems use relatively simple statistical models, and fail to generalize across multiple programs. In this work, we propose Tractable Fault Localization Models (TFLMs) that can be learned from data, and probabilistically infer the location of the bug. While most previous statistical debugging methods generalize over many executions of a single program, TFLMs are trained on a corpus of previously seen buggy programs, and learn to identify recurring patterns of bugs. Widely-used fault localization techniques such as TARANTULA evaluate the suspiciousness of each line in isolation; in contrast, a TFLM defines a joint probability distribution over buggy indicator variables for each line. Joint distributions with rich dependency structure are often computationally intractable; TFLMs avoid this by exploiting recent developments in tractable probabilistic models (specifically, Relational SPNs). Further, TFLMs can incorporate additional sources of information, including coverage-based features such as TARANTULA. We evaluate the fault localization performance of TFLMs that include TARANTULA scores as features in the probabilistic model. Our study shows that the learned TFLMs isolate bugs more effectively than previous statistical methods or using TARANTULA directly.Comment: Fifth International Workshop on Statistical Relational AI (StaR-AI 2015

Conditional Sum-Product Networks: Imposing Structure on Deep Probabilistic Architectures

Probabilistic graphical models are a central tool in AI; however, they are generally not as expressive as deep neural models, and inference is notoriously hard and slow. In contrast, deep probabilistic models such as sum-product networks (SPNs) capture joint distributions in a tractable fashion, but still lack the expressive power of intractable models based on deep neural networks. Therefore, we introduce conditional SPNs (CSPNs), conditional density estimators for multivariate and potentially hybrid domains which allow harnessing the expressive power of neural networks while still maintaining tractability guarantees. One way to implement CSPNs is to use an existing SPN structure and condition its parameters on the input, e.g., via a deep neural network. This approach, however, might misrepresent the conditional independence structure present in data. Consequently, we also develop a structure-learning approach that derives both the structure and parameters of CSPNs from data. Our experimental evidence demonstrates that CSPNs are competitive with other probabilistic models and yield superior performance on multilabel image classification compared to mean field and mixture density networks. Furthermore, they can successfully be employed as building blocks for structured probabilistic models, such as autoregressive image models.Comment: 13 pages, 6 figure