Einsum Networks: Fast and Scalable Learning of Tractable Probabilistic Circuits
Eindhoven University of Technology（埃因霍温理工大学）；Technical University Darmstadt（达姆施塔特技术大学）；Graz University of Technology（格拉茨理工大学）；University California Los Angeles（加州大学洛杉矶分校）；University of Cambridge（剑桥大学）
Robert Peharz, Steven Lang, Antonio Vergari, Karl Stelzner, Alejandro Molina, Martin Trapp, Guy Van den Broeck, Kristian Kersting, Zoubin Ghahramani
发表信息：
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:7563-7574, 2020
Probabilistic circuits (PCs) are a promising avenue for probabilistic modeling, as they permit a wide range of exact and efficient inference routines. Recent ``deep-learning-style'' implementations of PCs strive for a better scalability, but are still difficult to train on real-world data, due to their sparsely connected computational graphs. In this paper, we propose Einsum Networks (EiNets), a novel implementation design for PCs, improving prior art in several regards. At their core, EiNets combine a large number of arithmetic operations in a single monolithic einsum-operation, leading to speedups and memory savings of up to two orders of magnitude, in comparison to previous implementations. As an algorithmic contribution, we show that the implementation of Expectation-Maximization (EM) can be simplified for PCs, by leveraging automatic differentiation. Furthermore, we demonstrate that EiNets scale well to datasets which were previously out of reach, such as SVHN and CelebA, and that they can be used as faithful generative image models.
We present a new algorithm for approximate inference in probabilistic programs, based on a stochastic gradient for variational programs. This method is efficient without restrictions on the probabilistic program; it is particularly practical for distributions which are not analytically tractable, including highly structured distributions that arise in probabilistic programs. We show how to automatically derive mean-field probabilistic programs and optimize them, and demonstrate that our perspective improves inference efficiency over other algorithms.
We introduce a dynamic mechanism for the solution of analytically-tractable substructure in probabilistic programs, using conjugate priors and affine transformations to reduce variance in Monte Carlo estimators. For inference with Sequential Monte Carlo, this automatically yields improvements such as locally-optimal proposals and Rao-Blackwellization. The mechanism maintains a directed graph alongside the running program that evolves dynamically as operations are triggered upon it. Nodes of the graph represent random variables, edges the analytically-tractable relationships between them. Rando…
Many practical techniques for probabilistic inference require a sequence of distributions that interpolate between a tractable distribution and an intractable distribution of interest. Usually, the sequences used are simple, e.g., based on geometric averages between distributions. When models are expressed as probabilistic programs, the models themselves are highly structured objects that can be used to derive annealing sequences that are more sensitive to domain structure. We propose an algorithm for transforming probabilistic programs to coarse-to-fine programs which have the same marginal d…
Probabilistic models learned as density estimators can be exploited in representation learning beside being toolboxes used to answer inference queries only. However, how to extract useful representations highly depends on the particular model involved. We argue that tractable inference, i.e. inference that can be computed in polynomial time, can enable general schemes to extract features from black box models. We plan to investigate how Tractable Probabilistic Models (TPMs) can be exploited to generate embeddings by random query evaluations. We devise two experimental designs to assess and com…
Variational inference provides a powerful tool for approximate probabilistic in- ference on complex, structured models. Typical variational inference methods, however, require to use inference networks with computationally tractable proba- bility density functions. This largely limits the design and implementation of vari- ational inference methods. We consider wild variational inference methods that do not require tractable density functions on the inference networks, and hence can be applied in more challenging cases. As an example of application, we treat stochastic gradient Langevin dynami…
University of Edinburgh（爱丁堡大学）；Alan Turing Institute（艾伦图灵研究所）；University of Edinburgh & Alan Turing Institute（爱丁堡大学和艾伦图灵研究所（此处为合并表述，按原文提取））
Incorporating constraints is a major concern in probabilistic machine learning. A wide variety of problems require predictions to be integrated with reasoning about constraints, from modelling routes on maps to approving loan predictions. In the former, we may require the prediction model to respect the presence of physical paths between the nodes on the map, and in the latter, we may require that the prediction model respect fairness constraints that ensure that outcomes are not subject to bias. Broadly speaking, constraints may be probabilistic, logical or causal, but the overarching challen…
We consider the problem of explaining a class of tractable deep probabilistic models, the Sum-Product Networks (SPNs) and present an algorithm ExSPN to generate explanations. To this effect, we define the notion of a context-specific independence tree(CSI-tree) and present an iterative algorithm that converts an SPN to a CSI-tree. The resulting CSI-tree is both interpretable and explainable to the domain expert. We achieve this by extracting the conditional independencies encoded by the SPN and approximating the local context specified by the structure of the SPN. Our extensive empirical evalu…
Probabilistic Boolean Networks (PBNs) have been previously proposed so as to gain insights into complex dy- namical systems. However, identification of large networks and of the underlying discrete Markov Chain which describes their temporal evolution, still remains a challenge. In this paper, we introduce an equivalent representation for the PBN, the Stochastic Conjunctive Normal Form (SCNF), which paves the way to a scalable learning algorithm and helps predict long- run dynamic behavior of large-scale systems. Moreover, SCNF allows its efficient sampling so as to statistically infer multi- …
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 pa…
TU Darmstadt（达姆施塔特工业大学）；The University of Texas at Dallas（德克萨斯大学达拉斯分校）；Hessian Center for AI (hessian.AI)（黑森人工智能中心 (hessian.AI)）
While probabilistic models are an important tool for studying causality, doing so suffers from the intractability of inference. As a step towards tractable causal models, we consider the problem of learning interventional distributions using sum-product networks (SPNs) that are over-parameterized by gate functions, e.g., neural networks. Providing an arbitrarily intervened causal graph as input, effectively subsuming Pearl's do-operator, the gate function predicts the parameters of the SPN. The resulting interventional SPNs are motivated and illustrated by a structural causal model themed arou…
Tractable Representation Learning with Probabilistic Circuits
Technische Universität Darmstadt（达姆施塔特工业大学）；University of Texas, Dallas（德克萨斯大学达拉斯分校）；University of Edinburgh（爱丁堡大学）；Universität Bremen（不来梅大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
Probabilistic Circuits (PCs) are a promising avenue for probabilistic modeling. They combine advantages of probabilistic graphical models (PGMs) with those of neural networks (NNs). Crucially, however, they are tractable probabilistic models, supporting efficient and exact computation of many probabilistic inference queries, such as marginals and MAP. Further, since PCs are structured computation graphs, they can take advantage of deep-learning-style parameter updates, which greatly improves their scalability. However, this innovation also makes PCs prone to overfitting, which has been observe…
Counterfactual examples are an appealing class of post-hoc explanations for machine learning models. Given input $x$ of class $y_1$, its counterfactual is a contrastive example $x^\prime$ of another class $y_0$. Current approaches primarily solve this task by a complex optimization: define an objective function based on the loss of the counterfactual outcome $y_0$ with hard or soft constraints, then optimize this function as a black-box. This "deep learning" approach, however, is rather slow, sometimes tricky, and may result in unrealistic counterfactual examples. In this work, we propose a no…
Tsinghua University（清华大学）；University of California, Los Angeles（加州大学洛杉矶分校）；Peking University（北京大学）；Beijing Institute for General Artificial Intelligence（北京通用人工智能研究院）
Probabilistic Circuits (PCs) are a general and unified computational framework for tractable probabilistic models that support efficient computation of various inference tasks (e.g., computing marginal probabilities). Towards enabling such reasoning capabilities in complex real-world tasks, Liu et al. (2022) propose to distill knowledge (through latent variable assignments) from less tractable but more expressive deep generative models. However, it is still unclear what factors make this distillation work well. In this paper, we theoretically and empirically discover that the performance of a …
University of California Los Angeles（加州大学洛杉矶分校）；Washington University in St. Louis（圣路易斯华盛顿大学）
Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework …
University of Oxford（牛津大学）；University of Edinburgh（爱丁堡大学）；Alan Turing Institute（艾伦·图灵研究所）
Moral responsibility is a major concern in autonomous systems, with applications ranging from self-driving cars to kidney exchanges. Although there have been recent attempts to formalise responsibility and blame, among similar notions, the problem of learning within these formalisms has been unaddressed. From the viewpoint of such systems, the urgent questions are: (a) How can models of moral scenarios and blameworthiness be extracted and learnt automatically from data? (b) How can judgements be computed effectively and efficiently, given the split-second decision points faced by some systems?…
Bayesian structure learning allows one to capture uncertainty over the causal directed acyclic graph (DAG) responsible for generating given data. In this work, we present Tractable Uncertainty for STructure learning (TRUST), a framework for approximate posterior inference that relies on probabilistic circuits as the representation of our posterior belief. In contrast to sample-based posterior approximations, our representation can capture a much richer space of DAGs, while also being able to tractably reason about the uncertainty through a range of useful inference queries. We empirically show…
Sum-Product Networks (SPNs) are expressive probabilistic models that provide exact, tractable inference. They achieve this efficiency by making use of local independence. On the other hand, mixtures of exchangeable variable models (MEVMs) are a class of tractable probabilistic models that make use of exchangeability of discrete random variables to render inference tractable. Exchangeability, which arises naturally in relational domains, has not been considered for efficient representation and inference in SPNs yet. The contribution of this paper is a novel probabilistic model which we call Exc…
University of Edinburgh（爱丁堡大学）；Alan Turing Institute（艾伦·图灵研究所）
Probabilistic representations, such as Bayesian and Markov networks, are fundamental to much of statistical machine learning. Thus, learning probabilistic representations directly from data is a deep challenge, the main computational bottleneck being inference that is intractable. Tractable learning is a powerful new paradigm that attempts to learn distributions that support efficient probabilistic querying. By leveraging local structure, representations such as sum-product networks (SPNs) can capture high tree-width models with many hidden layers, essentially a deep architecture, while still …
Scaling Probabilistic Circuits via Monarch Matrices
Department of Computer Science, University of California, Los Angeles（加州大学洛杉矶分校计算机科学系）；Department of Computer Science, Stanford University（斯坦福大学计算机科学系）
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Scaling probabilistic models to large realistic problems and datasets is a key challenge in machine learning. Central to this effort is the development of tractable probabilistic models (TPMs): models whose structure guarantees efficient probabilistic inference algorithms. The current landscape of TPMs is fragmented: there exist various kinds of TPMs with different strengths and weaknesses. Two of the most prominent classes of TPMs are determinantal point processes (DPPs) and probabilistic circuits (PCs). This paper provides the first systematic study of their relationship. We propose a unifie…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Bari（巴里大学）
Sum-Product Networks (SPNs) are recently introduced deep tractable probabilistic models by which several kinds of inference queries can be answered exactly and in a tractable time. Up to now, they have been largely used as black box density estimators, assessed only by comparing their likelihood scores only. In this paper we explore and exploit the inner representations learned by SPNs. We do this with a threefold aim: first we want to get a better understanding of the inner workings of SPNs; secondly, we seek additional ways to evaluate one SPN model and compare it against other probabilistic…
Tsinghua University（清华大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Computing the expectation of kernel functions is a ubiquitous task in machine learning, with applications from classical support vector machines to exploiting kernel embeddings of distributions in probabilistic modeling, statistical inference, causal discovery, and deep learning. In all these scenarios, we tend to resort to Monte Carlo estimates as expectations of kernels are intractable in general. In this work, we characterize the conditions under which we can compute expected kernels exactly and efficiently, by leveraging recent advances in probabilistic circuit representations. We first co…
Probabilistic Circuits That Know What They Don't Know
TU Darmstadt（达姆施塔特工业大学）；Hessian Center for AI (hessian.AI)（黑森人工智能中心）；German Research Center for Artificial Intelligence (DFKI)（德国人工智能研究中心）；Centre for Cognitive Science TU Darmstadt（达姆施塔特工业大学认知科学中心）
Decision Trees are some of the most popular machine learning models today due to their out-of-the-box performance and interpretability. Often, Decision Trees models are constructed greedily in a top-down fashion via heuristic search criteria, such as Gini impurity or entropy. However, trees constructed in this manner are sensitive to minor fluctuations in training data and are prone to overfitting. In contrast, Bayesian approaches to tree construction formulate the selection process as a posterior inference problem; such approaches are more stable and provide greater theoretical guarantees. Ho…
A sequence of random variables is exchangeable if its joint distribution is invariant under variable permutations. We introduce exchangeable variable models (EVMs) as a novel class of probabilistic models whose basic building blocks are partially exchangeable sequences, a generalization of exchangeable sequences. We prove that a family of tractable EVMs is optimal under zero-one loss for a large class of functions, including parity and threshold functions, and strictly subsumes existing tractable independence-based model families. Extensive experiments show that EVMs outperform state of the ar…
University of Edinburgh（爱丁堡大学）；Alan Turing Institute（艾伦·图灵研究所）
Probabilistic representations, such as Bayesian and Markov networks, are fundamental to much of statistical machine learning. Thus, learning probabilistic representations directly from data is a deep challenge, the main computational bottleneck being inference that is intractable. Tractable learning is a powerful new paradigm that attempts to learn distributions that support efficient probabilistic querying. By leveraging local structure, representations such as sum-product networks (SPNs) can capture high tree-width models with many hidden layers, essentially a deep architecture, while still …
Probabilistic circuits (PCs) are a tractable representation of probability distributions allowing for exact and efficient computation of likelihoods and marginals. There has been significant recent progress on improving the scale and expressiveness of PCs. However, PC training performance plateaus as model size increases. We discover that most capacity in existing large PC structures is wasted: fully-connected parameter layers are only sparsely used. We propose two operations: pruning and growing, that exploit the sparsity of PC structures. Specifically, the pruning operation removes unimporta…
UCLA（加州大学洛杉矶分校）；TU Darmstadt（达姆施塔特工业大学）；University of Cambridge（剑桥大学）；TU Dortmund（多特蒙德工业大学）
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 maintainin…
Eindhoven University of Technology（埃因霍温理工大学）；KU Leuven（鲁汶大学）；Graz University of Technology（格拉茨理工大学）
Probabilistic circuits (PCs) are a prominent representation of probability distributions with tractable inference. While parameter learning in PCs is rigorously studied, structure learning is often more based on heuristics than on principled objectives. In this paper, we develop Bayesian structure scores for deterministic PCs, i.e., the structure likelihood with parameters marginalized out, which are well known as rigorous objectives for structure learning in probabilistic graphical models. When used within a greedy cutset algorithm, our scores effectively protect against overfitting and yield…
Scaling Probabilistic Circuits via Data Partitioning
TU Darmstadt（达姆施塔特工业大学）；Hessian Center for AI (hessian.AI)（黑森人工智能中心）；Eindhoven University of Technology（埃因霍温理工大学）；University of Bremen（不来梅大学）；German Research Center for AI (DFKI)（德国人工智能研究中心）
The Libra Toolkit is a collection of algorithms for learning and inference with discrete probabilistic models, including Bayesian networks, Markov networks, dependency networks, and sum-product networks. Compared to other toolkits, Libra places a greater emphasis on learning the structure of tractable models in which exact inference is efficient. It also includes a variety of algorithms for learning graphical models in which inference is potentially intractable, and for performing exact and approximate inference. Libra is released under a 2-clause BSD license to encourage broad use in academia…
University of Waterloo（滑铁卢大学）；University of Georgia（佐治亚大学）
Sum-Product Networks (SPN) have recently emerged as a new class of tractable probabilistic graphical models. Unlike Bayesian networks and Markov networks where inference may be exponential in the size of the network, inference in SPNs is in time linear in the size of the network. Since SPNs represent distributions over a fixed set of variables only, we propose dynamic sum product networks (DSPNs) as a generalization of SPNs for sequence data of varying length. A DSPN consists of a template network that is repeated as many times as needed to model data sequences of any length. We present a loca…
TU Darmstadt（达姆施塔特工业大学）；Max-Planck-Institute（马克斯-普朗克研究所）；University of Cambridge（剑桥大学）；University of Waterloo（滑铁卢大学）；University of Bari “Aldo Moro”（巴里大学“阿尔多·莫罗”）
We introduce SPFlow, an open-source Python library providing a simple interface to inference, learning and manipulation routines for deep and tractable probabilistic models called Sum-Product Networks (SPNs). The library allows one to quickly create SPNs both from data and through a domain specific language (DSL). It efficiently implements several probabilistic inference routines like computing marginals, conditionals and (approximate) most probable explanations (MPEs) along with sampling as well as utilities for serializing, plotting and structure statistics on an SPN. Moreover, many of the a…
Hyperparameter Optimization via Interacting with Probabilistic Circuits
Computer Science Department, Technical University Darmstadt（技术大学达姆施塔特计算机科学系）；Hessian Center for AI (hessian.AI)（黑森人工智能中心）；German Research Center for AI (DFKI)（德国人工智能研究中心）
Computer Science Department, University of California, Los Angeles（加州大学洛杉矶分校计算机科学系）
Tractable Boolean and arithmetic circuits have been studied extensively in AI for over two decades now. These circuits were initially proposed as "compiled objects," meant to facilitate logical and probabilistic reasoning, as they permit various types of inference to be performed in linear-time and a feed-forward fashion like neural networks. In more recent years, the role of tractable circuits has significantly expanded as they became a computational and semantical backbone for some approaches that aim to integrate knowledge, reasoning and learning. In this article, we review the foundations …
Probabilistic circuits (PCs) are a class of tractable probabilistic models that allow efficient, often linear-time, inference of queries such as marginals and most probable explanations (MPE). However, marginal MAP, which is central to many decision-making problems, remains a hard query for PCs unless they satisfy highly restrictive structural constraints. In this paper, we develop a pruning algorithm that removes parts of the PC that are irrelevant to a marginal MAP query, shrinking the PC while maintaining the correct solution. This pruning technique is so effective that we are able to build…
