Improving probabilistic inference in graphical models with determinism and cycles

Lifted probabilistic inference algorithms have been successfully applied to a large number of symmetric graphical models. Unfortunately, the majority of real-world graphical models is asymmetric. This is even the case for relational representations when evidence is given. Therefore, more recent work in the community moved to making the models symmetric and then applying existing lifted inference algorithms. However, this approach has two shortcomings. First, all existing over-symmetric approximations require a relational representation such as Markov logic networks. Second, the induced symmetries often change the distribution significantly, making the computed probabilities highly biased. We present a framework for probabilistic sampling-based inference that only uses the induced approximate symmetries to propose steps in a Metropolis-Hastings style Markov chain. The framework, therefore, leads to improved probability estimates while remaining unbiased. Experiments demonstrate that the approach outperforms existing MCMC algorithms.

The problem of separating superimposed action potentials in electromyographic (EMG) signals is considered. Based on a graphical model (factor graph), a new EMG signal decomposition algorithm that uses loopy belief propagation is presented. Results show that the algorithm is capable of decomposing multiple superpositions in simulated and measured EMG signals.

: Graphical models fuse probability theory and graph theory in such a way as to permit efficient representation and computation with probability distributions. They intuitively capture statistical relationships among random variables and provide a succinct formalism that allows for the development of tractable algorithms for statistical inference. In recent years, certain types of graphical models, particularly Bayesian networks and dynamic Bayesian networks (DBNs), have been applied to various problems in missile defense that involve decision making under uncertainty and estimation in dynamic systems, such as data association, multitarget tracking, and classification. While the set of problems addressed in the missile defense arena is quite diverse, all require mathematically sound machinery for dealing with uncertainty. The graphical model regime provides a robust, flexible framework for representing and computationally handling uncertainty in real-world problems. While the graphical model regime is relatively new, it has deep roots in many fields, as the formalism generalizes many commonly used stochastic models, including Kalman filters and hidden Markov models. In this report, we describe the mathematical foundations of graphical models and statistical inference, focusing on the concepts and techniques that are most useful to the problem of decision making in dynamic systems under uncertainty. In general, statistical inference on a graphical model is an NP-Hard problem, so there have been large research efforts that involve developing algorithms for performing inference efficiently for certain classes of models, or obtaining approximations for quantities of interest using algorithms for approximate inference. Due to the breadth of problems, a broad class of algorithms has been of interest to researchers over the past several years. As such, the need arose for an extensible and efficient software library for performing statistical inference on graphical models.

Estimating the partition function is a key computation in graphical models that is required for important tasks like parameter estimation, model selection and structure learning. However, this computation is intractable in general. Thus, developing efficient and accurate approximations is of considerable interest. Variational methods express the computation of the partition function as an optimization problem (solving which intractable in general) and then relax the optimization problem in various ways to obtain approximations to the partition function. Two popular algorithms belonging to this framework are loopy belief propagation (LBP) and tree-reweighted belief propagation (TRW-BP). Both these algorithms are so-called message passingalgorithms that work by updating local beliefs about probabilities at graph nodes by passing messages between the nodes in the graph until convergence is achieved. TRW-BP is guaranteed to give an upper bound on the partition function. However, neither algorithm is guaranteed to converge in general. Although convergent alternatives to TRW-BP have been proposed, they have no guarantees on the number of iterations required for convergence. Thus, these algorithms could be prohibitively expensive for applications requiring repeated inference on large graphs (like training large CRFs). In order to overcome this problem, Sutton et al. propose the piecewise approximation(PW) to the partition function. PW tends to be loose, but can be computed very fast, because it has a closed-form expression. Sutton et al. apply it successfully to common NLP tasks. In this paper, for a special class of potentials (that contain associative potentials), we prove that both LBP and TRW-BP converge in a single iteration. Using this fact, we obtain closed-form expressions for the TRW and LBP approximations to the partition function. In recent work, Wainwright et al. prove that LBP gives a lower bound on the partition function for binary attractive MRFs. We thus also get closed-form lower bounds for attractive associative MRFs. This enables us to obtain bounds on the error between the true partition function and these approximations for attractive associative MRFs. We also construct examples which show that TRW and LBP can give arbitrarily bad approximations to the marginal probabilities. Using the closed-form bounds for these special cases, we also develop novel closed-form upper bounds for arbitrary MRFs and closed-form lower bounds for associative binary MRFs. We also present experimental results showing that the new upper bounds are almost always tighter than the piecewise bound. The experiments also show that the novel lover bounds beat popular existing bounds like the mean-field bound on densely connected graphs.

While loopy belief propagation (LBP) performs reasonably well for inference in some Gaussian graphical models with cycles, its performance is unsatisfactory for many others. In particular for some models LBP does not converge, and in general when it does converge, the computed variances are incorrect (except for cycle-free graphs for which belief propagation (BP) is non-iterative and exact). In this paper we propose {\em feedback message passing} (FMP), a message-passing algorithm that makes use of a special set of vertices (called a {\em feedback vertex set} or {\em FVS}) whose removal results in a cycle-free graph. In FMP, standard BP is employed several times on the cycle-free subgraph excluding the FVS while a special message-passing scheme is used for the nodes in the FVS. The computational complexity of exact inference is $O(k^2n)$, where $k$ is the number of feedback nodes, and $n$ is the total number of nodes. When the size of the FVS is very large, FMP is intractable. Hence we propose {\em approximate FMP}, where a pseudo-FVS is used instead of an FVS, and where inference in the non-cycle-free graph obtained by removing the pseudo-FVS is carried out approximately using LBP. We show that, when approximate FMP converges, it yields exact means and variances on the pseudo-FVS and exact means throughout the remainder of the graph. We also provide theoretical results on the convergence and accuracy of approximate FMP. In particular, we prove error bounds on variance computation. Based on these theoretical results, we design efficient algorithms to select a pseudo-FVS of bounded size. The choice of the pseudo-FVS allows us to explicitly trade off between efficiency and accuracy. Experimental results show that using a pseudo-FVS of size no larger than $\log(n)$, this procedure converges much more often, more quickly, and provides more accurate results than LBP on the entire graph.

Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference in MLNs is probabilistic and it is often performed by approximate methods such as Markov chain Monte Carlo (MCMC) Gibbs sampling. An MLN has many regular, symmetric structures that can be exploited at both first-order level and in the generated Markov network. We analyze the graph structures that are produced by various lifting methods and investigate the extent to which quantum protocols can be used to speed up Gibbs sampling with state preparation and measurement schemes. We review different such approaches, discuss their advantages, theoretical limitations, and their appeal to implementations. We find that a straightforward application of a recent result yields exponential speedup compared to classical heuristics in approximate probabilistic inference, thereby demonstrating another example where advanced quantum resources can potentially prove useful in machine learning.

Accuracy of Loopy belief propagation in Gaussian models

In numerous real world applications, from sensor networks to computer vision to natural text processing, one needs to reason about the system in question in the face of uncertainty. A key problem in all those settings is to compute the probability distribution over the variables of interest (the query) given the observed values of other random variables (the evidence). Probabilistic graphical models (PGMs) have become the approach of choice for representing and reasoning with high-dimensional probability distributions. However, for most models capable of accurately representing real-life distributions, inference is fundamentally intractable. As a result, optimally balancing the expressive power and inference complexity of the models, as well as designing better approximate inference algorithms, remain important open problems with potential to significantly improve the quality of answers to probabilistic queries. This thesis contributes algorithms for learning and approximate inference in probabilistic graphical models that improve on the state of the art by emphasizing the computational aspects of inference over the representational properties of the models. Our contributions fall into two categories: learning accurate models where exact inference is tractable and speeding up approximate inference by focusing computation on the query variables and only spending as much effort on the remaining parts of the model as needed to answer the query accurately. First, for a case when the set of evidence variables is not known in advance and a single model is needed that can be used to answer any query well, we propose a polynomial time algorithm for learning the structure of tractable graphical models with quality guarantees, including PAC learnability and graceful degradation guarantees. Ours is the first efficient algorithm to provide this type of guarantees. A key theoretical insight of our approach is a tractable upper bound on the mutual information of arbitrarily large sets of random variables that yields exponential speedups over the exact computation. Second, for a setting where the set of evidence variables is known in advance, we propose an approach for learning tractable models that tailors the structure of the model for the particular value of evidence that become known at test time. By avoiding a commitment to a single tractable structure during learning, we are able to expand the representation power of the model without sacrificing efficient exact inference and parameter learning. We provide a general framework that allows one to leverage existing structure learning algorithms for discovering high-quality evidence-specific structures. Empirically, we demonstrate state of the art accuracy on real-life datasets and an order of magnitude speedup. Finally, for applications where the intractable model structure is a given and approximate inference is needed, we propose a principled way to speed up convergence of belief propagation by focusing the computation on the query variables and away from the variables that are of no direct interest to the user. We demonstrate significant speedups over the state of the art on large-scale relational models. Unlike existing approaches, ours does not involve model simplification, and thus has an advantage of converging to the fixed point of the full model. More generally, we argue that the common approach of concentrating on the structure of representation provided by PGMs, and only structuring the computation as representation allows, is suboptimal because of the fundamental computational problems. It is the computation that eventually yields answers to the queries, so directly focusing on structure of computation is a natural direction for improving the quality of the answers. The results of this thesis are a step towards adapting the structure of computation as a foundation of graphical models.

Many different algorithms developed in statistical physics, coding theory, signal processing, and artificial intelligence can be expressed by graphical models and solved (either exactly or approximately) with iterative message-passing algorithms on the model. One quantity of interest in these algorithms is the partition function. In graphical models without cycle (trees), the partition function can be computed efficiently by means of message-passing algorithms, like GDL or the sum-product algorithm. In contrast, when graphical models contain cycles, the computation of the partition function is in general intractable. Our contributions in this dissertation are: deriving deterministic upper and lower bounds on partition function, and developing methods to approximate this quantity with Monte Carlo methods. Specifically, we obtain subtree-based upper and lower bounds which lead to theoretical results on optimality properties of the minimum entropy sub-tree and finally lead to a greedy algorithm. At last, we introduce and analyze a number of estimators that use Gibbs sampling to draw samples from different target distributions to estimate the value of the partition function. In one estimator, we demonstrate a novel strategy which combines Gibbs sampling and message-passing algorithms on trees.

Despite great recent advances achieved by deep neural networks (DNNs), they are often vulnerable to adversarial attacks. Intensive research efforts have been made to improve the robustness of DNNs; however, most empirical defenses can be adaptively attacked again, and the theoretically certified robustness is limited, especially on large-scale datasets. One potential root cause of such vulnerabilities for DNNs is that although they have demonstrated powerful expressiveness, they lack the reasoning ability to make robust and reliable predictions. In this paper, we aim to integrate domain knowledge to enable robust learning with the reasoning paradigm. In particular, we propose a certifiably robust learning with reasoning pipeline (CARE), which consists of a learning component and a reasoning component. Concretely, we use a set of standard DNNs to serve as the learning component to make semantic predictions (e.g., whether the input is furry), and we leverage the probabilistic graphical models, such as Markov logic networks (MLN), to serve as the reasoning component to enable knowledge/logic reasoning (e.g., <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\text{IsPanda} \Longrightarrow \text{IsFurry}$</tex> ). However, it is known that the exact inference of MLN (reasoning) is #P-complete, which limits the scalability of the pipeline. To this end, we propose to approximate the MLN inference via variational inference based on an efficient expectation maximization algorithm. In particular, we leverage graph convolutional networks (GCNs) to encode the posterior distribution during variational inference and update the parameters of GCNs (E-step) and the weights of knowledge rules in MLN (M-step) iteratively. We conduct extensive experiments on different datasets such as AwA2, Word50, GTSRB, and PDF malware, and we show that CARE achieves significantly higher certified robustness (e.g., the certified accuracy is improved from 36.0% to 61.8% under <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\ell_{2}$</tex> radius 2.0 on AwA2) compared with the state-of-the-art baselines. We additionally conducted different ablation studies to demonstrate the empirical robustness of CARE and the effectiveness of different knowledge integration. The official code is available at https://github.com/javyduck/CARE.

Belief Propagation (BP) is a widely used approximation for exact probabilistic inference in graphical models, such as Markov Random Fields (MRFs). In graphs with cycles, however, no exact convergence guarantees for BP are known, in general. For the case when all edges in the MRF carry the same symmetric, doubly stochastic potential, recent works have proposed to approximate BP by linearizing the update equations around default values, which was shown to work well for the problem of node classification. The present paper generalizes all prior work and derives an approach that approximates loopy BP on any pairwise MRF with the problem of solving a linear equation system. This approach combines exact convergence guarantees and a fast matrix implementation with the ability to model heterogenous networks. Experiments on synthetic graphs with planted edge potentials show that the linearization has comparable labeling accuracy as BP for graphs with weak potentials, while speeding-up inference by orders of magnitude.

Table constraints are very important in constraint programming as they are present in many real problems from areas such as configuration and databases. As a result, numerous specialized algorithms that achieve generalized arc consistency (GAC) on table constraints have been proposed. Since these algorithms achieve GAC, they operate on one constraint at a time. In this paper we propose an efficient algorithm for table constraints that achieves a stronger local consistency than GAC. This algorithm, called maxRPWC+, is based on the local consistency maxRPWC and allows the efficient handling of intersecting table constraints. Experimental results from benchmark problems demonstrate that maxRPWC+ is clearly more robust than a state-of-the-art GAC algorithm in classes of problems with interleaved table constraints, being orders of magnitude faster in some of these classes.

In structured classification problems, there is a direct conflict between expressive models and efficient inference: while graphical models such as Markov random fields or factor graphs can represent arbitrary dependences among instance labels, the cost of inference via belief propagation in these models grows rapidly as the graph structure becomes more complicated. One important source of complexity in belief propagation is the need to marginalize large factors to compute messages. This operation takes time exponential in the number of variables in the factor, and can limit the expressiveness of the models we can use. In this paper, we study a new class of potential functions, which we call decomposable k-way potentials, and provide efficient algorithms for computing messages from these potentials during belief propagation. We believe these new potentials provide a good balance between expressive power and efficient inference in practical structured classification problems. We discuss three instances of decomposable potentials: the associative Markov network potential, the nested junction tree, and a new type of potential which we call the voting potential. We use these potentials to classify images of protein subcellular location patterns in groups of cells. Classifying subcellular location patterns can help us answer many important questions in computational biology, including questions about how various treatments affect the synthesis and behavior of proteins and networks of proteins within a cell. Our new representation and algorithm lead to substantial improvements in both inference speed and classification accuracy.

Information fusion in navigation systems via factor graph based incremental smoothing

Template matching of iris images for biometric recognition typically suffers from both local deformations between the template and query images and large occlusions from the eyelid. In this work, we model deformation and occlusion as a set of hidden variables for each iris comparison. We use afield of directional vectors to represent deformation and a field of binary variables to represent occlusion. We impose a probability distribution on these fields using a lattice-type undirected graphical model, in which the graph edges represent interdependencies between neighboring iris regions. Gabor wavelet-based similarity scores and intensity statistics are used as observations in the model. Loopy belief propagation is applied to estimate the conditional distributions on the hidden variables, which are in turn used to compute final match scores. We present underlying theory as well as experimental results from both the CASIA iris database and the database provided for the iris challenge evaluation (ICE). We show that our proposed method significantly improves recognition accuracy on these datasets over existing methods.