Reducing computational complexity of estimating multivariate histogram-based probabilistic model

The Estimation of Distribution Algorithms (EDAs) is a novel class of evolutionary algorithms which is motivated by the idea of building probabilistic graphical model of promising solutions to represent linkage information between variables in chromosome. Through learning of and sampling from probabilistic graphical model, new population is generated and optimization procedure is repeated until the stopping criteria are met. In this paper, the mechanism of the Estimation of Distribution Algorithms is analyzed. Currently existing EDAs are surveyed and categorized according to the probabilistic model they used, then the strengths and weakness and the future perspective of EDAs are concluded.

A probabilistic graphical model approach to stochastic multiscale partial differential equations

Model reference adaptive search (MRAS) for solving global optimization problems works with a parameterized probabilistic model on the solution space and generates at each iteration a group of candidate solutions. These candidate solutions are then used to update the parameters associated with the probabilistic model in such a way that the future search will be biased toward the region containing high-quality solutions. The parameter updating procedure in MRAS is guided by a sequence of implicit probabilistic models we call reference models. We provide a particular algorithm instantiation of the MRAS method, where the sequence of reference models can be viewed as the generalized probability distribution models for estimation of distribution algorithms (EDAs) with proportional selection scheme. In addition, we show that the model reference framework can also be used to describe the recently proposed cross-entropy (CE) method for optimization and to study its properties. Hence, this paper can also be seen as a study on the effectiveness of combining CE and EDAs. We prove global convergence of the proposed algorithm in both continuous and combinatorial domains, and we carry out numerical studies to illustrate the performance of the algorithm.

Estimation of distribution algorithms are a promising method in evolutionary computation. In the estimation of distribution algorithms, instead of using conventional crossover and mutation operations, probabilistic models are used to sample the genetic information in the next population. Although the use of probabilistic models enables EDAs to have several schema simultaneously, it sometimes causes worse performance due to converse schemata. In this paper, estimation of distribution algorithms with niche separation mechanism is proposed. Coexistence schemata are also newly introduced, where schemata at the same loci but converse information. The proposed method splits population into two subpopulations if such coexistence schemata are found. One of subpopulations is used to constitute the next generation while the probabilistic model of another subpopulation is enqueued. The queued probabilistic model is used if the current population is converged. The experimental results on max-sat problems and Ising spin glass problems show the effectiveness of the proposed method.

Problem structure, or linkage, refers to the interaction between variables in a black-box fitness function. Discovering structure is a feature of a range of algorithms, including estimation of distribution algorithms (EDAs) and perturbation methods (PMs). The complexity of structure has traditionally been used as a broad measure of problem difficulty, as the computational complexity relates directly to the complexity of structure. The EDA literature describes necessary and unnecessary interactions in terms of the relationship between problem structure and the structure of probabilistic graphical models discovered by the EDA. In this paper we introduce a classification of problems based on monotonicity invariance. We observe that the minimal problem structures for these classes often reveal that significant proportions of detected structures are unnecessary. We perform a complete classification of all functions on 3 bits. We consider nonmonotonicity linkage discovery using perturbation methods and derive a concept of directed ordinal linkage associated to optimization schedules. The resulting refined classification factored out by relabeling, shows a hierarchy of nine directed ordinal linkage classes for all 3-bit functions. We show that this classification allows precise analysis of computational complexity and parallelizability and conclude with a number of suggestions for future work.

The layout positioning problem of facilities on a straight line is known as Single Row Facility Layout Problem (SRFLP). The objective of SRFLP, categorized as NP-Complete problem, is to arrange the layout such that the sum of distances between all facilities’ pairs can be minimized. Extended Artificial Chromosome Genetic Algorithm (eACGA) is a promising algorithm that has been proposed recently. eACGA extends the probabilistic model in Estimation of Distribution Algorithms (EDAs) and then hybridize it with Genetic Algorithms (GAs). eACGA is proven to produce an excellent solution for scheduling problem. In this paper, we modify the eACGA to solve SRFLP. Computational results on benchmark problems show the effectiveness of eACGA for solving SRFLP. Key Words: single row facility layout, estimation distribution algorithm, genetic algorithm.

Probabilistic graphical modeling (PGM) provides a framework for formulating an interpretable generative process of data and expressing uncertainty about unknowns. This makes PGM very useful for understanding phenomena underlying data and for decision making. PGM has seen great success in domains where interpretable inferences are key, e.g. marketing, medicine, neuroscience, and social science. However, PGM tends to lack flexibility, which has hindered its use when it comes to modeling large scale high-dimensional complex data and performing tasks that require flexibility (e.g. in vision and language applications.)\n\nDeep learning (DL) is another framework for modeling and learning from data that has seen great empirical success in recent years. DL is very powerful and offers great flexibility, but it lacks the interpretability and calibration of PGM. \n\nThis thesis develops deep probabilistic graphical modeling (DPGM). DPGM consists in leveraging DL to make PGM more flexible. DPGM brings about new methods for learning from data that exhibit the advantages of both PGM and DL. \n\nWe use DL within PGM to build flexible models endowed with an interpretable latent structure. One family of models we develop extends exponential family principal component analysis (EF-PCA) using neural networks to improve predictive performance while enforcing the interpretability of the latent factors. Another model class we introduce enables accounting for long-term dependencies when modeling sequential data, which is a challenge when using purely DL or PGM approaches. This model class for sequential data was successfully applied to language modeling, unsupervised document representation learning for sentiment analysis, conversation modeling, and patient representation learning for hospital readmission prediction. Finally, DPGM successfully solves several outstanding problems of probabilistic topic models. \n\nLeveraging DL within PGM also brings about new algorithms for learning with complex data. For example, we develop entropy-regularized adversarial learning, a learning paradigm that deviates from the traditional maximum likelihood approach used in PGM. From the DL perspective, entropy-regularized adversarial learning provides a solution to the long-standing mode collapse problem of generative adversarial networks.

Probabilistic graphical models are a fundamental tool in probabilistic modeling, machine learning and artificial intelligence. They allow us to integrate in a natural way expert knowledge, physical modeling, heterogeneous and correlated data and quantities of interest. For exactly this reason, multiple sources of model uncertainty are inherent within the modular structure of the graphical model. In this paper we develop information-theoretic, robust uncertainty quantification methods and non-parametric stress tests for directed graphical models to assess the effect and the propagation through the graph of multi-sourced model uncertainties to quantities of interest. These methods allow us to rank the different sources of uncertainty and correct the graphical model by targeting its most impactful components with respect to the quantities of interest. Thus, from a machine learning perspective, we provide a mathematically rigorous approach to correctability that guarantees a systematic selection for improvement of components of a graphical model while controlling potential new errors created in the process in other parts of the model. We demonstrate our methods in two physico-chemical examples, namely quantum scale-informed chemical kinetics and materials screening to improve the efficiency of fuel cells.

Inference is a central problem in probabilistic graphical models, and is often the main sub-step in probabilistic learning procedures. Thus, accurate inference algorithms are essential to both answer queries on a learned model, as well as to learn a robust model. Gibbs sampling is arguably one of the most popular approximate inference methods that has been widely used for probabilistic inference in several different domains including natural language processing, computer vision. etc. Here, we develop an approach that improves the performance of blocked Gibbs sampling, an advanced variant of the Gibbs sampling algorithm. Specifically, we utilize correlation among variables in the probabilistic graphical model to develop an adaptive blocked Gibbs sampler that automatically tunes its proposal distribution based on statistics derived from previous samples. Specifically, we adapt the proposal such that we sample blocks containing highly correlated variables more often than the others. This in turn helps improve probability estimates given by the sampler, by selecting hard-to-sample variables more often during the sampling procedure. Further, since adaptation breaks the Markovian property of the sampler, we develop a method to guarantee that our sampler converges to the correct stationary distribution despite being non-Markovian, by diminishing the adaptation of the selection probabilities over time. We evaluate our method with several discrete probabilistic graphical models taken from UAI challenge problems corresponding to different domains, and show that our approach is superior in terms of accuracy as compared to methods that ignore correlation information in the proposal distribution of the sampler.

In an estimation of distribution algorithm (EDA), global population distribution is modeled by a probabilistic model, from which new trial solutions are sampled, whereas individual location information is not directly and fully exploited. In this paper, we suggest to combine an EDA with cheap and expensive local search (LS) methods for making use of both global statistical information and individual location information. In our approach, part of a new solution is sampled from a modified univariate histogram probabilistic model and the rest is generated by refining a parent solution through a cheap LS method that does not need any function evaluation. When the population has converged, an expensive LS method is applied to improve a promising solution found so far. Controlled experiments have been carried out to investigate the effects of the algorithm components and the control parameters, the scalability on the number of variables, and the running time. The proposed algorithm has been compared with two state-of-the-art algorithms on two test suites of 27 test instances. Experimental results have shown that, for simple test instances, our algorithm can produce better or similar solutions but with faster convergence speed than the compared methods and for some complicated test instances it can find better solutions.

Estimation of distribution algorithms (EDAs) are intelligent systems that take advantage of statistical learning techniques. The distribution of promising regions in the search space is estimated and probabilistically guide the new particles’ searching towards them in EDAs. But EDAs cannot solve complex optimisation problems reliably and efficiently because of premature convergence and difficulty of complex probabilistic model learning. This paper presents a PCA-EDA algorithm which introduces principal component analysis (PCA) into simple Gaussian model-based EDA. PCA-EDA is aimed to keep the balance of accuracy and efficiency of probabilistic model learning, as well as to avoid premature convergence by PCA’s ability of analysis of Gaussian model’s variance. From the results of numerical experiments, it is showed that the proposed method is feasible and effective for complex optimisation problems.

An introduction and survey of estimation of distribution algorithms

Artificial chromosomes with genetic algorithm 2 (ACGA2) for single machine scheduling problems with sequence-dependent setup times

Probabilistic graphical models (PGM) are one of the rich frameworks. These models are used over complex domains for coding probability distributions. The joint distributions interact with each other over large numbers of random variables and are the combination of statistics and computer science. These concepts are dependent on theories such as probability theory, graph algorithms, machine learning, which make a basic tool in devising many machine learning problems. These are the origin for the contemporary methods in an extensive range of applications. These applications range as medical diagnosis, image understanding, speech recognition, natural language processing, etc. Graphical models are one of dominant tools for handling image processing applications. On the other hand, the volume of image data gives rise to a problem. The representation of all possible graphical model node variables with that of discrete states heads to the number of states for the model. This leads to interpretation computationally obstinate. Many projects involve a human intervention or an automated system to obtain the consensus established on existing information. The PGM, discussed in this chapter, offers a variety of approaches. The approach is based on models and allows interpretable models to be built which then is employed by reasoning algorithms. These models are also studied significantly from data and allow the approaches for cases where the model is manually built. Most real-world applications are of uncertain data which makes a model building more challenging. This chapter emphasizes on PGM where the uncertainty of data is obvious. PGM provides models that are more realistic. These are extended from Bayesian networks, undirected Markov networks, discrete and continuous models, and extensions to deal with dynamical systems and relational data also. For each class of models, the chapter describes the fundamental bases: representation, inference, and learning. Finally, the chapter considers the decision making under the uncertainty of the data.

Most clustering and classification methods are based on the assumption that the objects to be clustered are independent. However, in more and more modern applications, data are structured in a way that makes this assumption not realistic and potentially misleading. A typical example that can be viewed as a clustering task is image segmentation where the objects are the pixels on a regular grid and depend on neighbouring pixels on this grid. Also, when data are geographically located, it is of interest to cluster data with an underlying dependence structure accounting for some spatial localisation. These spatial interactions can be naturally encoded via a graph not necessarily regular as a grid. Data sets can then be modelled via Markov random fields and mixture models (e.g. the so-called MRF and Hidden MRF). More generally, probabilistic graphical models are tools that can be used to represent and manipulate data in a structured way while modeling uncertainty. This chapter introduces the basic concepts. The two main classes of probabilistic graphical models are considered: Bayesian networks and Markov networks. The key concept of conditional independence and its link to Markov properties is presented. The main problems that can be solved with such tools are described. Some illustrations are given associated with some practical work.