We propose a scalable algorithm for model selection in sigmoid belief networks (SBNs), based on the factorized asymptotic Bayesian (FAB) framework. We derive the corresponding generalized factorized information criterion (gFIC) for the SBN, which is proven to be statistically consistent with the marginal log-likelihood. To capture the dependencies within hidden variables in SBNs, a recognition network is employed to model the variational distribution. The resulting algorithm, which we call FABIA, can simultaneously execute both model selection and inference by maximizing the lower bound of gFIC. On both synthetic and real data, our experiments suggest that FABIA, when compared to state-of-the-art algorithms for learning SBNs, $(i)$ produces a more concise model, thus enabling faster testing; $(ii)$ improves predictive performance; $(iii)$ accelerates convergence; and $(iv)$ prevents overfitting.

We propose a scalable algorithm for model selection in sigmoid belief networks (SBNs), based on the factorized asymptotic Bayesian (FAB) framework. We derive the corresponding generalized factorized information criterion (gFIC) for the SBN, which is proven to be statistically consistent with the marginal log-likelihood. To capture the dependencies within hidden variables in SBNs, a recognition network is employed to model the variational distribution. The resulting algorithm, which we call FABIA, can simultaneously execute both model selection and inference by maximizing the lower bound of gFIC. On both synthetic and real data, our experiments suggest that FABIA, when compared to state-of-the-art algorithms for learning SBNs, $(i)$ produces a more concise model, thus enabling faster testing; $(ii)$ improves predictive performance; $(iii)$ accelerates convergence; and $(iv)$ prevents overfitting.

The importance of interpretability of machine learning models has been increasing due to emerging enterprise predictive analytics, threat of data privacy, accountability of artificial intelligence in society, and so on. Piecewise linear models have been actively studied to achieve both accuracy and interpretability. They often produce competitive accuracy against state-of-the-art non-linear methods. In addition, their representations (i.e., rule-based segmentation plus sparse linear formula) are often preferred by domain experts. A disadvantage of such models, however, is high computational cost for simultaneous determinations of the number of "pieces" and cardinality of each linear predictor, which has restricted their applicability to middle-scale data sets. This paper proposes a distributed factorized asymptotic Bayesian (FAB) inference of learning piece-wise sparse linear models on distributed memory architectures. The distributed FAB inference solves the simultaneous model selection issue without communicating $O(N)$ data where N is the number of training samples and achieves linear scale-out against the number of CPU cores. Experimental results demonstrate that the distributed FAB inference achieves high prediction accuracy and performance scalability with both synthetic and benchmark data.

Sparsity learning with known grouping structures has received considerable attention due to wide modern applications in high-dimensional data analysis. Although advantages of using group information have been well-studied by shrinkage-based approaches, benefits of group sparsity have not been well-documented for greedy-type methods, which much limits our understanding and use of this important class of methods. In this paper, generalizing from a popular forward-backward greedy approach, we propose a new interactive greedy algorithm for group sparsity learning and prove that the proposed greedy-type algorithm attains the desired benefits of group sparsity under high dimensional settings. An estimation error bound refining other existing methods and a guarantee for group support recovery are also established simultaneously. In addition, an interactive feature is incorporated to allow extra algorithm flexibility without compromise in theoretical properties. The promising use of our proposal is demonstrated through numerical evaluations including a real industrial application in human activity recognition.

The cardinality constraint is an intrinsic way to restrict the solution structure in many domains, for example, sparse learning, feature selection, and compressed sensing. To solve a cardinality constrained problem, the key challenge is to solve the projection onto the cardinality constraint set, which is NP-hard in general when there exist multiple overlapped cardinality constraints. In this paper, we consider the scenario where the overlapped cardinality constraints satisfy a Three-view Cardinality Structure (TVCS), which reflects the natural restriction in many applications, such as identification of gene regulatory networks and task-worker assignment problem. We cast the projection into a linear programming, and show that for TVCS, the vertex solution of this linear programming is the solution for the original projection problem. We further prove that such solution can be found with the complexity proportional to the number of variables and constraints. We finally use synthetic experiments and two interesting applications in bioinformatics and crowdsourcing to validate the proposed TVCS model and method.

This paper deals with price optimization, which is to find the best pricing strategy that maximizes revenue or profit, on the basis of demand forecasting models. Though recent advances in regression technologies have made it possible to reveal price-demand relationship of a number of multiple products, most existing price optimization methods, such as mixed integer programming formulation, cannot handle tens or hundreds of products because of their high computational costs. To cope with this problem, this paper proposes a novel approach based on network flow algorithms. We reveal a connection between supermodularity of the revenue and cross elasticity of demand. On the basis of this connection, we propose an efficient algorithm that employs network flow algorithms. The proposed algorithm can handle hundreds or thousands of products, and returns an exact optimal solution under an assumption regarding cross elasticity of demand. Even in case in which the assumption does not hold, the proposed algorithm can efficiently find approximate solutions as good as can other state-of-the-art methods, as empirical results show.

Factorial hidden Markov models (FHMMs) are powerful tools of modeling sequential data. Learning FHMMs yields a challenging simultaneous model selection issue, i.e., selecting the number of multiple Markov chains and the dimensionality of each chain. Our main contribution is to address this model selection issue by extending Factorized Asymptotic Bayesian (FAB) inference to FHMMs. First, we offer a better approximation of marginal log-likelihood than the previous FAB inference. Our key idea is to integrate out transition probabilities, yet still apply the Laplace approximation to emission probabilities. Second, we prove that if there are two very similar hidden states in an FHMM, i.e. one is redundant, then FAB will almost surely shrink and eliminate one of them, making the model parsimonious. Experimental results show that FAB for FHMMs significantly outperforms state-of-the-art nonparametric Bayesian iFHMM and Variational FHMM in model selection accuracy, with competitive held-out perplexity.

Factorized information criterion (FIC) is a recently developed approximation technique for the marginal log-likelihood, which provides an automatic model selection framework for a few latent variable models (LVMs) with tractable inference algorithms. This paper reconsiders FIC and fills theoretical gaps of previous FIC studies. First, we reveal the core idea of FIC that allows generalization for a broader class of LVMs, including continuous LVMs, in contrast to previous FICs, which are applicable only to binary LVMs. Second, we investigate the model selection mechanism of the generalized FIC. Our analysis provides a formal justification of FIC as a model selection criterion for LVMs and also a systematic procedure for pruning redundant latent variables that have been removed heuristically in previous studies. Third, we provide an interpretation of FIC as a variational free energy and uncover a few previously-unknown their relationships. A demonstrative study on Bayesian principal component analysis is provided and numerical experiments support our theoretical results.

Group LASSO is widely used to enforce the structural sparsity, which achieves the sparsity at the inter-group level. In this paper, we propose a new formulation called "exclusive group LASSO", which brings out sparsity at intragroup level in the context of feature selection. The proposed exclusive group LASSO is applicable on any feature structures, regardless of their overlapping or non-overlapping structures. We provide analysis on the properties of exclusive group LASSO, and propose an effective iteratively re-weighted algorithm to solve the corresponding optimization problem with rigorous convergence analysis. We show applications of exclusive group LASSO for uncorrelated feature selection. Extensive experiments on both synthetic and real-world datasets validate the proposed method.

Region-specific linear models are widely used in practical applications because of their non-linear but highly interpretable model representations. One of the key challenges in their use is non-convexity in simultaneous optimization of regions and region-specific models. This paper proposes novel convex region-specific linear models, which we refer to as partition-wise linear models. Our key ideas are 1) assigning linear models not to regions but to partitions (region-specifiers) and representing region-specific linear models by linear combinations of partition-specific models, and 2) optimizing regions via partition selection from a large number of given partition candidates by means of convex structured regularizations. In addition to providing initialization-free globally-optimal solutions, our convex formulation makes it possible to derive a generalization bound and to use such advanced optimization techniques as proximal methods and decomposition of the proximal maps for sparsity-inducing regularizations. Experimental results demonstrate that our partition-wise linear models perform better than or are at least competitive with state-of-the-art region-specific or locally linear models.