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Register a new userModel Order Selection Based on Information Theoretic Criteria: Design of the Penalty
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Information theoretic criteria (ITC) have been widely adopted in engineering and statistics for selecting, among an ordered set of candidate models, the one that better fits the observed sample data. The selected model minimizes a penalized likelihood metric, where the penalty is determined by the criterion adopted. While rules for choosing a penalty that guarantees a consistent estimate of the model order are known, theoretical tools for its design with finite samples have never been provided in a general setting. In this paper, we study model order selection for finite samples under a design perspective, focusing on the generalized information criterion (GIC), which embraces the most common ITC. The theory is general, and as case studies we consider: a) the problem of estimating the number of signals embedded in additive white Gaussian noise (AWGN) by using multiple sensors; b) model selection for the general linear model (GLM), which includes e.g. the problem of estimating the number of sinusoids in AWGN. The analysis reveals a trade-off between the probabilities of overestimating and underestimating the order of the model. We then propose to design the GIC penalty to minimize underestimation while keeping the overestimation probability below a specified level. For the considered problems, this method leads to analytical derivation of the optimal penalty for a given sample size. A performance comparison between the penalty optimized GIC and common AIC and BIC is provided, demonstrating the effectiveness of the proposed design strategy.
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Selecting a minimal feature set that is maximally informative about a target variable is a central task in machine learning and statistics. Information theory provides a powerful framework for formulating feature selection algorithms -- yet, a rigorous, information-theoretic definition of feature relevancy, which accounts for feature interactions such as redundant and synergistic contributions, is still missing. We argue that this lack is inherent to classical information theory which does not provide measures to decompose the information a set of variables provides about a target into unique, redundant, and synergistic contributions. Such a decomposition has been introduced only recently by the partial information decomposition (PID) framework. Using PID, we clarify why feature selection is a conceptually difficult problem when approached using information theory and provide a novel definition of feature relevancy and redundancy in PID terms. From this definition, we show that the conditional mutual information (CMI) maximizes relevancy while minimizing redundancy and propose an iterative, CMI-based algorithm for practical feature selection. We demonstrate the power of our CMI-based algorithm in comparison to the unconditional mutual information on benchmark examples and provide corresponding PID estimates to highlight how PID allows to quantify information contribution of features and their interactions in feature-selection problems.
In high-dimensional and/or non-parametric regression problems, regularization (or penalization) is used to control model complexity and induce desired structure. Each penalty has a weight parameter that indicates how strongly the structure corresponding to that penalty should be enforced. Typically the parameters are chosen to minimize the error on a separate validation set using a simple grid search or a gradient-free optimization method. It is more efficient to tune parameters if the gradient can be determined, but this is often difficult for problems with non-smooth penalty functions. Here we show that for many penalized regression problems, the validation loss is actually smooth almost-everywhere with respect to the penalty parameters. We can therefore apply a modified gradient descent algorithm to tune parameters. Through simulation studies on example regression problems, we find that increasing the number of penalty parameters and tuning them using our method can decrease the generalization error.
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In this paper, we study the information theoretic bounds for exact recovery in sub-hypergraph models for community detection. We define a general model called the $m-$uniform sub-hypergraph stochastic block model ($m-$ShSBM). Under the $m-$ShSBM, we use Fanos inequality to identify the region of model parameters where any algorithm fails to exactly recover the planted communities with a large probability. We also identify the region where a Maximum Likelihood Estimation (MLE) algorithm succeeds to exactly recover the communities with high probability. Our bounds are tight and pertain to the community detection problems in various models such as the planted hypergraph stochastic block model, the planted densest sub-hypergraph model, and the planted multipartite hypergraph model.
Obtaining labels can be costly and time-consuming. Active learning allows a learning algorithm to intelligently query samples to be labeled for efficient learning. Fisher information ratio (FIR) has been used as an objective for selecting queries in active learning. However, little is known about the theory behind the use of FIR for active learning. There is a gap between the underlying theory and the motivation of its usage in practice. In this paper, we attempt to fill this gap and provide a rigorous framework for analyzing existing FIR-based active learning methods. In particular, we show that FIR can be asymptotically viewed as an upper bound of the expected variance of the log-likelihood ratio. Additionally, our analysis suggests a unifying framework that not only enables us to make theoretical comparisons among the existing querying methods based on FIR, but also allows us to give insight into the development of new active learning approaches based on this objective.
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