Problem Of Parameter Learning Research Articles

Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data

Article Parameter Learning of Probabilistic Boolean Control Networks with Input-Output Data Hongwei Chen 1,*, Qi Chen 1, Bo Shen 1, and Yang Liu 2 1 College of Information Science and Technology, Donghua University, Shanghai 201620, China 2 School of Mathematical Sciences, Zhejiang Normal University, Jinhua 321004, China * Correspondence: hongwei@dhu.edu.cn Received: 23 September 2023 Accepted: 27 November 2023 Published: 26 March 2024 Abstract: This paper investigates the parameter learning problem for the probabilistic Boolean control networks (PBCNs) with input-output data. Firstly, an algebraic expression of the PBCNs is obtained by taking advantage of the semi-tensor product technique, and then, the parameter learning problem is transformed into an optimal problem to reveal the parameter matrices of a linear system in a computationally efficient way. Secondly, two recursive semi-tensor product based algorithms are designed to calculate the forward and backward probabilities. Thirdly, the expectation maximization algorithm is proposed as an elaborate technique to address the parameter learning problem. In addition, a useful index is introduced to describe the performance of the proposed parameter learning algorithm. Finally, two numerical examples are employed to demonstrate the reliability of the proposed parameter learning approach.

Learning bayesian network parameters from limited data by integrating entropy and monotonicity

High-accuracy parameter learning in Bayesian Networks (BNs) is a key challenge in real-time decision support applications, particularly when the available data are limited. Prior/Expert knowledge was introduced to eliminate the drawbacks of insufficient information; however, this method is subjective. In this study, we explored the use of monotonicity constraints to control the causal relationships between the nodes and their parents-nodes in BNs and proposed a new learning algorithm called global domain monotonicity based on maximum information entropy (GDM-MIE), which was designed for parameter learning in uncertain discrete BNs with nonlinear equality constraints when only finite data are available. In the proposed algorithm, a class of monotonicity is encoded as a constraint on information entropy and the parameter learning problem is transformed into constraints among the node parameters based on a known network. Furthermore, we considered the parameters as uncertain entropy information and discussed the monotonicity among parameters in the global spatial domain, proving the accuracy of the logical relationships of the model, system reliability, and time complexity. Finally, the proposed method was validated using standard BNs, and its performance was analyzed by comparing the proposed method with the existing learning algorithms. The results showed that the proposed method is more accurate and has better Kullback–Leibler divergence. To revalidate the rationality of the proposed method, the Alarm and Asia networks were employed as special cases. The GDM-MIE was found to achieve the intended goal of observing the estimated parameters by closely approximating the original real parameters with a small sample size, indicating that the proposed algorithm can served as an efficient and feasible method for learning Bayesian parameter.

Optimality Conditions for Bilevel Imaging Learning Problems with Total Variation Regularization

We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.

Learning‐based robust control methodologies under information constraints

Learning‐based robust control methodologies under information constraints

A super resolution target separation and reconstruction approach for single channel sar against deceptive jamming

The excellent remote sensing ability of synthetic aperture radar (SAR) will be misled seriously when it encounters deceptive jamming which possesses high fidelity and fraudulence. In this paper, the dynamic synthetic aperture (DSA) scheme is used to extract the difference between the true and false targets. A simultaneous deceptive jamming suppression and target reconstruction method is proposed for a single channel SAR system to guarantee remote sensing ability. The system model is formulated as a sparse signal recovery problem with an unknown parametric dictionary to be estimated. An iterative re-weighted method is employed to jointly handle the dictionary parameter learning and target reconstruction problem in an majorization-minimization framework, where a surrogate function majorizing the Gaussian entropy in the objective function is introduced to circumvent its non-convexity. After dictionary parameter learning, the grid mismatching problem in a fixed grid based method is avoided. Therefore, the proposed method can reap a super resolution result. Besides, a simple yet effective DSA section scheme is developed for the SAR data excerpting, in which only two DSAs are required. Experimental results about location error and reconstruction power error reveal that the proposed method is able to achieve a good performance in deceptive jamming suppression.

Parameter learning of stochastic Boolean networks

AbstractIn this article, the parameter learning problem is studied for stochastic Boolean networks (SBNs). Both the measure noise and the system noise are assumed to be white and modeled by sequences of Bernoulli distributed stochastic variables which are mutually independent. An algebraic representation of the SBNs is obtained by taking advantage of vector expression of logic variable and applying the semi‐tensor product technique. Consequently, the parameter learning problem is reformulated as an optimization problem that makes it possible to identify the system matrices of SBNs in an efficient computation way. Subsequently, properties of forward and backward probabilities are investigated, and the EM algorithm is utilized to learn the model parameters from time series data. Finally, a numerical experiment is presented to show the usefulness of the designed parameter learning algorithm.

Hierarchical data fusion processes involving the Möbius representation of capacities

The use of the Choquet integral in data fusion processes allows for the effective modelling of interactions and dependencies between data features or criteria. Its application requires identification of the defining capacity (also known as fuzzy measure) values. The main limiting factor is the complexity of the underlying parameter learning problem, which grows exponentially in the number of variables. However, in practice we may have expert knowledge regarding which of the subsets of criteria interact with each other, and which groups are independent. In this paper we study hierarchical aggregation processes, architecturally similar to feed-forward neural networks, but which allow for the simplification of the fitting problem both in terms of the number of variables and monotonicity constraints. We note that the Möbius representation lets us identify a number of relationships between the overall fuzzy measure and the data pipeline structure. Included in our findings are simplified fuzzy measures that generalise both k-intolerant and k-interactive capacities.

Learning Adaptive Regularization for Image Labeling Using Geometric Assignment

We study the inverse problem of model parameter learning for pixelwise image labeling, using the linear assignment flow and training data with ground truth. This is accomplished by a Riemannian gradient flow on the manifold of parameters that determines the regularization properties of the assignment flow. Using the symplectic partitioned Runge–Kutta method for numerical integration, it is shown that deriving the sensitivity conditions of the parameter learning problem and its discretization commute. A convenient property of our approach is that learning is based on exact inference. Carefully designed experiments demonstrate the performance of our approach, the expressiveness of the mathematical model as well as its limitations, from the viewpoint of statistical learning and optimal control.

Modeling and parameter learning method for the Hammerstein–Wiener model with disturbance

In this paper, a novel modeling and parameter learning method for the Hammerstein–Wiener model with disturbance is proposed, and the Hammerstein–Wiener model is implemented to approximate complex nonlinear industrial processes. The proposed Hammerstein–Wiener model has two static nonlinear blocks represented by two independent neuro-fuzzy models that surround a dynamic linear block described by the finite impulse response model. The parameter learning method of the Hammerstein–Wiener model with disturbance can be summarized in the following three steps: First, the designed input signals are implemented to completely separate the parameter learning problem of output nonlinear block, linear block, and input nonlinear block. Meanwhile, the static output nonlinear block parameters can be learned based on input and output data of two sets of separable signals with different sizes. Second is to determine the dynamic linear block parameter using correlation analysis algorithm using one set of separable signal; thus, the process disturbance can be compensated by the calculation of correlation function. The final one is to achieve unbiased estimation of the static input nonlinear block parameters using least squares method according to the input–output data of random signal. Furthermore, with the parameter learning method, the proposed model can achieve less computation complexity and good robustness. The simulation results of two cases are provided to demonstrate the advantage of the proposed modeling and parameter learning method.

Learning General Latent-Variable Graphical Models with Predictive Belief Propagation

Learning general latent-variable probabilistic graphical models is a key theoretical challenge in machine learning and artificial intelligence. All previous methods, including the EM algorithm and the spectral algorithms, face severe limitations that largely restrict their applicability and affect their performance. In order to overcome these limitations, in this paper we introduce a novel formulation of message-passing inference over junction trees named predictive belief propagation, and propose a new learning and inference algorithm for general latent-variable graphical models based on this formulation. Our proposed algorithm reduces the hard parameter learning problem into a sequence of supervised learning problems, and unifies the learning of different kinds of latent graphical models into a single learning framework, which is local-optima-free and statistically consistent. We then give a proof of the correctness of our algorithm and show in experiments on both synthetic and real datasets that our algorithm significantly outperforms both the EM algorithm and the spectral algorithm while also being orders of magnitude faster to compute.

Automatic Parameter Tying: A New Approach for Regularized Parameter Learning in Markov Networks

Parameter tying is a regularization method in which parameters (weights) of a machine learning model are partitioned into groups by leveraging prior knowledge and all parameters in each group are constrained to take the same value. In this paper, we consider the problem of parameter learning in Markov networks and propose a novel approach called automatic parameter tying (APT) that uses automatic instead of a priori and soft instead of hard parameter tying as a regularization method to alleviate overfitting. The key idea behind APT is to set up the learning problem as the task of finding parameters and groupings of parameters such that the likelihood plus a regularization term is maximized. The regularization term penalizes models where parameter values deviate from their group mean parameter value. We propose and use a block coordinate ascent algorithm to solve the optimization task. We analyze the sample complexity of our new learning algorithm and show that it yields optimal parameters with high probability when the groups are well separated. Experimentally, we show that our method improves upon L2 regularization and suggest several pragmatic techniques for good practical performance.

A New Approach to Learning in Dynamic Bayesian Networks (DBNs)

In this paper, we revisit the parameter learning problem, namely the estimation of model parameters for Dynamic Bayesian Networks (DBNs). DBNs are directed graphical models of stochastic processes that encompasses and generalize Hidden Markov models (HMMs) and Linear Dynamical Systems (LDSs). Whenever we apply these models to economics and finance, we are forced to make some modeling assumptions about the state dynamics and the graph topology (the DBN structure). These assumptions may be incorrectly specified and contain some additional noise compared to reality. Trying to use a best fit approach through maximum likelihood estimation may miss this point and try to fit at any price these models on data. We present here a new methodology that takes a radical point of view and instead focus on the final efficiency of our model. Parameters are hence estimated in terms of their efficiency rather than their distributional fit to the data. The resulting optimization problem that consists in finding the optimal parameters is a hard problem. We rely on Covariance Matrix Adaptation Evolution Strategy (CMA-ES) method to tackle this issue. We apply this method to the seminal problem of trend detection in financial markets. We see on numerical results that the resulting parameters seem less error prone to over fitting than traditional moving average cross over trend detection and perform better. The method developed here for algorithmic trading is general. It can be applied to other real case applications whenever there is no physical law underlying our DBNs.

Latent Dependency Forest Models

Probabilistic modeling is one of the foundations of modern machine learning and artificial intelligence. In this paper, we propose a novel type of probabilistic models named latent dependency forest models (LDFMs). A LDFM models the dependencies between random variables with a forest structure that can change dynamically based on the variable values. It is therefore capable of modeling context-specific independence. We parameterize a LDFM using a first-order non-projective dependency grammar. Learning LDFMs from data can be formulated purely as a parameter learning problem, and hence the difficult problem of model structure learning is circumvented. Our experimental results show that LDFMs are competitive with existing probabilistic models.

A unified approach to universal prediction: generalized upper and lower bounds.

We study sequential prediction of real-valued, arbitrary, and unknown sequences under the squared error loss as well as the best parametric predictor out of a large, continuous class of predictors. Inspired by recent results from computational learning theory, we refrain from any statistical assumptions and define the performance with respect to the class of general parametric predictors. In particular, we present generic lower and upper bounds on this relative performance by transforming the prediction task into a parameter learning problem. We first introduce the lower bounds on this relative performance in the mixture of experts framework, where we show that for any sequential algorithm, there always exists a sequence for which the performance of the sequential algorithm is lower bounded by zero. We then introduce a sequential learning algorithm to predict such arbitrary and unknown sequences, and calculate upper bounds on its total squared prediction error for every bounded sequence. We further show that in some scenarios, we achieve matching lower and upper bounds, demonstrating that our algorithms are optimal in a strong minimax sense such that their performances cannot be improved further. As an interesting result, we also prove that for the worst case scenario, the performance of randomized output algorithms can be achieved by sequential algorithms so that randomized output algorithms do not improve the performance.

A stochastic hybrid model of a biological filter

We present a hybrid model of a biological filter, a genetic circuit which removes fast fluctuations in the cell's internal representation of the extra cellular environment. The model takes the classic feed-forward loop (FFL) motif and represents it as a network of continuous protein concentrations and binary, unobserved gene promoter states. We address the problem of statistical inference and parameter learning for this class of models from partial, discrete time observations. We show that the hybrid representation leads to an efficient algorithm for approximate statistical inference in this circuit, and show its effectiveness on a simulated data set.

A Bilevel Optimization Approach for Parameter Learning in Variational Models

In this work we consider the problem of parameter learning for variational image denoising models. The learning problem is formulated as a bilevel optimization problem, where the lower-level problem is given by the variational model and the higher-level problem is expressed by means of a loss function that penalizes errors between the solution of the lower-level problem and the ground truth data. We consider a class of image denoising models incorporating $\ell_p$-norm--based analysis priors using a fixed set of linear operators. We devise semismooth Newton methods for solving the resulting nonsmooth bilevel optimization problems and show that the optimized image denoising models can achieve state-of-the-art performance.

Machine learning for neuroscience

What is machine learning? Machine learning is a type of statistics that places particular emphasis on the use of advanced computational algorithms. As computers become more powerful, and modern experimental methods in areas such as imaging generate vast bodies of data, machine learning is becoming ever more important for extracting reliable and meaningful relationships and for making accurate predictions. Key strands of modern machine learning grew out of attempts to understand how large numbers of interconnected, more or less neuron-like elements could learn to achieve behaviourally meaningful computations and to extract useful features from images or sound waves. By the 1990s, key approaches had converged on an elegant framework called ‘graphical models’, explained in Koller and Friedman, in which the nodes of a graph represent variables such as edges and corners in an image, or phonemes and words in speech. The probabilistic relationships between nodes are represented by conditional probability tables or simple functions whose parameters are learned from the data. There are three main problems in fitting graphical models to data: inference, parameter learning and structure learning. The inference problem is how to infer the probable values of unobserved variables when the values of a subset of the variables have been observed, and is a problem that perceptual systems need to solve if they are to infer the hidden causes of their sensory input. The parameter-learning problem is how to adjust the parameters governing the way in which one variable influences another, so that the graphical model is a better fit to some observed data. In the brain, this is presumably done by changing synapse strengths. The structure-learning problem is how to decide which unobserved variables are needed and how they must be connected to model the correlations between observed variables. In the brain, evolution and early pruning of connections presumably have a large role to play in determining the structure. Could you provide a brief description of the methods of machine learning? Machine learning can be divided into three parts: 1) in supervised learning, the aim is to predict a class label or a real value from an input (classifying objects in images or predicting the future value of a stock are examples of this type of learning); 2) in unsupervised learning, the aim is to discover good features for representing the input data; and 3) in reinforcement learning, the aim is to discover what action should be performed next in order to maximize the eventual payoff.

Learning Bayesian network parameters under order constraints

We consider the problem of learning the parameters of a Bayesian network from data, while taking into account prior knowledge about the signs of influences between variables. Such prior knowledge can be readily obtained from domain experts. We show that this problem of parameter learning is a special case of isotonic regression and provide a simple algorithm for computing isotonic estimates. Our experimental results for a small Bayesian network in the medical domain show that taking prior knowledge about the signs of influences into account leads to an improved fit of the true distribution, especially when only a small sample of data is available. More importantly, however, the isotonic estimator provides parameter estimates that are consistent with the specified prior knowledge, thereby resulting in a network that is more likely to be accepted by experts in its domain of application.

Effect of wrong samples on the convergence of learning processes. II. A remedy

In our earlier work [1] on the problem of parameter learning in pattern recognition, it was found that estimates converged to nontrue values in the presence of labeling errors. The present work describes a possible remedy to this problem by rejecting those training samples that do not lie within a certain neighborhood of the current estimate of the mean. The convergence of this class of restrictive updating procedure in presence of wrong samples has been studied along with the comparison of its estimates to those in [1]. It is established that, in the presence of labeling errors, the estimates of the proposed restrictive updating procedure are always asymptotically closer to the respective true values than the estimates in [1], provided that certain conditions are satisfied. A set of three-class bivariate data and speech data are also used to demonstrate the above features.

Effect of wrong samples on the convergence of learning processes

For the problem of parameter learning in pattern recognition, when there is a possibility of training samples being mislabeled, the authors have investigated the convergence of stochastic-approximation-based learning algorithms. In the cases considered, it is found that estimates converge to nontrue values in the presence of labeling errors. The general m-class, N-feature pattern recognition problem is considered.