Generative neural samplers are probabilistic models that implement sampling using feedforward neural networks: they take a random input vector and produce a sample from a probability distribution defined by the network weights. These models are expressive and allow efficient computation of samples and derivatives, but cannot be used for computing likelihoods or for marginalization. The generative-adversarial training method allows to train such models through the use of an auxiliary discriminative neural network. We show that the generative-adversarial approach is a special case of an existing more general variational divergence estimation approach. We show that any f-divergence can be used for training generative neural samplers. We discuss the benefits of various choices of divergence functions on training complexity and the quality of the obtained generative models.
(NCE) @cite_19 is a method that estimates the parameters of unnormalized probabilistic models by performing non-linear logistic regression to discriminate the data from artificially generated noise. NCE can be viewed as a special case of GAN where the discriminator is constrained to a specific form that depends on the model (logistic regression classifier) and the generator (kept fixed) is providing the artificially generated noise (see supplementary material).
@cite_19
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Two recently introduced criteria for estimation of generative models are both based on a reduction to binary classification. Noise-contrastive estimation (NCE) is an estimation procedure in which a generative model is trained to be able to distinguish data samples from noise samples. Generative adversarial networks (GANs) are pairs of generator and discriminator networks, with the generator network learning to generate samples by attempting to fool the discriminator network into believing its samples are real data. Both estimation procedures use the same function to drive learning, which naturally raises questions about how they are related to each other, as well as whether this function is related to maximum likelihood estimation (MLE). NCE corresponds to training an internal data model belonging to the discriminator network but using a fixed generator network. We show that a variant of NCE, with a dynamic generator network, is equivalent to maximum likelihood estimation. Since pairing a learned discriminator with an appropriate dynamically selected generator recovers MLE, one might expect the reverse to hold for pairing a learned generator with a certain discriminator. However, we show that recovering MLE for a learned generator requires departing from the distinguishability game. Specifically: (i) The expected gradient of the NCE discriminator can be made to match the expected gradient of MLE, if one is allowed to use a non-stationary noise distribution for NCE, (ii) No choice of discriminator network can make the expected gradient for the GAN generator match that of MLE, and (iii) The existing theory does not guarantee that GANs will converge in the non-convex case. This suggests that the key next step in GAN research is to determine whether GANs converge, and if not, to modify their training algorithm to force convergence. We consider the task of estimating, from observed data, a probabilistic model that is parameterized by a finite number of parameters. In particular, we are considering the situation where the model probability density function is unnormalized. That is, the model is only specified up to the partition function. The partition function normalizes a model so that it integrates to one for any choice of the parameters. However, it is often impossible to obtain it in closed form. Gibbs distributions, Markov and multi-layer networks are examples of models where analytical normalization is often impossible. Maximum likelihood estimation can then not be used without resorting to numerical approximations which are often computationally expensive. We propose here a new objective function for the estimation of both normalized and unnormalized models. The basic idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise. With this approach, the normalizing partition function can be estimated like any other parameter. We prove that the new estimation method leads to a consistent (convergent) estimator of the parameters. For large noise sample sizes, the new estimator is furthermore shown to behave like the maximum likelihood estimator. In the estimation of unnormalized models, there is a trade-off between statistical and computational performance. We show that the new method strikes a competitive trade-off in comparison to other estimation methods for unnormalized models. As an application to real data, we estimate novel two-layer models of natural image statistics with spline nonlinearities. We present a new estimation principle for parameterized statistical models. The idea is to perform nonlinear logistic regression to discriminate between the observed data and some artificially generated noise, using the model log-density function in the regression nonlinearity. We show that this leads to a consistent (convergent) estimator of the parameters, and analyze the asymptotic variance. In particular, the method is shown to directly work for unnormalized models, i.e. models where the density function does not integrate to one. The normalization constant can be estimated just like any other parameter. For a tractable ICA model, we compare the method with other estimation methods that can be used to learn unnormalized models, including score matching, contrastive divergence, and maximum-likelihood where the normalization constant is estimated with importance sampling. Simulations show that noise-contrastive estimation offers the best trade-off between computational and statistical efficiency. The method is then applied to the modeling of natural images: We show that the method can successfully estimate a large-scale two-layer model and a Markov random field. We introduce the "Energy-based Generative Adversarial Network" model (EBGAN) which views the discriminator as an energy function that attributes low energies to the regions near the data manifold and higher energies to other regions. Similar to the probabilistic GANs, a generator is seen as being trained to produce contrastive samples with minimal energies, while the discriminator is trained to assign high energies to these generated samples. Viewing the discriminator as an energy function allows to use a wide variety of architectures and loss functionals in addition to the usual binary classifier with logistic output. Among them, we show one instantiation of EBGAN framework as using an auto-encoder architecture, with the energy being the reconstruction error, in place of the discriminator. We show that this form of EBGAN exhibits more stable behavior than regular GANs during training. We also show that a single-scale architecture can be trained to generate high-resolution images. Logistic regression and artificial neural networks are the models of choice in many medical data classification tasks. In this review, we summarize the differences and similarities of these models from a technical point of view, and compare them with other machine learning algorithms. We provide considerations useful for critically assessing the quality of the models and the results based on these models. Finally, we summarize our findings on how quality criteria for logistic regression and artificial neural network models are met in a sample of papers from the medical literature. This paper addresses the important tradeoff between privacy and learnability, when designing algorithms for learning from private databases. We focus on privacy-preserving logistic regression. First we apply an idea of [6] to design a privacy-preserving logistic regression algorithm. This involves bounding the sensitivity of regularized logistic regression, and perturbing the learned classifier with noise proportional to the sensitivity. We then provide a privacy-preserving regularized logistic regression algorithm based on a new privacy-preserving technique: solving a perturbed optimization problem. We prove that our algorithm preserves privacy in the model due to [6]. We provide learning guarantees for both algorithms, which are tighter for our new algorithm, in cases in which one would typically apply logistic regression. Experiments demonstrate improved learning performance of our method, versus the sensitivity method. Our privacy-preserving technique does not depend on the sensitivity of the function, and extends easily to a class of convex loss functions. Our work also reveals an interesting connection between regularization and privacy. Since EEG is one of the most important sources of information in therapy of epilepsy, several researchers tried to address the issue of decision support for such a data. In this paper, we introduce two fundamentally different approaches for designing classification models (classifiers); the traditional statistical method based on logistic regression and the emerging computationally powerful techniques based on artificial neural networks (ANNs). Logistic regression as well as feedforward error backpropagation artificial neural networks (FEBANN) and wavelet neural networks (WNN) based classifiers were developed and compared in relation to their accuracy in classification of EEG signals. In these methods we used FFT and autoregressive (AR) model by using maximum likelihood estimation (MLE) of EEG signals as an input to classification system with two discrete outputs: epileptic seizure or nonepileptic seizure. By identifying features in the signal we want to provide an automatic system that will support a physician in the diagnosing process. By applying AR with MLE in connection with WNN, we obtained novel and reliable classifier architecture. The network is constructed by the error backpropagation neural network using Morlet mother wavelet basic function as node activation function. The comparisons between the developed classifiers were primarily based on analysis of the receiver operating characteristic (ROC) curves as well as a number of scalar performance measures pertaining to the classification. The WNN-based classifier outperformed the FEBANN and logistic regression based counterpart. Within the same group, the WNN-based classifier was more accurate than the FEBANN-based classifier, and the logistic regression-based classifier. In the presence of a heavy-tail noise distribution, regression becomes much more difficult. Traditional robust regression methods assume that the noise distribution is symmetric, and they downweight the influence of so-called outliers. When the noise distribution is asymmetric, these methods yield biased regression estimators. Motivated by data-mining problems for the insurance industry, we propose a new approach to robust regression tailored to deal with asymmetric noise distribution. The main idea is to learn most of the parameters of the model using conditional quantile estimators (which are biased but robust estimators of the regression) and to learn a few remaining parameters to combine and correct these estimators, to minimize the average squared error in an unbiased way. Theoretical analysis and experiments show the clear advantages of the approach. Results are on artificial data as well as insurance data, using both linear and neural network predictors. Introduction to the Logistic Regression Model Multiple Logistic Regression Interpretation of the Fitted Logistic Regression Model Model-Building Strategies and Methods for Logistic Regression Assessing the Fit of the Model Application of Logistic Regression with Different Sampling Models Logistic Regression for Matched Case-Control Studies Special Topics References Index. We compare discriminative and generative learning as typified by logistic regression and naive Bayes. We show, contrary to a widely-held belief that discriminative classifiers are almost always to be preferred, that there can often be two distinct regimes of performance as the training set size is increased, one in which each algorithm does better. This stems from the observation—which is borne out in repeated experiments—that while discriminative learning has lower asymptotic error, a generative classifier may also approach its (higher) asymptotic error much faster. Logistic regression is a commonly used representation for aggregators in Bayesian belief networks when a child has multiple parents. In this paper we consider extending logistic regression to relational models, where we want to model varying populations and interactions among parents. In this paper, we first examine the representational problems caused by population variation. We show how these problems arise even in simple cases with a single parametrized parent, and propose a linear relational logistic regression which we show can represent arbitrary linear (in population size) decision thresholds, whereas the traditional logistic regression cannot. Then we examine representing interactions among the parents of a child node, and representing non-linear dependency on population size. We propose a multi-parent relational logistic regression which can represent interactions among parents and arbitrary polynomial decision thresholds. Finally, we show how other well-known aggregators can be represented using this relational logistic regression. NLP models have many and sparse features, and regularization is key for balancing model overfitting versus underfitting. A recently repopularized form of regularization is to generate fake training data by repeatedly adding noise to real data. We reinterpret this noising as an explicit regularizer, and approximate it with a second-order formula that can be used during training without actually generating fake data. We show how to apply this method to structured prediction using multinomial logistic regression and linear-chain CRFs. We tackle the key challenge of developing a dynamic program to compute the gradient of the regularizer efficiently. The regularizer is a sum over inputs, so we can estimate it more accurately via a semi-supervised or transductive extension. Applied to text classification and NER, our method provides a >1 absolute performance gain over use of standardL2 regularization. This article describes the gologit2 program for generalized ordered logit models. gologit2 is inspired by Vincent Fu’s gologit routine (Stata Technical Bulletin Reprints 8: 160–164) and is backward compatible with it but offers several additional powerful options. A major strength of gologit2 is that it can fit three special cases of the generalized model: the proportional odds parallel-lines model, the partial proportional odds model, and the logistic regression model. Hence, gologit2 can fit models that are less restrictive than the parallel-lines models fitted by ologit (whose assumptions are often violated) but more parsimonious and interpretable than those fitted by a nonordinal method, such as multinomial logistic regression (i.e., mlogit). Other key advantages of gologit2 include support for linear constraints, survey data estimation, and the computation of estimated probabilities via the predict command. Function estimation approximation is viewed from the perspective of numerical optimization in function space, rather than parameter space. A connection is made between stagewise additive expansions and steepest-descent minimization. A general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion. Specific algorithms are presented for least-squares, least absolute deviation, and Huber-M loss functions for regression, and multiclass logistic likelihood for classification. Special enhancements are derived for the particular case where the individual additive components are regression trees, and tools for interpreting such TreeBoost models are presented. Gradient boosting of regression trees produces competitive, highly robust, interpretable procedures for both regression and classification, especially appropriate for mining less than clean data. Connections between this approach and the boosting methods of Freund and Shapire and Friedman, Hastie and Tibshirani are discussed. It is difficult to apply machine learning to new domains because often we lack labeled problem instances. In this paper, we provide a solution to this problem that leverages domain knowledge in the form of affinities between input features and classes. For example, in a baseball vs. hockey text classification problem, even without any labeled data, we know that the presence of the word puck is a strong indicator of hockey. We refer to this type of domain knowledge as a labeled feature. In this paper, we propose a method for training discriminative probabilistic models with labeled features and unlabeled instances. Unlike previous approaches that use labeled features to create labeled pseudo-instances, we use labeled features directly to constrain the model's predictions on unlabeled instances. We express these soft constraints using generalized expectation (GE) criteria --- terms in a parameter estimation objective function that express preferences on values of a model expectation. In this paper we train multinomial logistic regression models using GE criteria, but the method we develop is applicable to other discriminative probabilistic models. The complete objective function also includes a Gaussian prior on parameters, which encourages generalization by spreading parameter weight to unlabeled features. Experimental results on text classification data sets show that this method outperforms heuristic approaches to training classifiers with labeled features. Experiments with human annotators show that it is more beneficial to spend limited annotation time labeling features rather than labeling instances. For example, after only one minute of labeling features, we can achieve 80 accuracy on the ibm vs. mac text classification problem using GE-FL, whereas ten minutes labeling documents results in an accuracy of only 77 Logistic regression is one core predictive modeling technique that has been used extensively in health and biomedical problems. Recently a lot of research has been focusing on enforcing sparsity on the learned model to enhance its effectiveness and interpretability, which results in sparse logistic regression model. However, no matter the original or sparse logistic regression, they require the inputs to be in vector form. This limits the applicability of logistic regression in the problems when the data cannot be naturally represented vectors (e.g., functional magnetic resonance imaging and electroencephalography signals). To handle the cases when the data are in the form of multi-dimensional arrays, we propose MulSLR: Multilinear Sparse Logistic Regression. MulSLR can be viewed as a high order extension of sparse logistic regression. Instead of solving one classification vector as in conventional logistic regression, we solve for K classification vectors in MulSLR (K is the number of modes in the data). We propose a block proximal descent approach to solve the problem and prove its convergence. The convergence rate of the proposed algorithm is also analyzed. Finally we validate the efficiency and effectiveness of MulSLR on predicting the onset risk of patients with Alzheimer's disease and heart failure. Machine learning techniques such as classification and regression trees (CART) have been suggested as promising alternatives to logistic regression for the estimation of propensity scores. The authors examined the performance of various CART-based propensity score models using simulated data. Hypothetical studies of varying sample sizes (n=500, 1000, 2000) with a binary exposure, continuous outcome, and ten covariates were simulated under seven scenarios differing by degree of non-linear and non-additive associations between covariates and the exposure. Propensity score weights were estimated using logistic regression (all main effects), CART, pruned CART, and the ensemble methods of bagged CART, random forests, and boosted CART. Performance metrics included covariate balance, standard error, percent absolute bias, and 95 confidence interval coverage. All methods displayed generally acceptable performance under conditions of either non-linearity or non-additivity alone. However, under conditions of both moderate non-additivity and moderate non-linearity, logistic regression had subpar performance, while ensemble methods provided substantially better bias reduction and more consistent 95 CI coverage. The results suggest that ensemble methods, especially boosted CART, may be useful for propensity score weighting. In this paper we upgrade linear logistic regression and boosting to multi-instance data, where each example consists of a labeled bag of instances. This is done by connecting predictions for individual instances to a bag-level probability estimate by simple averaging and maximizing the likelihood at the bag level—in other words, by assuming that all instances contribute equally and independently to a bag’s label. We present empirical results for artificial data generated according to the underlying generative model that we assume, and also show that the two algorithms produce competitive results on the Musk benchmark datasets. A successful approach to structured learning is to write the learning objective as a joint function of linear parameters and inference messages, and iterate between updates to each. This paper observes that if the inference problem is "smoothed" through the addition of entropy terms, for fixed messages, the learning objective reduces to a traditional (non-structured) logistic regression problem with respect to parameters. In these logistic regression problems, each training example has a bias term determined by the current set of messages. Based on this insight, the structured energy function can be extended from linear factors to any function class where an "oracle" exists to minimize a logistic loss. Abstract A number of tools are available today – and more are currently being developed – to discriminate, within a particular storm, areas with hail precipitation using data provided by conventional meteorological radar systems. However, at a supraregional level most of the methods that have traditionally been employed to identify hailstorms often obtain ambiguous results. On the other hand, many currently available systems for data extraction and processing make it relatively easy to calculate a large number of variables derived from radar parameters for each storm analyzed and at different stages in its development. The questions are now: is it possible to select and or classify these variables according to their ability to discriminate hailstorms from non-hail storms? And if so, would the combination of several of these variables enable us to develop new and improved discriminating tools? These questions have prompted the use of mathematical methods for discrimination, such as logistic regression and linear discriminant analysis. In both models the stepwise method was used to construct the mathematical equation automatically. Data from a C-band radar system installed in the NW of the Iberian Peninsula were used to set up the two models. The two models share a number of variables (VIL, maximum reflectivity, height of the maximum reflectivity and ddBZ_max dt.), but the discriminant model makes use additionally of the top and the tilt of the storm. The results have been assessed not only in terms of the precision indices (the models have a probability of detection of 84.9 and 86.8 , respectively), but also with respect to their accuracy, sufficiency relation, resolution and discrimination. Finally, the logistic model has been calibrated for S-band radar systems. The results show a very satisfactory probability of detection, demonstrating that these methods may also be effective in this type of system.