There have been different strategies to improve the performance of a machine learning model, e.g., increasing the depth, width, and/or nonlinearity of the model, and using ensemble learning to aggregate multiple base/weak learners in parallel or in series. This paper proposes a novel strategy called patch learning (PL) for this problem. It consists of three steps: 1) train an initial global model using all training data; 2) identify from the initial global model the patches which contribute the most to the learning error, and train a (local) patch model for each such patch; and, 3) update the global model using training data that do not fall into any patch. To use a PL model, we first determine if the input falls into any patch. If yes, then the corresponding patch model is used to compute the output. Otherwise, the global model is used. We explain in detail how PL can be implemented using fuzzy systems. Five regression problems on 1D/2D/3D curve fitting, nonlinear system identification, and chaotic time-series prediction, verified its effectiveness. To our knowledge, the PL idea has not appeared in the literature before, and it opens up a promising new line of research in machine learning.

The increasing impact of algorithmic decisions on people's lives compels us to scrutinize their fairness and, in particular, the disparate impacts that ostensibly-color-blind algorithms can have on different groups. Examples include credit decisioning, hiring, advertising, criminal justice, personalized medicine, and targeted policymaking, where in some cases legislative or regulatory frameworks for fairness exist and define specific protected classes. In this paper we study a fundamental challenge to assessing disparate impacts in practice: protected class membership is often not observed in the data. This is particularly a problem in lending and healthcare. We consider the use of an auxiliary dataset, such as the US census, that includes class labels but not decisions or outcomes. We show that a variety of common disparity measures are generally unidentifiable aside for some unrealistic cases, providing a new perspective on the documented biases of popular proxy-based methods. We provide exact characterizations of the sharpest-possible partial identification set of disparities either under no assumptions or when we incorporate mild smoothness constraints. We further provide optimization-based algorithms for computing and visualizing these sets, which enables reliable and robust assessments -- an important tool when disparity assessment can have far-reaching policy implications. We demonstrate this in two case studies with real data: mortgage lending and personalized medicine dosing.

Conditional kernel mean embeddings form an attractive nonparametric framework for representing conditional means of functions, describing the observation processes for many complex models. However, the recovery of the original underlying function of interest whose conditional mean was observed is a challenging inference task. We formalize deconditional kernel mean embeddings as a solution to this inverse problem, and show that it can be naturally viewed as a nonparametric Bayes' rule. Critically, we introduce the notion of task transformed Gaussian processes and establish deconditional kernel means as their posterior predictive mean. This connection provides Bayesian interpretations and uncertainty estimates for deconditional kernel mean embeddings, explains their regularization hyperparameters, and reveals a marginal likelihood for kernel hyperparameter learning. These revelations further enable practical applications such as likelihood-free inference and learning sparse representations for big data.

Recovering sparse conditional independence graphs from data is a fundamental problem in machine learning with wide applications. A popular formulation of the problem is an $\ell_1$ regularized maximum likelihood estimation. Many convex optimization algorithms have been designed to solve this formulation to recover the graph structure. Recently, there is a surge of interest to learn algorithms directly based on data, and in this case, learn to map empirical covariance to the sparse precision matrix. However, it is a challenging task in this case, since the symmetric positive definiteness (SPD) and sparsity of the matrix are not easy to enforce in learned algorithms, and a direct mapping from data to precision matrix may contain many parameters. We propose a deep learning architecture, GLAD, which uses an Alternating Minimization (AM) algorithm as our model inductive bias, and learns the model parameters via supervised learning. We show that GLAD learns a very compact and effective model for recovering sparse graph from data.

This article explores the concepts of ocean wave multivariate multistep forecasting, reconstruction and feature selection. We introduce recurrent neural network frameworks, integrated with Bayesian hyperparameter optimization and Elastic Net methods. We consider both short- and long-term forecasts and reconstruction, for significant wave height and output power of the ocean waves. Sequence-to-sequence neural networks are being developed for the first time to reconstruct the missing characteristics of ocean waves based on information from nearby wave sensors. Our results indicate that the Adam and AMSGrad optimization algorithms are the most robust ones to optimize the sequence-to-sequence network. For the case of significant wave height reconstruction, we compare the proposed methods with alternatives on a well-studied dataset. We show the superiority of the proposed methods considering several error metrics. We design a new case study based on measurement stations along the east coast of the United States and investigate the feature selection concept. Comparisons substantiate the benefit of utilizing Elastic Net. Moreover, case study results indicate that when the number of features is considerable, having deeper structures improves the performance.

In this research work, probabilistic decision-making approaches are studied, e.g. Bayesian and Boltzmann strategies, along with various deterministic exploration strategies, e.g. greedy, epsilon-Greedy and random approaches. In this research work, a comparative study has been done between probabilistic and deterministic decision-making approaches, the experiments are performed in OpenAI gym environment, solving Cart Pole problem. This research work discusses about the Bayesian approach to decision-making in deep reinforcement learning, and about dropout, how it can reduce the computational cost. All the exploration approaches are compared. It also discusses about the importance of exploration in deep reinforcement learning, and how improving exploration strategies may help in science and technology. This research work shows how probabilistic decision-making approaches are better in the long run as compared to the deterministic approaches. When there is uncertainty, Bayesian dropout approach proved to be better than all other approaches in this research work.

We present a new generalization error bound, the \emph{PAC-Bayesian transportation bound}, unifying the PAC-Bayesian analysis and the generic chaining method in view of the optimal transportation. The proposed bound is the first PAC-Bayesian framework that characterizes the cost of de-randomization of stochastic predictors facing any Lipschitz loss functions. As an example, we give an upper bound on the de-randomization cost of spectrally normalized neural networks~(NNs) to evaluate how much randomness contributes to the generalization of NNs.

We propose a framework for the greedy approximation of high-dimensional Bayesian inference problems, through the composition of multiple \emph{low-dimensional} transport maps or flows. Our framework operates recursively on a sequence of ``residual'' distributions, given by pulling back the posterior through the previously computed transport maps. The action of each map is confined to a low-dimensional subspace that we identify by minimizing an error bound. At each step, our approach thus identifies (i) a relevant subspace of the residual distribution, and (ii) a low-dimensional transformation between a restriction of the residual onto this subspace and a standard Gaussian. We prove weak convergence of the approach to the posterior distribution, and we demonstrate the algorithm on a range of challenging inference problems in differential equations and spatial statistics.

Many mathematical imaging problems are posed as non-convex optimization problems. When numerically tractable global optimization procedures are not available, one is often interested in testing ex post facto whether or not a locally convergent algorithm has found the globally optimal solution. If the problem has a statistical maximum likelihood formulation, a local test of global optimality can be constructed. In this paper, we develop an improved test, based on a global maximum validation function proposed by Biernacki, under the assumption that the statistical distribution is in the generalized location family, a condition often satisfied in imaging problems. In addition, a new reparameterization and embedding procedure is presented that exploits knowledge about the forward operator to improve the global maximum validation function. Finally, the reparameterized embedding technique is applied to a physically-motivated joint-inverse problem arising in camera blur estimation. The advantages of the proposed global optimum testing techniques are numerically demonstrated in terms of increased detection accuracy and reduced computation.

In recent years, impressive progress has been made in the design of implicit probabilistic models via Generative Adversarial Networks (GAN) and its extension, the Conditional GAN (CGAN). Excellent solutions have been demonstrated mostly in image processing applications which involve large, continuous output spaces. There is almost no application of these powerful tools to problems having small dimensional output spaces. Regression problems involving the inductive learning of a map, $y=f(x,z)$, $z$ denoting noise, $f:\mathbb{R}^n\times \mathbb{R}^k \rightarrow \mathbb{R}^m$, with $m$ small (e.g., $m=1$ or just a few) is one good case in point. The standard approach to solve regression problems is to probabilistically model the output $y$ as the sum of a mean function $m(x)$ and a noise term $z$; it is also usual to take the noise to be a Gaussian. These are done for convenience sake so that the likelihood of observed data is expressible in closed form. In the real world, on the other hand, stochasticity of the output is usually caused by missing or noisy input variables. Such a real world situation is best represented using an implicit model in which an extra noise vector, $z$ is included with $x$ as input. CGAN is naturally suited to design such implicit models. This paper makes the first step in this direction. Through several artificial and real world datasets, we demonstrate CGAN to be an effective approach for solving regression problems. We compare against Gaussian Processes and show that CGAN has excellent output likelihood properties and possesses the ability to model complex noise forms in a better way.