Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the "knot" points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.

We formulate natural gradient variational inference (VI), expectation propagation (EP), and posterior linearisation (PL) as extensions of Newton's method for optimising the parameters of a Bayesian posterior distribution. This viewpoint explicitly casts inference algorithms under the framework of numerical optimisation. We show that common approximations to Newton's method from the optimisation literature, namely Gauss-Newton and quasi-Newton methods (e.g., the BFGS algorithm), are still valid under this'Bayes-Newton' framework. This leads to a suite of novel algorithms which are guaranteed to result in positive semi-definite covariance matrices, unlike standard VI and EP. Our unifying viewpoint provides new insights into the connections between various inference schemes. All the presented methods apply to any model with a Gaussian prior and non-conjugate likelihood, which we demonstrate with (sparse) Gaussian processes and state space models. Keywords: Approximate Bayesian inference, optimisation, variational inference, expectation propagation, Gaussian processes.

Battery performance datasets are typically non-normal and multicollinear. Extrapolating such datasets for model predictions needs attention to such characteristics. This study explores the impact of data normality in building machine learning models. In this work, tree-based regression models and multiple linear regressions models are each built from a highly skewed non-normal dataset with multicollinearity and compared. Several techniques are necessary, such as data transformation, to achieve a good multiple linear regression model with this dataset; the most useful techniques are discussed. With these techniques, the best multiple linear regression model achieved an R^2 = 81.23% and exhibited no multicollinearity effect for the dataset used in this study. Tree-based models perform better on this dataset, as they are non-parametric, capable of handling complex relationships among variables and not affected by multicollinearity. We show that bagging, in the use of Random Forests, reduces overfitting. Our best tree-based model achieved accuracy of R^2 = 97.73%. This study explains why tree-based regressions promise as a machine learning model for non-normally distributed, multicollinear data.

Imitation learning (IL) is a popular approach in the continuous control setting as among other reasons it circumvents the problems of reward mis-specification and exploration in reinforcement learning (RL). In IL from demonstrations, an important challenge is to obtain agent policies that are smooth with respect to the inputs. Learning through imitation a policy that is smooth as a function of a large state-action ($s$-$a$) space (typical of high dimensional continuous control environments) can be challenging. We take a first step towards tackling this issue by using smoothness inducing regularizers on \textit{both} the policy and the cost models of adversarial imitation learning. Our regularizers work by ensuring that the cost function changes in a controlled manner as a function of $s$-$a$ space; and the agent policy is well behaved with respect to the state space. We call our new smooth IL algorithm \textit{Smooth Policy and Cost Imitation Learning} (SPaCIL, pronounced 'Special'). We introduce a novel metric to quantify the smoothness of the learned policies. We demonstrate SPaCIL's superior performance on continuous control tasks from MuJoCo. The algorithm not just outperforms the state-of-the-art IL algorithm on our proposed smoothness metric, but, enjoys added benefits of faster learning and substantially higher average return.

Attentional blink (AB) is a biological effect, showing that for 200 to 500ms after paying attention to one visual target, it is difficult to notice another target that appears next, and attentional blink magnitude (ABM) is a indicating parameter to measure the degree of this effect. Researchers have shown that different categories of images can access the consciousness of human mind differently, and produce different ranges of ABM values. So in this paper, we compare two different types of images, categorized as animal and object, by predicting ABM values directly from image features extracted from convolutional neural network (CNN), and indirectly from functional magnetic resonance imaging (fMRI) data. First, for two sets of images, we separately extract their average features from layers of Alexnet, a classic model of CNN, then input the features into a trained linear regression model to predict ABM values, and we find higher-level instead of lower-level image features determine the categorical difference in AB effect, and mid-level image features predict ABM values more correctly than low-level and high-level image features. Then we employ fMRI data from different brain regions collected when the subjects viewed 50 test images to predict ABM values, and conclude that brain regions covering relatively broader areas, like LVC, HVC and VC, perform better than other smaller brain regions, which means AB effect is more related to synthetic impact of several visual brain regions than only one particular visual regions.

Artificial Intelligence (AI) techniques continue to broaden across governmental and public sectors, such as power and energy - which serve as critical infrastructures for most societal operations. However, due to the requirements of reliability, accountability, and explainability, it is risky to directly apply AI-based methods to power systems because society cannot afford cascading failures and large-scale blackouts, which easily cost billions of dollars. To meet society requirements, this paper proposes a methodology to develop, deploy, and evaluate AI systems in the energy sector by: (1) understanding the power system measurements with physics, (2) designing AI algorithms to forecast the need, (3) developing robust and accountable AI methods, and (4) creating reliable measures to evaluate the performance of the AI model. The goal is to provide a high level of confidence to energy utility users. For illustration purposes, the paper uses power system event forecasting (PEF) as an example, which carefully analyzes synchrophasor patterns measured by the Phasor Measurement Units (PMUs). Such a physical understanding leads to a data-driven framework that reduces the dimensionality with physics and forecasts the event with high credibility. Specifically, for dimensionality reduction, machine learning arranges physical information from different dimensions, resulting inefficient information extraction. For event forecasting, the supervised learning model fuses the results of different models to increase the confidence. Finally, comprehensive experiments demonstrate the high accuracy, efficiency, and reliability as compared to other state-of-the-art machine learning methods.

Machine learning is a large field of study that overlaps with and inherits ideas from many related fields such as artificial intelligence. The focus of the field is learning, that is, acquiring skills or knowledge from experience. Most commonly, this means synthesizing useful concepts from historical data. As such, there are many different types of learning that you may encounter as a practitioner in the field of machine learning: from whole fields of study to specific techniques. Supervised learning describes a class of problems that involves using a model to learn a mapping between input examples and the target variable.

This paper aims to propose and theoretically analyze a new distributed scheme for sparse linear regression and feature selection. The primary goal is to learn the few causal features of a high-dimensional dataset based on noisy observations from an unknown sparse linear model. However, the presumed training set which includes $n$ data samples in $\mathbb{R}^p$ is already distributed over a large network with $N$ clients connected through extremely low-bandwidth links. Also, we consider the asymptotic configuration of $1\ll N\ll n\ll p$. In order to infer the causal dimensions from the whole dataset, we propose a simple, yet effective method for information sharing in the network. In this regard, we theoretically show that the true causal features can be reliably recovered with negligible bandwidth usage of $O\left(N\log p\right)$ across the network. This yields a significantly lower communication cost in comparison with the trivial case of transmitting all the samples to a single node (centralized scenario), which requires $O\left(np\right)$ transmissions. Even more sophisticated schemes such as ADMM still have a communication complexity of $O\left(Np\right)$. Surprisingly, our sample complexity bound is proved to be the same (up to a constant factor) as the optimal centralized approach for a fixed performance measure in each node, while that of a na\"{i}ve decentralized technique grows linearly with $N$. Theoretical guarantees in this paper are based on the recent analytic framework of debiased LASSO in Javanmard et al. (2019), and are supported by several computer experiments performed on both synthetic and real-world datasets.

The dual normal factor graph and the factor graph duality theorem have been considered for discrete graphical models. In this paper, we show an application of the factor graph duality theorem to continuous graphical models. Specifically, we propose a method to solve exactly the Gaussian graphical models defined on the ladder graph if certain conditions on the local covariance matrices are satisfied. Unlike the conventional approaches, the efficiency of the method depends on the position of the zeros in the local covariance matrices. The method and details of the dualization are illustrated on two toy examples.

In the analysis of data sets consisting of (X, Y)-pairs, a tacit assumption is that each pair corresponds to the same observation unit. If, however, such pairs are obtained via record linkage of two files, this assumption can be violated as a result of mismatch error rooting, for example, in the lack of reliable identifiers in the two files. Recently, there has been a surge of interest in this setting under the term "Shuffled data" in which the underlying correct pairing of (X, Y)-pairs is represented via an unknown index permutation. Explicit modeling of the permutation tends to be associated with substantial overfitting, prompting the need for suitable methods of regularization. In this paper, we propose a flexible exponential family prior on the permutation group for this purpose that can be used to integrate various structures such as sparse and locally constrained shuffling. This prior turns out to be conjugate for canonical shuffled data problems in which the likelihood conditional on a fixed permutation can be expressed as product over the corresponding (X,Y)-pairs. Inference is based on the EM algorithm in which the intractable E-step is approximated by the Fisher-Yates algorithm. The M-step is shown to admit a significant reduction from $n^2$ to $n$ terms if the likelihood of (X,Y)-pairs has exponential family form as in the case of generalized linear models. Comparisons on synthetic and real data show that the proposed approach compares favorably to competing methods.