Deep learning has been the engine powering many successes of data science. However, the deep neural network (DNN), as the basic model of deep learning, is often excessively over-parameterized, causing many difficulties in training, prediction and interpretation. We propose a frequentist-like method for learning sparse DNNs and justify its consistency under the Bayesian framework: the proposed method could learn a sparse DNN with at most $O(n/\log(n))$ connections and nice theoretical guarantees such as posterior consistency, variable selection consistency and asymptotically optimal generalization bounds. In particular, we establish posterior consistency for the sparse DNN with a mixture Gaussian prior, show that the structure of the sparse DNN can be consistently determined using a Laplace approximation-based marginal posterior inclusion probability approach, and use Bayesian evidence to elicit sparse DNNs learned by an optimization method such as stochastic gradient descent in multiple runs with different initializations. The proposed method is computationally more efficient than standard Bayesian methods for large-scale sparse DNNs. The numerical results indicate that the proposed method can perform very well for large-scale network compression and high-dimensional nonlinear variable selection, both advancing interpretable machine learning.

Artificial intelligence systems, which are designed with a capability to learn from the data presented to them, are used throughout society. These systems are used to screen loan applicants, make sentencing recommendations for criminal defendants, scan social media posts for disallowed content and more. Because these systems don't assign meaning to their complex learned correlation network, they can learn associations that don't equate to causality, resulting in non-optimal and indefensible decisions being made. In addition to making decisions that are sub-optimal, these systems may create legal liability for their designers and operators by learning correlations that violate anti-discrimination and other laws regarding what factors can be used in different types of decision making. This paper presents the use of a machine learning expert system, which is developed with meaning-assigned nodes (facts) and correlations (rules). Multiple potential implementations are considered and evaluated under different conditions, including different network error and augmentation levels and different training levels. The performance of these systems is compared to random and fully connected networks.

Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature method that learns parsimonious random feature models utilizing techniques from compressive sensing. We provide uniform bounds on the approximation error for functions in a reproducing kernel Hilbert space depending on the number of samples and the distribution of features. The error bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay.

"The key distinguishing property of a Bayesian approach is marginalization instead of optimization, where we represent solutions given by all settings of parameters weighted by their posterior probabilities, rather than bet everything on a single setting of parameters." The time is ripe to dig into marginalization vs optimization, and broaden our general understanding of the Bayesian approach.

The arguments / discussions between the Bayesian vs frequentist approaches in statistics are long running. I am interested in how these approaches impact machine learning. Often, books on machine learning combine the two approaches, or in some cases, take only one approach. This does not help from a learning standpoint. So, in this two-part blog we first discuss the differences between the Frequentist and Bayesian approaches.

Human emotion recognition has become a need for more realistic and interactive machines and computer systems. The greatest challenge is the availability of high-performance algorithms to effectively manage individual differences and nonstationarities in physiological data streams, i.e., algorithms that self-customize to a user with no subject-specific calibration data. We describe an evolving Gaussian Fuzzy Classifier (eGFC), which is supported by an online semi-supervised learning algorithm to recognize emotion patterns from electroencephalogram (EEG) data streams. We extract features from the Fourier spectrum of EEG data. The data are provided by 28 individuals playing the games 'Train Sim World', 'Unravel', 'Slender The Arrival', and 'Goat Simulator' - a public dataset. Different emotions prevail, namely, boredom, calmness, horror and joy. We analyze the effect of individual electrodes, time window lengths, and frequency bands on the accuracy of user-independent eGFCs. We conclude that both brain hemispheres may assist classification, especially electrodes on the frontal (Af3-Af4), occipital (O1-O2), and temporal (T7-T8) areas. We observe that patterns may be eventually found in any frequency band; however, the Alpha (8-13Hz), Delta (1-4Hz), and Theta (4-8Hz) bands, in this order, are the highest correlated with emotion classes. eGFC has shown to be effective for real-time learning of EEG data. It reaches a 72.2% accuracy using a variable rule base, 10-second windows, and 1.8ms/sample processing time in a highly-stochastic time-varying 4-class classification problem.

As machine learning systems become more powerful they also become increasingly unpredictable and opaque. Yet, finding human-understandable explanations of how they work is essential for their safe deployment. This technical report illustrates a methodology for investigating the causal mechanisms that drive the behaviour of artificial agents. Six use cases are covered, each addressing a typical question an analyst might ask about an agent. In particular, we show that each question cannot be addressed by pure observation alone, but instead requires conducting experiments with systematically chosen manipulations so as to generate the correct causal evidence.

This paper presents a machine learning framework (GP-NODE) for Bayesian systems identification from partial, noisy and irregular observations of nonlinear dynamical systems. The proposed method takes advantage of recent developments in differentiable programming to propagate gradient information through ordinary differential equation solvers and perform Bayesian inference with respect to unknown model parameters using Hamiltonian Monte Carlo sampling and Gaussian Process priors over the observed system states. This allows us to exploit temporal correlations in the observed data, and efficiently infer posterior distributions over plausible models with quantified uncertainty. Moreover, the use of sparsity-promoting priors such as the Finnish Horseshoe for free model parameters enables the discovery of interpretable and parsimonious representations for the underlying latent dynamics. A series of numerical studies is presented to demonstrate the effectiveness of the proposed GP-NODE method including predator-prey systems, systems biology, and a 50-dimensional human motion dynamical system. Taken together, our findings put forth a novel, flexible and robust workflow for data-driven model discovery under uncertainty. All code and data accompanying this manuscript are available online at \url{https://github.com/PredictiveIntelligenceLab/GP-NODEs}.

Gaussian processes (GPs) are frequently used in machine learning and statistics to construct powerful models. In a Bayesian setting, GPs provide a probabilistic approach to model unknown functions; specifically, the GP prior assumes that the function evaluated at any finite set of inputs has a Gaussian distribution with consistent parameters, specified by the mean function and symmetric positive definite covariance (or kernel) function. The flexible, probabilistic and nonparametric nature of GP models makes them appropriate and useful in a wide range of applications, including geostatistics [Matheron, 1973], atmospheric sciences [Berrocal et al., 2010], biology [Stathopoulos et al., 2014], inverse problems [Kaipio and Somersalo, 2006], and more. However, when employing GPs in practice, important considerations must be made, specifically, to address the computational burden, approximation of the posterior, form of the covariance function and inference of its hyperparmeters. First, GPs suffer from a high computational burden, due to the need to store and invert large and dense covariance matrices.

Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature method that learns parsimonious random feature models utilizing techniques from compressive sensing. We provide uniform bounds on the approximation error for functions in a reproducing kernel Hilbert space depending on the number of samples and the distribution of features. The error bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature method outperforms shallow networks for well-structured functions and applications to scientific machine learning tasks.