Bayesian Nonparametrics is a class of models with a potentially infinite number of parameters. High flexibility and expressive power of this approach enables better data modelling compared to parametric methods. Bayesian Nonparametrics is used in problems where a dimension of interest grows with data, for example, in problems where the number of features is not fixed but allowed to vary as we observe more data. Another example is clustering where the number of clusters is automatically inferred from data. The Statsbot team asked a data scientist, Vadim Smolyakov, to introduce us to Bayesian Nonparametric models.

We start out by covering the topic of clustering. Clustering is essentially grouping pieces of information by similarity. For instance, imagine a large set of images that you want to group based on their similarities; clustering would be our primary choice for such a process. The chapter also covers high dimensionality and features, which are essential for understanding later chapters. We also cover kmeans and DBSCAN clustering algorithms in some depth to help show the reader how they work under the hood.

Algorithmic decision making process now affects many aspects of our lives. Standard tools for machine learning, such as classification and regression, are subject to the bias in data, and thus direct application of such off-the-shelf tools could lead to a specific group being unfairly discriminated. Removing sensitive attributes of data does not solve this problem because a \textit{disparate impact} can arise when non-sensitive attributes and sensitive attributes are correlated. Here, we study a fair machine learning algorithm that avoids such a disparate impact when making a decision. Inspired by the two-stage least squares method that is widely used in the field of economics, we propose a two-stage algorithm that removes bias in the training data. The proposed algorithm is conceptually simple. Unlike most of existing fair algorithms that are designed for classification tasks, the proposed method is able to (i) deal with regression tasks, (ii) combine explanatory attributes to remove reverse discrimination, and (iii) deal with numerical sensitive attributes. The performance and fairness of the proposed algorithm are evaluated in simulations with synthetic and real-world datasets.

Regularization for matrix factorization (MF) and approximation problems has been carried out in many different ways. Due to its popularity in deep learning, dropout has been applied also for this class of problems. Despite its solid empirical performance, the theoretical properties of dropout as a regularizer remain quite elusive for this class of problems. In this paper, we present a theoretical analysis of dropout for MF, where Bernoulli random variables are used to drop columns of the factors. We demonstrate the equivalence between dropout and a fully deterministic model for MF in which the factors are regularized by the sum of the product of squared Euclidean norms of the columns. Additionally, we inspect the case of a variable sized factorization and we prove that dropout achieves the global minimum of a convex approximation problem with (squared) nuclear norm regularization. As a result, we conclude that dropout can be used as a low-rank regularizer with data dependent singular-value thresholding.

Recent work on imitation learning has generated policies that reproduce expert behavior from multi-modal data. However, past approaches have focused only on recreating a small number of distinct, expert maneuvers, or have relied on supervised learning techniques that produce unstable policies. This work extends InfoGAIL, an algorithm for multi-modal imitation learning, to reproduce behavior over an extended period of time. Our approach involves reformulating the typical imitation learning setting to include "burn-in demonstrations" upon which policies are conditioned at test time. We demonstrate that our approach outperforms standard InfoGAIL in maximizing the mutual information between predicted and unseen style labels in road scene simulations, and we show that our method leads to policies that imitate expert autonomous driving systems over long time horizons.

Machine learning with big data often involves large optimization models. For distributed optimization over a cluster of machines, frequent communication and synchronization of all model parameters (optimization variables) can be very costly. A promising solution is to use parameter servers to store different subsets of the model parameters, and update them asynchronously at different machines using local datasets. In this paper, we focus on distributed optimization of large linear models with convex loss functions, and propose a family of randomized primal-dual block coordinate algorithms that are especially suitable for asynchronous distributed implementation with parameter servers. In particular, we work with the saddle-point formulation of such problems which allows simultaneous data and model partitioning, and exploit its structure by doubly stochastic coordinate optimization with variance reduction (DSCOVR). Compared with other first-order distributed algorithms, we show that DSCOVR may require less amount of overall computation and communication, and less or no synchronization. We discuss the implementation details of the DSCOVR algorithms, and present numerical experiments on an industrial distributed computing system.

The automation of posterior inference in Bayesian data analysis has enabled experts and nonexperts alike to use more sophisticated models, engage in faster exploratory modeling and analysis, and ensure experimental reproducibility. However, standard automated posterior inference algorithms are not tractable at the scale of massive modern datasets, and modifications to make them so are typically model-specific, require expert tuning, and can break theoretical guarantees on inferential quality. Building on the Bayesian coresets framework, this work instead takes advantage of data redundancy to shrink the dataset itself as a preprocessing step, providing fully-automated, scalable Bayesian inference with theoretical guarantees. We begin with an intuitive reformulation of Bayesian coreset construction as sparse vector sum approximation, and demonstrate that its automation and performance-based shortcomings arise from the use of the supremum norm. To address these shortcomings we develop Hilbert coresets, i.e., Bayesian coresets constructed under a norm induced by an inner-product on the log-likelihood function space. We propose two Hilbert coreset construction algorithms---one based on importance sampling, and one based on the Frank-Wolfe algorithm---along with theoretical guarantees on approximation quality as a function of coreset size. Since the exact computation of the proposed inner-products is model-specific, we automate the construction with a random finite-dimensional projection of the log-likelihood functions. The resulting automated coreset construction algorithm is simple to implement, and experiments on a variety of models with real and synthetic datasets show that it provides high-quality posterior approximations and a significant reduction in the computational cost of inference.

Reliable measures of statistical dependence could be useful tools for learning independent features and performing tasks like source separation using Independent Component Analysis (ICA). Unfortunately, many of such measures, like the mutual information, are hard to estimate and optimize directly. We propose to learn independent features with adversarial objectives which optimize such measures implicitly. These objectives compare samples from the joint distribution and the product of the marginals without the need to compute any probability densities. We also propose two methods for obtaining samples from the product of the marginals using either a simple resampling trick or a separate parametric distribution. Our experiments show that this strategy can easily be applied to different types of model architectures and solve both linear and non-linear ICA problems.

The conditional mutual information I(X;Y|Z) measures the average information that X and Y contain about each other given Z. This is an important primitive in many learning problems including conditional independence testing, graphical model inference, causal strength estimation and time-series problems. In several applications, it is desirable to have a functional purely of the conditional distribution p_{Y|X,Z} rather than of the joint distribution p_{X,Y,Z}. We define the potential conditional mutual information as the conditional mutual information calculated with a modified joint distribution p_{Y|X,Z} q_{X,Z}, where q_{X,Z} is a potential distribution, fixed airport. We develop K nearest neighbor based estimators for this functional, employing importance sampling, and a coupling trick, and prove the finite k consistency of such an estimator. We demonstrate that the estimator has excellent practical performance and show an application in dynamical system inference.

In this paper, we study the Nystr{\"o}m type subsampling for large scale kernel methods to reduce the computational complexities of big data. We discuss the multi-penalty regularization scheme based on Nystr{\"o}m type subsampling which is motivated from well-studied manifold regularization schemes. We develop a theoretical analysis of multi-penalty least-square regularization scheme under the general source condition in vector-valued function setting, therefore the results can also be applied to multi-task learning problems. We achieve the optimal minimax convergence rates of multi-penalty regularization using the concept of effective dimension for the appropriate subsampling size. We discuss an aggregation approach based on linear function strategy to combine various Nystr{\"o}m approximants. Finally, we demonstrate the performance of multi-penalty regularization based on Nystr{\"o}m type subsampling on Caltech-101 data set for multi-class image classification and NSL-KDD benchmark data set for intrusion detection problem.