Bayesian optimization forms a set of powerful tools that allows efficient black-box optimization and has been applied in a large variety of fields. In this thesis we first seek to advance Bayesian optimization by using estimated derivative observations. Later, we seek to tackle down the issues in Bayesian optimization when a large number of derivative observations and/or function observations are present. We start to describe our motivations in Chapter 1. We then give a broad review of Bayesian optimization in Chapter 2, where we start by covering the history of Bayesian optimization and its components.

It is widely known that several machine learning models are susceptible to adversarial attacks i.e., small adversarial perturbations applied to data points causing the model to misclassify the data. Adversarial training using empirical risk minimization methods, is the state-of-the-art method for defense against adversarial attacks. Despite being successful, several problems in understanding generalization performance of adversarial training remain open. In this paper, we derive precise theoretical predictions for the performance of adversarial training in binary linear classification. We consider the modern high-dimensional regime where the dimension of data grows with the size of the training dataset at a constant ratio. Our results provide exact asymptotics for the performance of estimators obtained by adversarial training with $\ell_q$-norm bounded perturbations ($q \ge 1$) and for binary labels and Gaussian features. These sharp predictions enable us to explore the role of various factors including over-parametrization ratio, data model and attack budget on the performance of adversarial training.

Network estimation and variable selection have been extensively studied in the statistical literature, but only recently have those two challenges been addressed simultaneously. In this paper, we seek to develop a novel method to simultaneously estimate network interactions and associations to relevant covariates for count data, and specifically for compositional data, which have a fixed sum constraint. We use a hierarchical Bayesian model with latent layers and employ spike-and-slab priors for both edge and covariate selection. For posterior inference, we develop a variational inference scheme with an expectation maximization step, to enable efficient estimation. Through simulation studies, we demonstrate that the proposed model outperforms existing methods in its accuracy of network recovery. We show the practical utility of our model via an application to microbiome data. The human microbiome has been shown to contribute to many of the functions of the human body, and also to be linked with a number of diseases. In our application, we seek to better understand the interaction between microbes and relevant covariates, as well as the interaction of microbes with each other. We provide a Python implementation of our algorithm, called SINC (Simultaneous Inference for Networks and Covariates), available online.

When equipped with efficient optimization algorithms, the over-parameterized neural networks have demonstrated high level of performance even though the loss function is non-convex and non-smooth. While many works have been focusing on understanding the loss dynamics by training neural networks with the gradient descent (GD), in this work, we consider a broad class of optimization algorithms that are commonly used in practice. For example, we show from a dynamical system perspective that the Heavy Ball (HB) method can converge to global minimum on mean squared error (MSE) at a linear rate (similar to GD); however, the Nesterov accelerated gradient descent (NAG) only converges to global minimum sublinearly. Our results rely on the connection between neural tangent kernel (NTK) and finite over-parameterized neural networks with ReLU activation, which leads to analyzing the limiting ordinary differential equations (ODE) for optimization algorithms. We show that, optimizing the non-convex loss over the weights corresponds to optimizing some strongly convex loss over the prediction error. As a consequence, we can leverage the classical convex optimization theory to understand the convergence behavior of neural networks. We believe our approach can also be extended to other loss functions and network architectures.

A systematic approach to finding variational approximation in an otherwise intractable non-conjugate model is to exploit the general principle of convex duality by minorizing the marginal likelihood that renders the problem tractable. While such approaches are popular in the context of variational inference in non-conjugate Bayesian models, theoretical guarantees on statistical optimality and algorithmic convergence are lacking. Focusing on logistic regression models, we provide mild conditions on the data generating process to derive non-asymptotic upper bounds to the risk incurred by the variational optima. We demonstrate that these assumptions can be completely relaxed if one considers a slight variation of the algorithm by raising the likelihood to a fractional power. Next, we utilize the theory of dynamical systems to provide convergence guarantees for such algorithms in logistic and multinomial logit regression. In particular, we establish local asymptotic stability of the algorithm without any assumptions on the data-generating process. We explore a special case involving a semi-orthogonal design under which a global convergence is obtained. The theory is further illustrated using several numerical studies.

The importance of asymmetries in prediction problems arising in economics has been recognized for a long time. In this paper, we focus on binary choice problems in a data-rich environment with general loss functions. In contrast to the asymmetric regression problems, the binary choice with general loss functions and high-dimensional datasets is challenging and not well understood. Econometricians have studied binary choice problems for a long time, but the literature does not offer computationally attractive solutions in data-rich environments. In contrast, the machine learning literature has many computationally attractive algorithms that form the basis for much of the automated procedures that are implemented in practice, but it is focused on symmetric loss functions that are independent of individual characteristics. One of the main contributions of our paper is to show that the theoretically valid predictions of binary outcomes with arbitrary loss functions can be achieved via a very simple reweighting of the logistic regression, or other state-of-the-art machine learning techniques, such as boosting or (deep) neural networks. We apply our analysis to racial justice in pretrial detention.

It is well known that machine learning methods can be vulnerable to adversarially-chosen perturbations of their inputs. Despite significant progress in the area, foundational open problems remain. Here we address several of these key questions. We derive exact and approximate Bayes-optimal robust classifiers for the important setting of two- and three-class Gaussian classification problems with arbitrary imbalance, for $\ell_2$ and $\ell_\infty$ adversaries. In contrast to classical Bayes-optimal classifiers, decisions here cannot be made pointwise and new theoretical approaches are needed. We develop and leverage new tools, including recent breakthroughs from probability theory on robust isoperimetry (Cianci et al, 2011, Mossel and Neeman 2015), which, to our knowledge, have not yet been used in the area. Our results reveal tradeoffs between standard and robust accuracy that grow when data is imbalanced. We also show further foundational results, including an analysis of the loss landscape, classification calibration for convex losses in certain models, and finite sample rates for the robust risk.

The industry-scale ranking systems are typically applied to millions of items in a personalized way for billions of users. To We investigate the Plackett-Luce (PL) model meet the need of scalability and to exploit a huge based listwise learning-to-rank (LTR) on amount of user feedback data, learning-to-rank (LTR) data with partitioned preference, where a set has been the most popular paradigm for building the of items are sliced into ordered and disjoint ranking system. Existing LTR approaches can be categorized partitions, but the ranking of items within a into three groups: pointwise (Gey, 1994), pairwise partition is unknown. Given N items with (Burges et al., 2005), and listwise (Cao et al., M partitions, calculating the likelihood of 2007; Taylor et al., 2008) methods. The pointwise and data with partitioned preference under the pairwise LTR methods convert the ranking problem PL model has a time complexity of O(N S!), into regression or classification tasks on single or pairs where S is the maximum size of the top M 1 of items respectively.

Curriculum learning (CL) is a training strategy that trains a machine learning model from easier data to harder data, which imitates the meaningful learning order in human curricula. As an easy-to-use plug-in tool, the CL strategy has demonstrated its power in improving the generalization capacity and convergence rate of various models in a wide range of scenarios such as computer vision and natural language processing, etc. In this survey article, we comprehensively review CL from various aspects including motivations, definitions, theories, and applications. We discuss works on curriculum learning within a general CL framework, elaborating on how to design a manually predefined curriculum or an automatic curriculum. In particular, we summarize existing CL designs based on the general framework of Difficulty Measurer + Training Scheduler and further categorize the methodologies for automatic CL into four groups, i.e., Self-paced Learning, Transfer Teacher, RL Teacher, and Other Automatic CL. Finally, we present brief discussions on the relationships between CL and other methods, and point out potential future research directions deserving further investigations.

Nonparametric's took me a while to get my head around. On the one hand, all I had ever studied involved making the formulae of a predictive model differentiable and optimizing in regards to the individual or set of parameters (think linear regression or GMM). On the other hand, the majority of the nonparametric methods were being used in classification (Random Forests, KNN, etc). But some of the best methods are nonparametric. They do not assume a particular family of distributions and try to select the best-fit ones, they make judgments without assuming a distribution. Keep up to date with my latest articles here!