Commonly used classification algorithms in machine learning, such as support vector machines, minimize a convex surrogate loss on training examples. In practice, these algorithms are surprisingly robust to errors in the training data. In this work, we identify a set of conditions on the data under which such surrogate loss minimization algorithms provably learn the correct classifier. This allows us to establish, in a unified framework, the robustness of these algorithms under various models on data as well as error. In particular, we show that if the data is linearly classifiable with a slightly non-trivial margin (i.e. a margin at least $C\div\sqrt{d}$ for $d$-dimensional unit vectors), and the class-conditional distributions are near isotropic and logconcave, then surrogate loss minimization has negligible error on the uncorrupted data even when a constant fraction of examples are adversarially mislabeled.

Traditional methods for kernel selection rely on parametric kernel functions or a combination thereof and although the kernel hyperparameters are tuned, these methods often provide sub-optimal results due to the limitations induced by the parametric forms. In this paper, we propose a novel formulation for kernel selection using efficient Bayesian optimisation to find the best fitting non-parametric kernel. The kernel is expressed using a linear combination of functions sampled from a prior Gaussian Process (GP) defined by a hyperkernel. We also provide a mechanism to ensure the positive definiteness of the Gram matrix constructed using the resultant kernels. Our experimental results on GP regression and Support Vector Machine (SVM) classification tasks involving both synthetic functions and several real-world datasets show the superiority of our approach over the state-of-the-art.

Wasserstein \textbf{D}istributionally \textbf{R}obust \textbf{O}ptimization (DRO) is concerned with finding decisions that perform well on data that are drawn from the worst probability distribution within a Wasserstein ball centered at a certain nominal distribution. In recent years, it has been shown that various DRO formulations of learning models admit tractable convex reformulations. However, most existing works propose to solve these convex reformulations by general-purpose solvers, which are not well-suited for tackling large-scale problems. In this paper, we focus on a family of Wasserstein distributionally robust support vector machine (DRSVM) problems and propose two novel epigraphical projection-based incremental algorithms to solve them. The updates in each iteration of these algorithms can be computed in a highly efficient manner. Moreover, we show that the DRSVM problems considered in this paper satisfy a Hölderian growth condition with explicitly determined growth exponents. Consequently, we are able to establish the convergence rates of the proposed incremental algorithms. Our numerical results indicate that the proposed methods are orders of magnitude faster than the state-of-the-art, and the performance gap grows considerably as the problem size increases.

Sufficient dimension reduction is used pervasively as a supervised dimension reduction approach. Most existing sufficient dimension reduction methods are developed for data with a continuous response and may have an unsatisfactory performance for the categorical response, especially for the binary-response. To address this issue, we propose a novel estimation method of sufficient dimension reduction subspace (SDR subspace) using optimal transport. The proposed method, named principal optimal transport direction (POTD), estimates the basis of the SDR subspace using the principal directions of the optimal transport coupling between the data respecting different response categories. The proposed method also reveals the relationship among three seemingly irrelevant topics, i.e., sufficient dimension reduction, support vector machine, and optimal transport. We study the asymptotic properties of POTD and show that in the cases when the class labels contain no error, POTD estimates the SDR subspace exclusively. Empirical studies show POTD outperforms most of the state-of-the-art linear dimension reduction methods.

We study fast algorithms for statistical regression problems under the strong contamination model, where the goal is to approximately optimize a generalized linear model (GLM) given adversarially corrupted samples. Prior works in this line of research were based on the \emph{robust gradient descent} framework of \cite{PrasadSBR20}, a first-order method using biased gradient queries, or the \emph{Sever} framework of \cite{DiakonikolasKK019}, an iterative outlier-removal method calling a stationary point finder. We present nearly-linear time algorithms for robust regression problems with improved runtime or estimation guarantees compared to the state-of-the-art. For the general case of smooth GLMs (e.g.\ logistic regression), we show that the robust gradient descent framework of \cite{PrasadSBR20} can be \emph{accelerated}, and show our algorithm extends to optimizing the Moreau envelopes of Lipschitz GLMs (e.g.\ support vector machines), answering several open questions in the literature. For the well-studied case of robust linear regression, we present an alternative approach obtaining improved estimation rates over prior nearly-linear time algorithms. Interestingly, our algorithm starts with an identifiability proof introduced in the context of the sum-of-squares algorithm of \cite{BakshiP21}, which achieved optimal error rates while requiring large polynomial runtime and sample complexity. We reinterpret their proof within the Sever framework and obtain a dramatically faster and more sample-efficient algorithm under fewer distributional assumptions.

In this paper, we propose a new covering technique localized for the trajectories of SGD. This localization provides an algorithm-specific complexity measured by the covering number, which can have dimension-independent cardinality in contrast to standard uniform covering arguments that result in exponential dimension dependency. Based on this localized construction, we show that if the objective function is a finite perturbation of a piecewise strongly convex and smooth function with $P$ pieces, i.e., non-convex and non-smooth in general, the generalization error can be upper bounded by $O(\sqrt{(\log n\log(nP))/n})$, where $n$ is the number of data samples. In particular, this rate is independent of dimension and does not require early stopping and decaying step size. Finally, we employ these results in various contexts and derive generalization bounds for multi-index linear models, multi-class support vector machines, and $K$-means clustering for both hard and soft label setups, improving the previously known state-of-the-art rates.

Several recent publications report advances in training optimal decision trees (ODTs) using mixed-integer programs (MIPs), due to algorithmic advances in integer programming and a growing interest in addressing the inherent suboptimality of heuristic approaches such as CART. In this paper, we propose a novel MIP formulation, based on 1-norm support vector machine model, to train a binary oblique ODT for classification problems. We further present techniques, such as cutting planes, to tighten its linear relaxation, to improve run times to reach optimality. Using 36 datasets from the University of California Irvine Machine Learning Repository, we demonstrate that our training approach outperforms its counterparts from literature in terms of out-of-sample performance (around 10% improvement in mean out-of-sample testing accuracy). Towards our goal of developing a scalable framework to train multivariate ODT on large datasets, we propose a new linear programming based data selection method to choose a subset of the data, and use it to train a decision tree through our proposed MIP model. We conclude this paper with extensive numerical testing results, that showcase the generalization performance of our new MIP formulation, and the improvement in mean out-of-sample accuracy on large datasets.

We propose a consolidated cross-validation (CV) algorithm for training and tuning the support vector machines (SVM) on reproducing kernel Hilbert spaces. Our consolidated CV algorithm utilizes a recently proposed exact leave-one-out formula for the SVM and accelerates the SVM computation via a data reduction strategy. In addition, to compute the SVM with the bias term (intercept), which is not handled by the existing data reduction methods, we propose a novel two-stage consolidated CV algorithm. With numerical studies, we demonstrate that our algorithm is about an order of magnitude faster than the two mainstream SVM solvers, kernlab and LIBSVM, with almost the same accuracy.

While large margin classifiers are originally an outcome of an optimization framework, support vectors (SVs) can be obtained from geometric approaches. This article presents advances in the use of Gabriel graphs (GGs) in binary and multiclass classification problems. For Chipclass, a hyperparameter-less and optimization-less GG-based binary classifier, we discuss how activation functions and support edge (SE)-centered neurons affect the classification, proposing smoother functions and structural SV (SSV)-centered neurons to achieve margins with low probabilities and smoother classification contours. We extend the neural network architecture, which can be trained with backpropagation with a softmax function and a cross-entropy loss, or by solving a system of linear equations. A new subgraph-/distance-based membership function for graph regularization is also proposed, along with a new GG recomputation algorithm that is less computationally expensive than the standard approach. Experimental results with the Friedman test show that our method was better than previous GG-based classifiers and statistically equivalent to tree-based models.

Input features are conventionally represented as vectors, matrices, or third order tensors in the real field, for color image classification. Inspired by the success of quaternion data modeling for color images in image recovery and denoising tasks, we propose a novel classification method for color image classification, named as the Low-rank Support Quaternion Matrix Machine (LSQMM), in which the RGB channels are treated as pure quaternions to effectively preserve the intrinsic coupling relationships among channels via the quaternion algebra. For the purpose of promoting low-rank structures resulting from strongly correlated color channels, a quaternion nuclear norm regularization term, serving as a natural extension of the conventional matrix nuclear norm to the quaternion domain, is added to the hinge loss in our LSQMM model. An Alternating Direction Method of Multipliers (ADMM)-based iterative algorithm is designed to effectively resolve the proposed quaternion optimization model. Experimental results on multiple color image classification datasets demonstrate that our proposed classification approach exhibits advantages in classification accuracy, robustness and computational efficiency, compared to several state-of-the-art methods using support vector machines, support matrix machines, and support tensor machines.