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 robusttoerrors inthe training data.

However,mostexistingworkspropose 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.

We introduce a novel machine learning method called the Penalized Profile Support Vector Machine based on the Gabriel edited set for the computation of the probability of failure for a complex system as determined by a threshold condition on a computer model of system behavior. The method is designed to minimize the number of evaluations of the computer model while preserving the geometry of the decision boundary that determines the probability. It employs an adaptive sampling strategy designed to strategically allocate points near the boundary determining failure and builds a locally linear surrogate boundary that remains consistent with its geometry by strategic clustering of training points. We prove two convergence results and we compare the performance of the method against a number of state of the art classification methods on four test problems. We also apply the method to determine the probability of survival using the Lotka--Volterra model for competing species.

Foundation vision, audio, and language models enable zero-shot performance on downstream tasks via their latent representations. Recently, unsupervised learning of data group structure with deep learning methods has gained popularity. TURTLE, a state of the art deep clustering algorithm, uncovers data labeling without supervision by alternating label and hyperplane updates, maximizing the hyperplane margin, in a similar fashion to support vector machines (SVMs). However, TURTLE assumes clusters are balanced; when data is imbalanced, it yields non-ideal hyperplanes that cause higher clustering error. We propose PET-TURTLE, which generalizes the cost function to handle imbalanced data distributions by a power law prior. Additionally, by introducing sparse logits in the labeling process, PET-TURTLE optimizes a simpler search space that in turn improves accuracy for balanced datasets. Experiments on synthetic and real data show that PET-TURTLE improves accuracy for imbalanced sources, prevents over-prediction of minority clusters, and enhances overall clustering.

In this paper, we design a regularization-free algorithm for high-dimensional support vector machines (SVMs) by integrating over-parameterization with Nesterov's smoothing method, and provide theoretical guarantees for the induced implicit regularization phenomenon. In particular, we construct an over-parameterized hinge loss function and estimate the true parameters by leveraging regularization-free gradient descent on this loss function. The utilization of Nesterov's method enhances the computational efficiency of our algorithm, especially in terms of determining the stopping criterion and reducing computational complexity. With appropriate choices of initialization, step size, and smoothness parameter, we demonstrate that unregularized gradient descent achieves a near-oracle statistical convergence rate. Additionally, we verify our theoretical findings through a variety of numerical experiments and compare the proposed method with explicit regularization. Our results illustrate the advantages of employing implicit regularization via gradient descent in conjunction with over-parameterization in sparse SVMs.