One widely recognized interpretation for their empirical success is their ability to perform in-context learning (ICL): pretrained transformers are capable of performing previously unseen tasks based on demonstrations and examples in the prompt, without requiring any additional task-specific fine-tuning (Brown et al., 2020). A line of recent works interpret the in-context learning (ICL) capability of transformers from an algorithmic perspective, viewing transformers as models that can implicitly execute certain learning algorithms on the context examples. Specifically, Garg et al. (2022) proposes a theoretical framework for ICL in terms of learning a hypothesis class, and empirically shows that transformers can in-context learn the linear function class. Motivated by this empirical finding, several recent works attempt to theoretically study how transformers perform in-context learning on linear regression tasks. Aky urek et al. (2022); Von Oswald et al. (2023) construct multi-layer transformers with linear attention that can execute gradient descent on the an "in-context loss" defined on the context data, thereby enabling in-context learning of linear regression.

We develop graph-based methods for semi-supervised learning based on label propagation on a data similarity graph. When data is abundant or arrive in a stream, the problems of computation and data storage arise for any graph-based method. We propose a fast approximate online algorithm that solves for the harmonic solution on an approximate graph. We show, both empirically and theoretically, that good behavior can be achieved by collapsing nearby points into a set of local representative points that minimize distortion. Moreover, we regularize the harmonic solution to achieve better stability properties. We also present graph-based methods for detecting conditional anomalies and apply them to the identification of unusual clinical actions in hospitals. Our hypothesis is that patient-management actions that are unusual with respect to the past patients may be due to errors and that it is worthwhile to raise an alert if such a condition is encountered. Conditional anomaly detection extends standard unconditional anomaly framework but also faces new problems known as fringe and isolated points. We devise novel nonparametric graph-based methods to tackle these problems. Our methods rely on graph connectivity analysis and soft harmonic solution. Finally, we conduct an extensive human evaluation study of our conditional anomaly methods by 15 experts in critical care.

This paper introduces the use of per-class regularization hyperparameters in Gabriel graph-based binary classifiers. We demonstrate how the quality index used for regularization behaves both in the margin region and in the presence of outliers, and how incorporating this regularization flexibility can lead to solutions that effectively eliminate outliers while training the classifier. We also show how it can address class imbalance by generating higher and lower thresholds for the majority and minority classes, respectively. Thus, rather than having a single solution based on fixed thresholds, flexible thresholds expand the solution space and can be optimized through hyperparameter tuning algorithms. Friedman test shows that flexible thresholds are capable of improving Gabriel graph-based classifiers.

Modern machine learning systems such as deep neural networks are often highly over-parameterized so that they can fit the noisy training data exactly, yet they can still achieve small test errors in practice. In this paper, we study this "benign overfitting" phenomenon of the maximum margin classifier for linear classification problems. Specifically, we consider data generated from sub-Gaussian mixtures, and provide a tight risk bound for the maximum margin linear classifier in the over-parameterized setting. Our results precisely characterize the condition under which benign overfitting can occur in linear classification problems, and improve on previous work. They also have direct implications for over-parameterized logistic regression.

Figure 8: Visualizations of accuracy and bias flows for the smaller ANN trained on the synthetic dataset. Note how the most dominant accuracy flows arise from X3, which is the only bias-free feature in the dataset. In contrast, the largest bias flows arise from X1 and X2, both of which are heavily biased features. It is intuitively clear from these pictures which edges have the largest bias-to-accuracy flow ratios, and hence which edges would be the first to be pruned.

In this paper we establish a duality between boosting and SVM, and use this to derive a novel discriminant dimensionality reduction algorithm. In particular, using the multiclass formulation of boosting and SVM we note that both use a combination of mapping and linear classification to maximize the multiclass margin. In SVM this is implemented using a pre-defined mapping (induced by the kernel) and optimizing the linear classifiers. In boosting the linear classifiers are pre-defined and the mapping (predictor) is learned through a combination of weak learners. We argue that the intermediate mapping, i.e. boosting predictor, is preserving the discriminant aspects of the data and that by controlling the dimension of this mapping it is possible to obtain discriminant low dimensional representations for the data. We use the aforementioned duality and propose a new method, Large Margin Discriminant Dimensionality Reduction (LADDER) that jointly learns the mapping and the linear classifiers in an efficient manner. This leads to a data-driven mapping which can embed data into any number of dimensions. Experimental results show that this embedding can significantly improve performance on tasks such as hashing and image/scene classification.

The support vector machine (SVM) and minimum Euclidean norm least squares regression are two fundamentally different approaches to fitting linear models, but they have recently been connected in models for very high-dimensional data through a phenomenon of support vector proliferation, where every training example used to fit an SVM becomes a support vector. In this paper, we explore the generality of this phenomenon and make the following contributions. First, we prove a super-linear lower bound on the dimension (in terms of sample size) required for support vector proliferation in independent feature models, matching the upper bounds from previous works. We further identify a sharp phase transition in Gaussian feature models, bound the width of this transition, and give experimental support for its universality. Finally, we hypothesize that this phase transition occurs only in much higher-dimensional settings in the $\ell_1$ variant of the SVM, and we present a new geometric characterization of the problem that may elucidate this phenomenon for the general $\ell_p$ case.

In this paper we establish a duality between boosting and SVM, and use this to derive a novel discriminant dimensionality reduction algorithm. In particular, using the multiclass formulation of boosting and SVM we note that both use a combination of mapping and linear classification to maximize the multiclass margin. In SVM this is implemented using a pre-defined mapping (induced by the kernel) and optimizing the linear classifiers. In boosting the linear classifiers are pre-defined and the mapping (predictor) is learned through combination of weak learners. We argue that the intermediate mapping, e.g.