We study the asymptotic generalization of an overparameterized linear model for multiclass classification under the Gaussian covariates bi-level model introduced in Subramanian et al. (NeurIPS'22), where the number of data points, features, and classes all grow together. We fully resolve the conjecture posed in Subramanian et al. '22, matching the predicted regimes for which the model does and does not generalize. Furthermore, our new lower bounds are akin to an information-theoretic strong converse: they establish that the misclassification rate goes to 0 or 1 asymptotically. One surprising consequence of our tight results is that the min-norm interpolating classifier can be asymptotically suboptimal relative to noninterpolating classifiers in the regime where the min-norm interpolating regressor is known to be optimal. The key to our tight analysis is a new variant of the Hanson-Wright inequality which is broadly useful for multiclass problems with sparse labels. As an application, we show that the same type of analysis can be used to analyze the related multi-label classification problem under the same bi-level ensemble.

Via an overparameterized linear model with Gaussian features, we provide conditions for good generalization for multiclass classification of minimum-norm interpolating solutions in an asymptotic setting where both the number of underlying features and the number of classes scale with the number of training points. The survival/contamination analysis framework for understanding the behavior of overparameterized learning problems is adapted to this setting, revealing that multiclass classification qualitatively behaves like binary classification in that, as long as there are not too many classes (made precise in the paper), it is possible to generalize well even in settings where regression tasks would not generalize. Besides various technical challenges, it turns out that the key difference from the binary classification setting is that there are relatively fewer training examples of each class in the multiclass setting as the number of classes increases, making the multiclass problem ``harder'' than the binary one.

Hence, R5 is wary of the relevance of our examples to deep learning.

We study the asymptotic generalization of an overparameterized linear model for multiclass classification under the Gaussian covariates bi-level model introduced in Subramanian et al. (NeurIPS'22), where the number of data points, features, and classes all grow together. We fully resolve the conjecture posed in Subramanian et al. '22, matching the predicted regimes for which the model does and does not generalize. Furthermore, our new lower bounds are akin to an information-theoretic strong converse: they establish that the misclassification rate goes to 0 or 1 asymptotically. One surprising consequence of our tight results is that the min-norm interpolating classifier can be asymptotically suboptimal relative to noninterpolating classifiers in the regime where the min-norm interpolating regressor is known to be optimal. The key to our tight analysis is a new variant of the Hanson-Wright inequality which is broadly useful for multiclass problems with sparse labels.

Via an overparameterized linear model with Gaussian features, we provide conditions for good generalization for multiclass classification of minimum-norm interpolating solutions in an asymptotic setting where both the number of underlying features and the number of classes scale with the number of training points. The survival/contamination analysis framework for understanding the behavior of overparameterized learning problems is adapted to this setting, revealing that multiclass classification qualitatively behaves like binary classification in that, as long as there are not too many classes (made precise in the paper), it is possible to generalize well even in settings where regression tasks would not generalize. Besides various technical challenges, it turns out that the key difference from the binary classification setting is that there are relatively fewer training examples of each class in the multiclass setting as the number of classes increases, making the multiclass problem harder'' than the binary one.

Abstract: We develop a Distributionally Robust Optimization (DRO) formulation for Multiclass Logistic Regression (MLR), which could tolerate data contaminated by outliers. The DRO framework uses a probabilistic ambiguity set defined as a ball of distributions that are close to the empirical distribution of the training set in the sense of the Wasserstein metric. We relax the DRO formulation into a regularized learning problem whose regularizer is a norm of the coefficient matrix. We establish out-of-sample performance guarantees for the solutions to our model, offering insights on the role of the regularizer in controlling the prediction error. We apply the proposed method in rendering deep Vision Transformer (ViT)-based image classifiers robust to random and adversarial attacks.

A common way to avoid overfitting in supervised learning is early stopping, where a held-out set is used for iterative evaluation during training to find a sweet spot in the number of training steps that gives maximum generalization. However, such a method requires a disjoint validation set, thus part of the labeled data from the training set is usually left out for this purpose, which is not ideal when training data is scarce. Furthermore, when the training labels are noisy, the performance of the model over a validation set may not be an accurate proxy for generalization. In this paper, we propose a method to spot an early stopping point in the training iterations without the need for a validation set. We first show that in the overparameterized regime the randomly initialized weights of a linear model converge to the same direction during training. Using this result, we propose to train two parallel instances of a linear model, initialized with different random seeds, and use their intersection as a signal to detect overfitting. In order to detect intersection, we use the cosine distance between the weights of the parallel models during training iterations. Noticing that the final layer of a NN is a linear map of pre-last layer activations to output logits, we build on our criterion for linear models and propose an extension to multi-layer networks, using the new notion of counterfactual weights. We conduct experiments on two areas that early stopping has noticeable impact on preventing overfitting of a NN: (i) learning from noisy labels; and (ii) learning to rank in IR. Our experiments on four widely used datasets confirm the effectiveness of our method for generalization. For a wide range of learning rates, our method, called Cosine-Distance Criterion (CDC), leads to better generalization on average than all the methods that we compare against in almost all of the tested cases.

Via an overparameterized linear model with Gaussian features, we provide conditions for good generalization for multiclass classification of minimum-norm interpolating solutions in an asymptotic setting where both the number of underlying features and the number of classes scale with the number of training points. The survival/contamination analysis framework for understanding the behavior of overparameterized learning problems is adapted to this setting, revealing that multiclass classification qualitatively behaves like binary classification in that, as long as there are not too many classes (made precise in the paper), it is possible to generalize well even in some settings where the corresponding regression tasks would not generalize. Besides various technical challenges, it turns out that the key difference from the binary classification setting is that there are relatively fewer positive training examples of each class in the multiclass setting as the number of classes increases, making the multiclass problem "harder" than the binary one.