We study quantile regression in an extrapolation regime where the covariate takes unusually large values. Under regular variation assumptions, extreme observations can be effectively characterized through their angular components, enabling learning strategies that focus on the angle of the most extreme observations. This approach is formalized through the minimization of an asymptotic conditional risk that localizes learning in the tail of the covariate distribution. We propose a novel Support Vector Machine (SVM) framework for extreme quantile regression, leveraging reproducing kernel Hilbert spaces to handle high-dimensional and nonlinear settings. Our method also accommodates unbounded response variables and avoids restrictive transformations. We establish finite-sample learning guarantees under mild regularity assumptions. The proposed framework unifies ideas from statistical learning and multivariate extremes, providing a tractable and theoretically grounded approach to extrapolation. We complement our theoretical findings with an empirical study on river flow data from the Danube, demonstrating the practical relevance of our methods.

As a consequence, empirical risk minimizers generally perform very poorly in extreme regions. It is the purpose of this paper to develop a general framework for classification in the extremes.

Neural Information Processing Systems http://nips.cc/

We study the problem of testing the goodness of fit of a discrete sample from many categories to the uniform distribution over the categories. As a class of alternative hypotheses, we consider the removal of an $\ell_p$ ball of radius $\epsilon$ around the uniform rate sequence for $p \leq 2$. When the number of samples $n$ and number of categories $N$ go to infinity while $\epsilon$ is small, the minimax risk $R_\epsilon^*$ in testing based on the sample's histogram (number of absent categories, singletons, collisions, ...) asymptotes to $2\Phi(-n N^{2-2/p} \epsilon^2/\sqrt{8N})$; $\Phi(x)$ is the normal CDF. This result allows the comparison of the many estimators previously proposed for this problem at the constant level, rather than at the rate of convergence of the risk or the scaling order of the sample complexity. The minimax test mostly relies on collisions in the very small sample limit but otherwise behaves like the chisquared test. Empirical studies over a range of problem parameters show that our estimate is accurate in finite samples and that the minimax test is significantly better than the chisquared test or a test that only uses collisions. Our analysis relies on the asymptotic normality of histogram ordinates, the equivalence between the minimax setting and a Bayesian setting, and the characterization of the least favorable prior by reducing a multi-dimensional optimization problem to a one-dimensional problem.

We investigate the asymptotic risk of a general class of overparameterized likelihood models, including deep models. The recent empirical success of large-scale models has motivated several theoretical studies to investigate a scenario wherein both the number of samples, $n$, and parameters, $p$, diverge to infinity and derive an asymptotic risk at the limit. However, these theorems are only valid for linear-in-feature models, such as generalized linear regression, kernel regression, and shallow neural networks. Hence, it is difficult to investigate a wider class of nonlinear models, including deep neural networks with three or more layers. In this study, we consider a likelihood maximization problem without the model constraints and analyze the upper bound of an asymptotic risk of an estimator with penalization. Technically, we combine a property of the Fisher information matrix with an extended Marchenko-Pastur law and associate the combination with empirical process techniques. The derived bound is general, as it describes both the double descent and the regularized risk curves, depending on the penalization. Our results are valid without the linear-in-feature constraints on models and allow us to derive the general spectral distributions of a Fisher information matrix from the likelihood. We demonstrate that several explicit models, such as parallel deep neural networks, ensemble learning, and residual networks, are in agreement with our theory. This result indicates that even large and deep models have a small asymptotic risk if they exhibit a specific structure, such as divisibility. To verify this finding, we conduct a real-data experiment with parallel deep neural networks. Our results expand the applicability of the asymptotic risk analysis, and may also contribute to the understanding and application of deep learning.

The Bezier simplex fitting is a novel data modeling technique which exploits geometric structures of data to approximate the Pareto front of multi-objective optimization problems. There are two fitting methods based on different sampling strategies. The inductive skeleton fitting employs a stratified subsampling from each skeleton of a simplex, whereas the all-at-once fitting uses a non-stratified sampling which treats a simplex as a whole. In this paper, we analyze the asymptotic risks of those B\'ezier simplex fitting methods and derive the optimal subsample ratio for the inductive skeleton fitting. It is shown that the inductive skeleton fitting with the optimal ratio has a smaller risk when the degree of a Bezier simplex is less than three. Those results are verified numerically under small to moderate sample sizes. In addition, we provide two complementary applications of our theory: a generalized location problem and a multi-objective hyper-parameter tuning of the group lasso. The former can be represented by a Bezier simplex of degree two where the inductive skeleton fitting outperforms. The latter can be represented by a Bezier simplex of degree three where the all-at-once fitting gets an advantage.

We study least squares linear regression over $N$ uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features $p$ is at most the sample size $n$, the estimator under consideration coincides with the principal component regression estimator; when $p>n$, the estimator is the least $\ell_2$ norm solution over the selected features. We give an average-case analysis of the out-of-sample prediction error as $p,n,N \to \infty$ with $p/N \to \alpha$ and $n/N \to \beta$, for some constants $\alpha \in [0,1]$ and $\beta \in (0,1)$. In this average-case setting, the prediction error exhibits a `double descent' shape as a function of $p$.