A.1 Equations for Gaussian design In this Appendix we discuss the derivation of eqs. Exact asymptotic formulas for the excess prediction error of least-squares and ridge regression are a classic result in high-dimensional statistics, and have been derived in many different works [23, 32, 52, 53]. In this manuscript, we follow the presentation given in [25], which is particularly adapted to our derivation and has the advantage to hold rigorously at large but finite number of samples nand features p. We start by reviewing the formulas in [25]. Note that the risk considered in eq.

In this manuscript we consider Kernel Ridge Regression (KRR) under the Gaussian design. Exponents for the decay of the excess generalization error of KRR have been reported in various works under the assumption of power-law decay of eigenvalues of the features co-variance. These decays were, however, provided for sizeably different setups, namely in the noiseless case with constant regularization and in the noisy optimally regularized case. Intermediary settings have been left substantially uncharted. In this work, we unify and extend this line of work, providing characterization of all regimes and excess error decay rates that can be observed in terms of the interplay of noise and regularization. In particular, we show the existence of a transition in the noisy setting between the noiseless exponents to its noisy values as the sample complexity is increased. Finally, we illustrate how this crossover can also be observed on real data sets.

In this manuscript we consider Kernel Ridge Regression (KRR) under the Gaussian design. Exponents for the decay of the excess generalization error of KRR have been reported in various works under the assumption of power-law decay of eigenvalues of the features co-variance. These decays were, however, provided for sizeably different setups, namely in the noiseless case with constant regularization and in the noisy optimally regularized case. Intermediary settings have been left substantially uncharted. In this work, we unify and extend this line of work, providing characterization of all regimes and excess error decay rates that can be observed in terms of the interplay of noise and regularization.

In this paper, we study the generalization performance of global minima for implementing empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving almost optimal generalization error bounds for numerous types of data under mild conditions. Since over-parameterization is crucial to guarantee that the global minima of ERM on deep ReLU nets can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results indeed fill a gap between optimization and generalization.

Large sparse datasets are common in many real-world applications. Linear SVM has been shown to be very efficient for classifying such datasets. However, it is still unknown how data sparseness would affect its convergence behavior. To study this problem in a systematic manner, we propose a novel approach to generate large and sparse data from real-world datasets, using statistical inference and the data sampling process in the PAC framework. We first study the convergence behavior of linear SVM experimentally, and make several observations, useful for real-world applications. We then offer theoretical proofs for our observations by studying the Bayes risk and PAC bound. Our experiment and theoretic results are valuable for learning large sparse datasets with linear SVM.