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We consider the problem of learning an unknown function $f_{\star}$ on the $d$-dimensional sphere with respect to the square loss, given i.i.d. samples $\{(y_i,{\boldsymbol x}_i)\}_{i\le n}$ where ${\boldsymbol x}_i$ is a feature vector uniformly distributed on the sphere and $y_i=f_{\star}({\boldsymbol x}_i)+\varepsilon_i$. We study two popular classes of models that can be regarded as linearizations of two-layers neural networks around a random initialization: the random features model of Rahimi-Recht (RF); the neural tangent kernel model of Jacot-Gabriel-Hongler (NT). Both these approaches can also be regarded as randomized approximations of kernel ridge regression (with respect to different kernels), and enjoy universal approximation properties when the number of neurons $N$ diverges, for a fixed dimension $d$. We consider two specific regimes: the approximation-limited regime, in which $n=\infty$ while $d$ and $N$ are large but finite; and the sample size-limited regime in which $N=\infty$ while $d$ and $n$ are large but finite. In the first regime we prove that if $d^{\ell + \delta} \le N\le d^{\ell+1-\delta}$ for small $\delta > 0$, then \RF\, effectively fits a degree-$\ell$ polynomial in the raw features, and \NT\, fits a degree-$(\ell+1)$ polynomial. In the second regime, both RF and NT reduce to kernel methods with rotationally invariant kernels. We prove that, if the number of samples is $d^{\ell + \delta} \le n \le d^{\ell +1-\delta}$, then kernel methods can fit at most a a degree-$\ell$ polynomial in the raw features. This lower bound is achieved by kernel ridge regression. Optimal prediction error is achieved for vanishing ridge regularization., Comment: 65 pages; 17 pdf figures

Most high-dimensional estimation methods propose to minimize a cost function (empirical risk) that is a sum of losses associated to each data point (each example). In this paper, we focus on the case of nonconvex losses. Classical empirical process theory implies uniform convergence of the empirical (or sample) risk to the population risk. While under additional assumptions, uniform convergence implies consistency of the resulting M-estimator, it does not ensure that the latter can be computed efficiently. ¶ In order to capture the complexity of computing M-estimators, we study the landscape of the empirical risk, namely its stationary points and their properties. We establish uniform convergence of the gradient and Hessian of the empirical risk to their population counterparts, as soon as the number of samples becomes larger than the number of unknown parameters (modulo logarithmic factors). Consequently, good properties of the population risk can be carried to the empirical risk, and we are able to establish one-to-one correspondence of their stationary points. We demonstrate that in several problems such as nonconvex binary classification, robust regression and Gaussian mixture model, this result implies a complete characterization of the landscape of the empirical risk, and of the convergence properties of descent algorithms. ¶ We extend our analysis to the very high-dimensional setting in which the number of parameters exceeds the number of samples, and provides a characterization of the empirical risk landscape under a nearly information-theoretically minimal condition. Namely, if the number of samples exceeds the sparsity of the parameters vector (modulo logarithmic factors), then a suitable uniform convergence result holds. We apply this result to nonconvex binary classification and robust regression in very high-dimension.

Multiple hypothesis testing is a core problem in statistical inference and arises in almost every scientific field. Given a set of null hypotheses $\mathcal{H}(n) = (H_1,\dotsc, H_n)$, Benjamini and Hochberg introduced the false discovery rate (FDR), which is the expected proportion of false positives among rejected null hypotheses, and proposed a testing procedure that controls FDR below a pre-assigned significance level. Nowadays FDR is the criterion of choice for large scale multiple hypothesis testing. In this paper we consider the problem of controlling FDR in an "online manner". Concretely, we consider an ordered --possibly infinite-- sequence of null hypotheses $\mathcal{H} = (H_1,H_2,H_3,\dots )$ where, at each step $i$, the statistician must decide whether to reject hypothesis $H_i$ having access only to the previous decisions. This model was introduced by Foster and Stine. We study a class of "generalized alpha-investing" procedures and prove that any rule in this class controls online FDR, provided $p$-values corresponding to true nulls are independent from the other $p$-values. (Earlier work only established mFDR control.) Next, we obtain conditions under which generalized alpha-investing controls FDR in the presence of general $p$-values dependencies. Finally, we develop a modified set of procedures that also allow to control the false discovery exceedance (the tail of the proportion of false discoveries). Numerical simulations and analytical results indicate that online procedures do not incur a large loss in statistical power with respect to offline approaches, such as Benjamini-Hochberg., Comment: 44 pages, 9 figures, to appear in Annals of Statistics