Publications by Year: Submitted

H. Hu and Y. M. Lu, “Universality Laws for High-Dimensional Learning with Random Features,” Submitted. arXiv:2009.07669 [cs.IT]Abstract
We prove a universality theorem for learning with random features. Our result shows that, in terms of training and generalization errors, the random feature model with a nonlinear activation function is asymptotically equivalent to a surrogate Gaussian model with a matching covariance matrix. This settles a conjecture based on which several recent papers develop their results. Our method for proving the universality builds on the classical Lindeberg approach. Major ingredients of the proof include a leave-one-out analysis for the optimization problem associated with the training process and a central limit theorem, obtained via Stein's method, for weakly correlated random variables.
O. Dhifallah and Y. M. Lu, “A Precise Performance Analysis of Learning with Random Features,” Submitted. arXiv:2008.11904 [cs.IT]Abstract
We study the problem of learning an unknown function using random feature models. Our main contribution is an exact asymptotic analysis of such learning problems with Gaussian data. Under mild regularity conditions for the feature matrix, we provide an exact characterization of the asymptotic training and generalization errors, valid in both the under-parameterized and over-parameterized regimes. The analysis presented in this paper holds for general families of feature matrices, activation functions, and convex loss functions. Numerical results validate our theoretical predictions, showing that our asymptotic findings are in excellent agreement with the actual performance of the considered learning problem, even in moderate dimensions. Moreover, they reveal an important role played by the regularization, the loss function and the activation function in the mitigation of the "double descent phenomenon" in learning.
H. Hu and Y. M. Lu, “Asymptotics and Optimal Designs of SLOPE for Sparse Linear Regression,” Submitted. arXiv:1903.11582 [cs.IT]Abstract
In sparse linear regression, the SLOPE estimator generalizes LASSO by assigning magnitude-dependent regularizations to different coordinates of the estimate. In this paper, we present an asymptotically exact characterization of the performance of SLOPE in the high-dimensional regime where the number of unknown parameters grows in proportion to the number of observations. Our asymptotic characterization enables us to derive optimal regularization sequences to either minimize the MSE or to maximize the power in variable selection under any given level of Type-I error. In both cases, we show that the optimal design can be recast as certain infinite-dimensional convex optimization problems, which have efficient and accurate finite-dimensional approximations. Numerical simulations verify our asymptotic predictions. They also demonstrate the superiority of our optimal design over LASSO and a regularization sequence previously proposed in the literature.