C. Wang, H. Hu, and Y. M. Lu, “
A Solvable High-Dimensional Model of GAN,” in
Proc. Thirty-third Conference on Neural Information Processing Systems (NeurIPS), 2019.
arXiv:1805.08349 [cs.LG]AbstractDespite the remarkable successes of generative adversarial networks (GANs) in many applications, theoretical understandings of their performance is still limited. In this paper, we present a simple shallow GAN model fed by high-dimensional input data. The dynamics of the training process of the proposed model can be exactly analyzed in the high-dimensional limit. In particular, by using the tool of scaling limits of stochastic processes, we show that the macroscopic quantities measuring the quality of the training process converge to a deterministic process that is characterized as the unique solution of a finite-dimensional ordinary differential equation (ODE). The proposed model is simple, but its training process already exhibits several different phases that can mimic the behaviors of more realistic GAN models used in practice. Specifically, depending on the choice of the learning rates, the training process can reach either a successful, a failed, or an oscillating phase. By studying the steady-state solutions of the limiting ODEs, we obtain a phase diagram that precisely characterizes the conditions under which each phase takes place. Although this work focuses on a simple GAN model, the analysis methods developed here might prove useful in the theoretical understanding of other variants of GANs with more advanced training algorithms.
L. Saglietti, Y. M. Lu, and C. Lucibello, “
Generalized Approximate Survey Propagation for High-Dimensional Estimation,” in
Proc. International Conference on Machine Learning (ICML), 2019.
arXiv:1905.05313 [cond-mat.dis-nn]AbstractIn Generalized Linear Estimation (GLE) problems, we seek to estimate a signal that is observed through a linear transform followed by a component-wise, possibly nonlinear and noisy, channel. In the Bayesian optimal setting, Generalized Approximate Message Passing (GAMP) is known to achieve optimal performance for GLE. However, its performance can significantly degrade whenever there is a mismatch between the assumed and the true generative model, a situation frequently encountered in practice. In this paper, we propose a new algorithm, named Generalized Approximate Survey Propagation (GASP), for solving GLE in the presence of prior or model mis-specifications. As a prototypical example, we consider the phase retrieval problem, where we show that GASP outperforms the corresponding GAMP, reducing the reconstruction threshold and, for certain choices of its parameters, approaching Bayesian optimal performance. Furthermore, we present a set of State Evolution equations that exactly characterize the dynamics of GASP in the high-dimensional limit.
H. Hong and Y. M. Lu, “
Asymptotics and optimal designs of SLOPE for sparse linear regression,” in
Prof. International Symposium on Information Theory (ISIT), 2019.
Longer version with full technical detailsAbstractIn sparse linear regression, the SLOPE estimator generalizes LASSO by assigning magnitude-dependent regular- izations 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 superi- ority of our optimal design over LASSO and a regularization sequence previously proposed in the literature.