Learning One-hidden-layer Neural Networks under General Input Distributions
This work addresses a fundamental bottleneck in neural network optimization for researchers and practitioners by enabling robust training across diverse data distributions, representing a significant advance beyond incremental improvements.
The paper tackles the problem of training one-hidden-layer neural networks under general input distributions, overcoming the restrictive Gaussian assumption of prior methods, and shows that stochastic gradient descent with their designed loss functions recovers true parameters globally and empirically outperforms existing approaches.
Significant advances have been made recently on training neural networks, where the main challenge is in solving an optimization problem with abundant critical points. However, existing approaches to address this issue crucially rely on a restrictive assumption: the training data is drawn from a Gaussian distribution. In this paper, we provide a novel unified framework to design loss functions with desirable landscape properties for a wide range of general input distributions. On these loss functions, remarkably, stochastic gradient descent theoretically recovers the true parameters with global initializations and empirically outperforms the existing approaches. Our loss function design bridges the notion of score functions with the topic of neural network optimization. Central to our approach is the task of estimating the score function from samples, which is of basic and independent interest to theoretical statistics. Traditional estimation methods (example: kernel based) fail right at the outset; we bring statistical methods of local likelihood to design a novel estimator of score functions, that provably adapts to the local geometry of the unknown density.