Search

Determining whether deep neural network (DNN) models can reliably recover target functions at overparameterization is a critical yet complex issue in the theory of deep learning. To advance understanding in this area, we introduce a concept we term "local linear recovery" (LLR), a weaker form of target function recovery that renders the problem more amenable to theoretical analysis. In the sense of LLR, we prove that functions expressible by narrower DNNs are guaranteed to be recoverable from fewer samples than model parameters. Specifically, we establish upper limits on the optimistic sample sizes, defined as the smallest sample size necessary to guarantee LLR, for functions in the space of a given DNN. Furthermore, we prove that these upper bounds are achieved in the case of two-layer tanh neural networks. Our research lays a solid groundwork for future investigations into the recovery capabilities of DNNs in overparameterized scenarios., Comment: arXiv admin note: text overlap with arXiv:2211.11623

This paper presents a comprehensive analysis of critical point sets in two-layer neural networks. To study such complex entities, we introduce the critical embedding operator and critical reduction operator as our tools. Given a critical point, we use these operators to uncover the whole underlying critical set representing the same output function, which exhibits a hierarchical structure. Furthermore, we prove existence of saddle branches for any critical set whose output function can be represented by a narrower network. Our results provide a solid foundation to the further study of optimization and training behavior of neural networks.

This report is actually the in-class project of MATH489 FA2022, Dynamics and Differential Equations in University of Illinois Urbana-Champaign. The report is mainly about gradient flow of a Morse-type functions (Morse functions, Morse--Bott functions, etc.) and analytic functions. Several small results that I develop during the reading are included as well. They have a focus on the convergence, limiting rate and direction of gradient flow in either setting., Comment: 14 pages in-class project report

Under mild assumptions, we investigate the geometry of the loss landscape for two-layer neural networks in the vicinity of global minima. Utilizing novel techniques, we demonstrate: (i) how global minima with zero generalization error become geometrically separated from other global minima as the sample size grows; and (ii) the local convergence properties and rate of gradient flow dynamics. Our results indicate that two-layer neural networks can be locally recovered in the regime of overparameterization.

We propose an optimistic estimate to evaluate the best possible fitting performance of nonlinear models. It yields an optimistic sample size that quantifies the smallest possible sample size to fit/recover a target function using a nonlinear model. We estimate the optimistic sample sizes for matrix factorization models, deep models, and deep neural networks (DNNs) with fully-connected or convolutional architecture. For each nonlinear model, our estimates predict a specific subset of targets that can be fitted at overparameterization, which are confirmed by our experiments. Our optimistic estimate reveals two special properties of the DNN models -- free expressiveness in width and costly expressiveness in connection. These properties suggest the following architecture design principles of DNNs: (i) feel free to add neurons/kernels; (ii) restrain from connecting neurons. Overall, our optimistic estimate theoretically unveils the vast potential of nonlinear models in fitting at overparameterization. Based on this framework, we anticipate gaining a deeper understanding of how and why numerous nonlinear models such as DNNs can effectively realize their potential in practice in the near future.

Models with nonlinear architectures/parameterizations such as deep neural networks (DNNs) are well known for their mysteriously good generalization performance at overparameterization. In this work, we tackle this mystery from a novel perspective focusing on the transition of the target recovery/fitting accuracy as a function of the training data size. We propose a rank stratification for general nonlinear models to uncover a model rank as an "effective size of parameters" for each function in the function space of the corresponding model. Moreover, we establish a linear stability theory proving that a target function almost surely becomes linearly stable when the training data size equals its model rank. Supported by our experiments, we propose a linear stability hypothesis that linearly stable functions are preferred by nonlinear training. By these results, model rank of a target function predicts a minimal training data size for its successful recovery. Specifically for the matrix factorization model and DNNs of fully-connected or convolutional architectures, our rank stratification shows that the model rank for specific target functions can be much lower than the size of model parameters. This result predicts the target recovery capability even at heavy overparameterization for these nonlinear models as demonstrated quantitatively by our experiments. Overall, our work provides a unified framework with quantitative prediction power to understand the mysterious target recovery behavior at overparameterization for general nonlinear models.

In recent years, understanding the implicit regularization of neural networks (NNs) has become a central task in deep learning theory. However, implicit regularization is itself not completely defined and well understood. In this work, we attempt to mathematically define and study implicit regularization. Importantly, we explore the limitations of a common approach to characterizing implicit regularization using data-independent functions. We propose two dynamical mechanisms, i.e., Two-point and One-point Overlapping mechanisms, based on which we provide two recipes for producing classes of one-hidden-neuron NNs that provably cannot be fully characterized by a type of or all data-independent functions. Following the previous works, our results further emphasize the profound data dependency of implicit regularization in general, inspiring us to study in detail the data dependency of NN implicit regularization in the future., Comment: Revised the structure of paper and added results about implicit regularization in training two-layer network or even more general "activation function-related" models