Gradient Descent Robustly Learns the Intrinsic Dimension of Data in Training Convolutional Neural Networks

Abstract: Modern neural networks are usually highly over-parameterized. Behind the wide usage of over-parameterized networks is the belief that, if the data are simple, then the trained network will be automatically equivalent to a simple predictor. Following this intuition, many existing works have studied different notions of "ranks" of neural networks and their relation to the rank of data. In this work, we study the rank of convolutional neural networks (CNNs) trained by gradient descent, with a specific focus on the robustness of the rank to image background noises. Specifically, we point out that, when adding background noises to images, the rank of the CNN trained with gradient descent is affected far less compared with the rank of the data. We support our claim with a theoretical case study, where we consider a particular data model to characterize low-rank clean images with added background noises. We prove that CNNs trained by gradient descent can learn the intrinsic dimension of clean images, despite the presence of relatively large background noises. We also conduct experiments on synthetic and real datasets to further validate our claim.

An Examination of Rank Robustness in CNNs Under Noisy Conditions

The paper, "Gradient Descent Robustly Learns the Intrinsic Dimension of Data in Training Convolutional Neural Networks" by Zhang et al., offers a thorough exploration of how convolutional neural networks (CNNs), particularly those trained using gradient descent, maintain robustness to the rank of the input data even in the presence of substantial noise. This work delves into the implicit bias in neural networks, a topic of significant interest in understanding how over-parameterized models manage to generalize effectively despite being fit to complex data.

Theoretical Contributions

A central focus of the paper is the concept of "rank robustness," which posits that CNNs manage to preserve the intrinsic rank structure of the data despite added noise. The authors theoretically substantiate this claim using a specific data model that simulates low-rank images with noisy backgrounds. Through detailed analysis, they demonstrate that:

1. Stable Rank Resistance: For CNNs, the stable rank of the learned filters remains close to the rank of the clean data across a range of noise levels. This is a notable discovery, considering that the stable rank of the input data itself tends to escalate rapidly as noise increases.

2. Dynamics of Training: The analysis hinges on dissecting the training dynamics of two-layer convolutional networks. By observing gradient descent behavior, they assert that filters adapt in such a way that each filter learns predominantly along the direction of a randomly initialized basis vector. This adaptation facilitates the learning of clean data features amidst noise.

3. Generalization and Convergence: Despite noise, CNNs trained using gradient descent show not only robustness in rank but also convergence in training loss. This robustness translates into low test error, suggesting an inherent capability of CNNs to generalize well from data corrupted with high levels of noise.

Experimental Validation

Experiments conducted on MNIST, CIFAR-10, and synthetic datasets underscore the theoretical findings. These experiments reveal that the data rank explodes with increasing noise, but the rank of the CNN filters remains consistently low and reflective of the clean data’s rank. The experiments thus provide empirical support for the theoretical assertions regarding noise resistance in CNN rank.

Implications and Future Directions

The implications of this study are profound for the development of neural networks, particularly those used for image recognition and similar tasks. It underscores the notion that CNNs are not just powerful tools due to their vast number of parameters but also because of their ability to distill essential features from data with inherent noise.

The framework established for assessing rank robustness could potentially be extended to more intricate network architectures beyond two-layer CNNs. Further research might explore whether similar phenomena occur in deeper networks or those with different activation functions. Additionally, investigating whether other forms of implicitly learned structure, such as sparsity, follow similar robustness principles could provide comprehensive insights into model generalization capabilities.

In conclusion, this paper by Zhang et al. makes a valuable contribution to our understanding of how convolutional networks maintain efficiency and accuracy in real-world scenarios where data often comprises noise. By elucidating the stability of rank in the presence of noise, they pave the way for more robust network designs and training methodologies in the field of deep learning.