On the Joint Interaction of Models, Data, and Features

Learning features from data is one of the defining characteristics of deep learning, but our theoretical understanding of the role features play in deep learning is still rudimentary. To address this gap, we introduce a new tool, the interaction tensor, for empirically analyzing the interaction between data and model through features. With the interaction tensor, we make several key observations about how features are distributed in data and how models with different random seeds learn different features. Based on these observations, we propose a conceptual framework for feature learning. Under this framework, the expected accuracy for a single hypothesis and agreement for a pair of hypotheses can both be derived in closed-form. We demonstrate that the proposed framework can explain empirically observed phenomena, including the recently discovered Generalization Disagreement Equality (GDE) that allows for estimating the generalization error with only unlabeled data. Further, our theory also provides explicit construction of natural data distributions that break the GDE. Thus, we believe this work provides valuable new insight into our understanding of feature learning. Related Works Feature learning. Representation learning [6] is the practice of discovering useful features from raw data directly instead of using hand-crafted features. Deep learning is the de facto approach for learning features from a large amount of data [13, 9, 53 ]. Yet, one of the most popular frameworks for understanding deep learning, neural tangent kernel (NTK) [28] cannot account for feature learning from data because it models deep learning as learning a linear classifier on top of random features defined by the gradient of a random neural network. Recent works have started to incorporate feature learning into theoretical analysis [37, 1, 62, 31, 59, 2, 3] . This paper is most immediately related to Allen-Zhu and Li [1] who propose the multi-view data structure where there exist two types of data: multi-view data which contain all the features of a class and single-view data which contain only one feature. They showed that a single two-layer CNN will only learn one feature for each class. In this work, we investigate whether this structure of features holds in practice by treating features as first-class citizens in both empirical investigation and theoretical analysis. Our experimental results reveal a more nuanced perspective on the structure of data and features. Based on these observations, we propose an abstract theoretical model that better reflects how features, data, and models behave in reality. We analyze the generalization property of the model and also its agreement property [43] . We show that this feature learning model provides an alternative condition under which the curious GDE phenomena observed in Jiang et al. [29] can arise.