The Interaction Bottleneck of Deep Neural Networks: Discovery, Proof, and Modulation

TL;DR

This paper uncovers a universal interaction bottleneck in deep neural networks, showing they favor low and high-order interactions over mid-order ones, and demonstrates how modulating these interactions affects model behavior and capabilities.

Contribution

It introduces a stratified analysis of interaction structures in DNNs, proves the intrinsic difficulty of mid-order interactions, and proposes methods to modulate these interactions for desired model properties.

Findings

DNNs under-represent mid-order interactions across architectures and tasks.

Mid-order interactions have high variability, making them difficult to learn.

Modulating interaction emphasis influences generalization, robustness, and fitting capacity.

Abstract

Understanding what kinds of cooperative structures deep neural networks (DNNs) can represent remains a fundamental yet insufficiently understood problem. In this work, we treat interactions as the fundamental units of such structure and investigate a largely unexplored question: how DNNs encode interactions under different levels of contextual complexity, and how these microscopic interaction patterns shape macroscopic representation capacity. To quantify this complexity, we use multi-order interactions [57], where each order reflects the amount of contextual information required to evaluate the joint interaction utility of a variable pair. This formulation enables a stratified analysis of cooperative patterns learned by DNNs. Building on this formulation, we develop a comprehensive study of interaction structure in DNNs. (i) We empirically discover a universal interaction bottleneck: across architectures and tasks, DNNs easily learn low-order and high-order interactions but consistently under-represent mid-order ones. (ii) We theoretically explain this bottleneck by proving that mid-order interactions incur the highest contextual variability, yielding large gradient variance and making them intrinsically difficult to learn. (iii) We further modulate the bottleneck by introducing losses that steer models toward emphasizing interactions of selected orders. Finally, we connect microscopic interaction structures with macroscopic representational behavior: low-order-emphasized models exhibit stronger generalization and robustness, whereas high-order-emphasized models demonstrate greater structural modeling and fitting capability. Together, these results uncover an inherent representational bias in modern DNNs and establish interaction order as a powerful lens for interpreting and guiding deep representations.