The Graphon Limit Hypothesis: Understanding Neural Network Pruning via Infinite Width Analysis
This work addresses a fundamental problem in machine learning for researchers and practitioners by providing a systematic theoretical approach to analyze sparse network trainability, though it is incremental as it builds on existing pruning and graphon theories.
The paper tackles the challenge of understanding why some sparse neural network structures are more trainable than others by proposing a theoretical framework based on graph limits, showing that connectivity patterns from pruning converge to graphons in infinite width, and deriving a Graphon NTK that correlates with training dynamics to explain convergence behaviors.
Sparse neural networks promise efficiency, yet training them effectively remains a fundamental challenge. Despite advances in pruning methods that create sparse architectures, understanding why some sparse structures are better trainable than others with the same level of sparsity remains poorly understood. Aiming to develop a systematic approach to this fundamental problem, we propose a novel theoretical framework based on the theory of graph limits, particularly graphons, that characterizes sparse neural networks in the infinite-width regime. Our key insight is that connectivity patterns of sparse neural networks induced by pruning methods converge to specific graphons as networks' width tends to infinity, which encodes implicit structural biases of different pruning methods. We postulate the Graphon Limit Hypothesis and provide empirical evidence to support it. Leveraging this graphon representation, we derive a Graphon Neural Tangent Kernel (Graphon NTK) to study the training dynamics of sparse networks in the infinite width limit. Graphon NTK provides a general framework for the theoretical analysis of sparse networks. We empirically show that the spectral analysis of Graphon NTK correlates with observed training dynamics of sparse networks, explaining the varying convergence behaviours of different pruning methods. Our framework provides theoretical insights into the impact of connectivity patterns on the trainability of various sparse network architectures.