Disentangle Sample Size and Initialization Effect on Perfect Generalization for Single-Neuron Target
This provides insights into the generalization behavior of overparameterized neural networks, though it is incremental as it builds on existing theoretical frameworks in a simplified setting.
The paper investigates how initialization and sample size affect perfect generalization in two-layer neural networks recovering a single-neuron target, finding that smaller initialization improves generalization and identifying critical sample size thresholds where target recovery transitions from impossible to possible.
Overparameterized models like deep neural networks have the intriguing ability to recover target functions with fewer sampled data points than parameters (see arXiv:2307.08921). To gain insights into this phenomenon, we concentrate on a single-neuron target recovery scenario, offering a systematic examination of how initialization and sample size influence the performance of two-layer neural networks. Our experiments reveal that a smaller initialization scale is associated with improved generalization, and we identify a critical quantity called the "initial imbalance ratio" that governs training dynamics and generalization under small initialization, supported by theoretical proofs. Additionally, we empirically delineate two critical thresholds in sample size--termed the "optimistic sample size" and the "separation sample size"--that align with the theoretical frameworks established by (see arXiv:2307.08921 and arXiv:2309.00508). Our results indicate a transition in the model's ability to recover the target function: below the optimistic sample size, recovery is unattainable; at the optimistic sample size, recovery becomes attainable albeit with a set of initialization of zero measure. Upon reaching the separation sample size, the set of initialization that can successfully recover the target function shifts from zero to positive measure. These insights, derived from a simplified context, provide a perspective on the intricate yet decipherable complexities of perfect generalization in overparameterized neural networks.