When the Left Foot Leads to the Right Path: Bridging Initial Prejudice and Trainability

Abstract: Understanding the statistical properties of deep neural networks (DNNs) at initialization is crucial for elucidating both their trainability and the intrinsic architectural biases they encode prior to data exposure. Mean-field (MF) analyses have demonstrated that the parameter distribution in randomly initialized networks dictates whether gradients vanish or explode. Concurrently, untrained DNNs were found to exhibit an initial-guessing bias (IGB), in which large regions of the input space are assigned to a single class. In this work, we derive a theoretical proof establishing the correspondence between IGB and previous MF theories, thereby connecting a network prejudice toward specific classes with the conditions for fast and accurate learning. This connection yields the counter-intuitive conclusion: the initialization that optimizes trainability is necessarily biased, rather than neutral. Furthermore, we extend the MF/IGB framework to multi-node activation functions, offering practical guidelines for designing initialization schemes that ensure stable optimization in architectures employing max- and average-pooling layers.