Weak Correlations as the Underlying Principle for Linearization of Gradient-Based Learning Systems

Deep learning models, such as wide neural networks, can be viewed as nonlinear dynamical systems composed of numerous interacting degrees of freedom. When such systems approach the limit of infinite number of degrees of freedom, their dynamics tend to simplify. This paper investigates gradient descent-based learning algorithms that exhibit linearization in their parameters. We establish that this apparent linearity, arises from weak correlations between the first, and higher-order derivatives of the hypothesis function with respect to the parameters, at initialization. Our findings indicate that these weak correlations fundamentally underpin the observed linearization phenomenon of wide neural networks. Leveraging this connection, we derive bounds on the deviation from linearity during stochastic gradient descent training. To support our analysis, we introduce a novel technique for characterizing the asymptotic behavior of random tensors. We validate our theoretical insights through empirical studies, comparing the linearized dynamics to the observed correlations.