On the Variance of the Fisher Information for Deep Learning

The Fisher information matrix (FIM) has been applied to the realm of deep learning. It is closely related to the loss landscape, the variance of the parameters, second order optimization, and deep learning theory. However, the exact FIM is either unavailable in closed form or too expensive to compute. In practice, it is almost always estimated based on empirical samples. We investigate two such estimators based on two equivalent representations of the FIM -- both unbiased and consistent with respect to the underlying "true" FIM. Their estimation quality is characterized by their variance given in closed form. We bound their variances and analyze how the parametric structure of a deep neural network can impact the variance. We discuss the meaning of this variance measure and our bounds in the context of deep learning.