On the Variance of the Fisher Information for Deep Learning
This work addresses a theoretical bottleneck for researchers in optimization and geometric learning, but it is incremental as it builds on existing FIM representations without introducing new methods.
The paper tackled the problem of estimating the Fisher information matrix (FIM) in deep learning, which is computationally expensive, by analyzing the variance of two unbiased estimators and deriving closed-form expressions and upper bounds for this variance.
In the realm of deep learning, the Fisher information matrix (FIM) gives novel insights and useful tools to characterize the loss landscape, perform second-order optimization, and build geometric learning theories. 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. Their estimation quality is naturally gauged by their variance given in closed form. We analyze how the parametric structure of a deep neural network can affect the variance. The meaning of this variance measure and its upper bounds are then discussed in the context of deep learning.