Fishers for Free? Approximating the Fisher Information Matrix by Recycling the Squared Gradient Accumulator

The diagonal of a model’s Fisher Information Matrix (the "Fisher") has frequently been used as a way to measure parameter sensitivity. Typically, the Fisher is estimated by computing the squared gradient of the model’s outputs with respect to its parameters, averaged over a few hundred or thousand examples — a process which incurs nontrivial computational costs. At the same time, adaptive gradient methods like the ubiquitous Adam optimizer compute a moving average of the squared gradient over the course of training. This paper therefore explores whether an approximation of the Fisher can be obtained "for free" by recycling the squared gradient accumulator that has already been computed over the course of training. Through a comprehensive set of experiments covering five applications of the Fisher, we demonstrate that the "Squisher" (Squared gradient accumulator as an approximation of the Fisher) consistently performs similarly to the Fisher while outperforming baseline methods. Additionally, we clarify the exact differences between the Squisher and the Fisher and provide empirical quantification of their respective impact.