Online Natural Gradient as a Kalman Filter
This work offers a new interpretation of natural gradient hyperparameters, potentially aiding in tuning for machine learning practitioners, but it is incremental as it reframes existing methods.
The paper shows that Amari's natural gradient in statistical learning is equivalent to Kalman filtering, providing exact algebraic correspondences for parameter estimation in both i.i.d. and recurrent cases.
We cast Amari's natural gradient in statistical learning as a specific case of Kalman filtering. Namely, applying an extended Kalman filter to estimate a fixed unknown parameter of a probabilistic model from a series of observations, is rigorously equivalent to estimating this parameter via an online stochastic natural gradient descent on the log-likelihood of the observations. In the i.i.d. case, this relation is a consequence of the "information filter" phrasing of the extended Kalman filter. In the recurrent (state space, non-i.i.d.) case, we prove that the joint Kalman filter over states and parameters is a natural gradient on top of real-time recurrent learning (RTRL), a classical algorithm to train recurrent models. This exact algebraic correspondence provides relevant interpretations for natural gradient hyperparameters such as learning rates or initialization and regularization of the Fisher information matrix.