Online Natural Gradient as a Kalman Filter
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We establish a full relationship between Kalman filtering and Amari's natural gradient in statistical learning. Namely, using an online natural gradient descent on data log-likelihood to evaluate the parameter of a probabilistic model from a series of observations, is exactly equivalent to using an extended Kalman filter to estimate the parameter (assumed to have constant dynamics). 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 settings for natural gradient hyperparameters such as learning rates or initialization and regularization of the Fisher information matrix.
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