Electronic Journal of Statistics

Online natural gradient as a Kalman filter

Yann Ollivier

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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.

Article information

Electron. J. Statist., Volume 12, Number 2 (2018), 2930-2961.

Received: June 2017
First available in Project Euclid: 18 September 2018

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Digital Object Identifier

Primary: 68T05: Learning and adaptive systems [See also 68Q32, 91E40] 65K10: Optimization and variational techniques [See also 49Mxx, 93B40]
Secondary: 93E35: Stochastic learning and adaptive control 90C26: Nonconvex programming, global optimization 93E11: Filtering [See also 60G35] 49M15: Newton-type methods

Statistical learning natural gradient Kalman filter stochastic gradient descent

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Ollivier, Yann. Online natural gradient as a Kalman filter. Electron. J. Statist. 12 (2018), no. 2, 2930--2961. doi:10.1214/18-EJS1468. https://projecteuclid.org/euclid.ejs/1537257630

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