Report

Natural Gradient Works Efficiently in Learning S Amari 11.03.18.(Fri) Computational Modeling of Intelligence Summarized by Joon Shik Kim Abstract • The ordinary gradient of a function does not represent its steepest direction, but the natural gradient does. • The dynamical behavior of natural gradient online learning is analyzed and is proved to be Fisher efficient. • The plateau phenomenon, which appears in the backpropagation learning algorithm of multilayer perceptron, might disappear or might not be so serious when the natural gradient is used. Introduction (1/2) • The stochastic gradient method is a popular learning method in the general nonlinear optimization framework. • The parameter space is not Euclidean but has a Riemannian metric structure in many cases. • In these cases, the ordinary gradient does not give the steepest direction of target function. Introduction (2/2) • Barkai, Seung, and Sompolisky (1995) proposed an adaptive method of adjusting the learning rate. We generalize their idea and evaluate its performance based on the Riemannian metric of errors. Natural Gradient (1/5) • The squared length of a small incremental vector dw, • When the coordinate system is nonorthogonal, the squared length is given by the quadratic form, Natural Gradient (2/5) • The steepest descent direction of a function L(w) at w is defined by the vector dw has that minimizes L(w+dw) where |dw| has a fixed length, that is, under the constant, Natural Gradient (3/5) • The steepest descent direction of L(w) in a Riemannian space is given by, Natural Gradient (4/5) Natural Gradient (5/5) Natural Gradient Learning • Risk function or average loss, • Learning is a procedure to search for the optimal w* that minimizes L(w). • Stochastic gradient descent learning Statistical Estimation of Probability Density Function (1/2) • In the case of statistical estimation, we assume a statistical model {p(z,w)}, and the problem is to obtain the probability distribution p ( z , wˆ ) that approximates the unknown density function q(z) in the best way. • Loss function is Statistical Estimation of Probability Density Function (2/2) • The expected loss is then given by Hz is the entropy of q(z) not depending on w. • Riemannian metric is Fisher information Fisher Information as the Metric of Kullback-Leibler Divergence (1/2) • p=q(θ+h) D ( q || p ) q q ln d p q ln p d q 2 p 1 p q 1 1 d 2 q q lim h D ( q ( ) || q ( h )) h 2 1 1 q 2 q lim h 2 ( p q )( p q ) d 1 q ( h ) q ( ) q ( h ) q ( ) q q 2 h h d Fisher Information as the Metric of Kullback-Leibler Divergence (2/2) lim h D ( q ( ) || q ( h )) h 2 lim h 1 2 2 q 2 q q 1 1 2 q I 1 q ( h ) q ( ) q ( h ) q ( ) 1 2 h 1 q q q 2 d ln q ln q d I: Fisher information h d Multilayer Neural Network (1/2) Multilayer Neural Network (2/2) c is a normalizing constant Natural Gradient Gives FisherEfficient Online Learning Algorithms (1/4) • DT = {(x1,y1),…,(xT,yT)} is T-independent input-output examples generated by the teacher network having parameter w*. • Minimizing the log loss over the training data DT is to obtain wˆ T that minimizes the training error Natural Gradient Gives FisherEfficient Online Learning Algorithms (2/4) • The Cramér-Rao theorem states that the expected squared error of an unbiased estimator satisfies E [( wˆ T w *) ( wˆ T w *)] T 1 I • An estimator is said to be efficient or Fisher efficient when it satisfies above equation. Natural Gradient Gives FisherEfficient Online Learning Algorithms (3/4) • Theorem 2. The natural gradient online estimator is Fisher efficient. • Proof Natural Gradient Gives FisherEfficient Online Learning Algorithms (4/4)