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Declarative Machine Learning
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#-------------------------------------------------------------
/*
* Stochastic Gradient Descent with Nesterov momentum (SGD-Nesterov) optimizer.
*/
update = function(matrix[double] X, matrix[double] dX, double lr, double mu, matrix[double] v)
return (matrix[double] X, matrix[double] v) {
/*
* Performs an SGD update with Nesterov momentum.
*
* As with regular SGD with momentum, in SGD with Nesterov momentum,
* we assume that the parameters have a velocity that continues
* with some momentum, and that is influenced by the gradient.
* In this view specifically, we perform the position update from the
* position that the momentum is about to carry the parameters to,
* rather than from the previous position. Additionally, we always
* store the parameters in their position after momentum.
*
* Reference:
* - Advances in optimizing Recurrent Networks, Bengio et al.,
* section 3.5.
* - http://arxiv.org/abs/1212.0901
*
* Inputs:
* - X: Parameters to update, of shape (any, any).
* - dX: Gradient wrt `X` of a loss function being optimized, of
* same shape as `X`.
* - lr: Learning rate.
* - mu: Momentum value.
* Typical values are in the range of [0.5, 0.99], usually
* started at the lower end and annealed towards the higher end.
* - v: State maintaining the velocity of the parameters `X`, of same
* shape as `X`.
*
* Outputs:
* - X: Updated parameters X, of same shape as input X.
* - v: Updated velocity of the parameters X, of same shape as
* input v.
*/
v_prev = v
v = mu*v - lr*dX # update velocity
X = X - mu*v_prev + (1+mu)*v # update position, including momentum
}
init = function(matrix[double] X)
return (matrix[double] v) {
/*
* Initialize the state for this optimizer.
*
* Note: This is just a convenience function, and state
* may be initialized manually if needed.
*
* Inputs:
* - X: Parameters to update, of shape (any, any).
*
* Outputs:
* - v: Initial velocity of the parameters `X`.
*/
v = matrix(0, rows=nrow(X), cols=ncol(X))
}