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Declarative Machine Learning
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/*
 * Stochastic Gradient Descent with momentum (SGD-momentum) 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 momentum.
   *
   * In SGD with momentum, we assume that the parameters have a velocity
   * that continues with some momentum, and that is influenced by the
   * gradient.
   *
   * 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 `X`.
   */
  v = mu*v - lr*dX  # update velocity
  X = X + v  # update position
}

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))
}