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/*
 * Adagrad optimizer.
 */

update = function(matrix[double] X, matrix[double] dX, double lr, double epsilon,
                  matrix[double] cache)
    return (matrix[double] X, matrix[double] cache) {
  /*
   * Performs an Adagrad update.
   *
   * This is an adaptive learning rate optimizer that maintains the
   * sum of squared gradients to automatically adjust the effective
   * learning rate.
   *
   * Reference:
   *  - Adaptive Subgradient Methods for Online Learning and Stochastic
   *    Optimization, Duchi et al.
   *      - http://jmlr.org/papers/v12/duchi11a.html
   *
   * 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.
   *  - epsilon: Smoothing term to avoid divide by zero errors.
   *      Typical values are in the range of [1e-8, 1e-4].
   *  - cache: State that maintains per-parameter sum of squared
   *      gradients, of same shape as `X`.
   *
   * Outputs:
   *  - X: Updated parameters `X`, of same shape as input `X`.
   *  - cache: State that maintains per-parameter sum of squared
   *      gradients, of same shape as `X`.
   */
  cache = cache + dX^2
  X = X - (lr * dX / (sqrt(cache)+epsilon))
}

init = function(matrix[double] X)
    return (matrix[double] cache) {
  /*
   * 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:
   *  - cache: State that maintains per-parameter sum of squared
   *      gradients, of same shape as `X`.
   */
  cache = matrix(0, rows=nrow(X), cols=ncol(X))
}