scripts/nn/layers/l2_loss.dml

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/*
 * L2 loss function.
 */

forward = function(matrix[double] pred, matrix[double] y)
    return (double loss) {
  /*
   * Computes the forward pass for an L2 loss function.  The inputs
   * consist of N examples, each with M dimensions to predict.
   *
   *   ```
   *   L_i = (1/2) norm(pred_i - y_i)^2
   *   L = (1/N) sum(L_i) for i=1 to N
   *   ```
   *
   * In these equations, `L` is the total loss, `L_i` is the loss for
   * example `i`, `y_i` is the scalar target, `pred_i` is the scalar
   * prediction, and `N` is the number of examples.
   *
   * This can be interpreted as the negative log-likelihood assuming
   * a Gaussian distribution.
   *
   * Inputs:
   *  - pred: Predictions, of shape (N, M).
   *  - y: Targets, of shape (N, M).
   *
   * Outputs:
   *  - loss: Average loss.
   */
  N = nrow(y)
  losses = 0.5 * rowSums((pred-y)^2)
  loss = sum(losses) / N
}

backward = function(matrix[double] pred, matrix[double] y)
    return (matrix[double] dpred) {
  /*
   * Computes the backward pass for an L2 loss function.  The inputs
   * consist of N examples, each with M dimensions to predict.
   *
   * Inputs:
   *  - pred: Predictions, of shape (N, M).
   *  - y: Targets, of shape (N, M).
   *
   * Outputs:
   *  - dpred: Gradient wrt `pred`, of shape (N, M).
   */
  N = nrow(y)
  dpred = (pred-y) / N
}