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Declarative Machine Learning
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/*
 * Log loss function.
 */

forward = function(matrix[double] pred, matrix[double] y)
    return (double loss) {
  /*
   * Computes the forward pass for a log loss function.
   *
   *   ```
   *   L_i = -y_i*log(pred_i) - (1-y_i)*log(1-pred_i)
   *   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 binary target, `pred_i` is probability
   * of the true class (i.e. `y=1`), and `N` is the number of examples.
   *
   * This can be interpreted as the negative log-likelihood assuming
   * a Bernoulli distribution.
   *
   * Inputs:
   *  - pred: Predictions, of shape (N, 1).
   *      Predictions should be probabilities of the true
   *      class (i.e. probability of `y=1`).
   *  - y: Targets, of shape (N, 1).
   *      Targets should be binary in the set {0, 1}.
   *
   * Outputs:
   *  - loss: Average loss.
   */
  N = nrow(y)
  losses = -y*log(pred) - (1-y)*log(1-pred)
  loss = sum(losses) / N
}

backward = function(matrix[double] pred, matrix[double] y)
    return (matrix[double] dpred) {
  /*
   * Computes the backward pass for a log loss function.
   *
   * Inputs:
   *  - pred: Predictions, of shape (N, 1).
   *      Predictions should be probabilities of the true
   *      class (i.e. probability of `y=1`).
   *  - y: Targets, of shape (N, 1).
   *      Targets should be binary in the set {0, 1}.
   *
   * Outputs:
   *  - dpred: Gradient wrt `pred`, of shape (N, 1).
   */
  N = nrow(y)
  dpred = (1/N) * (pred-y) / (pred*(1-pred))
}