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Declarative Machine Learning
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/*
 * Softmax classifier layer.
 */

forward = function(matrix[double] scores)
    return (matrix[double] probs) {
  /*
   * Computes the forward pass for a softmax classifier.  The input
   * has N examples, each with D values that are interpreted as
   * unnormalized, log-probabilities for each of D classes.  The softmax
   * function transforms these values to normalized probabilities across
   * the D classes, for every example.
   *
   * This can be interpreted as a generalization of the sigmoid
   * function to multiple classes.
   *
   *   `probs_ij = e^scores_ij / sum(e^scores_i)`
   *
   * Inputs:
   *  - scores: Inputs, of shape (N, D).
   *
   * Outputs:
   *  - probs: Outputs, of shape (N, D).
   */
  # For numerical stability, we subtract the max score of an example from all scores for that
  # example.  This is equivalent to the original formulation:
  # e^scores_i / sum(e^scores_i) == C*e^scores_i / C*sum(e^scores_i)
  #                              == e^(scores_i+log(C)) / sum(e^(scores_i+log(C))
  # set log(C) = -max(scores_i):
  #                              == e^(scores_i-max(scores_i)) / sum(e^(scores_i-max(scores_i))
  scores = scores - rowMaxs(scores)  # numerical stability
  unnorm_probs = exp(scores)  # unnormalized probabilities
  probs = unnorm_probs / rowSums(unnorm_probs)  # normalized probabilities
}

backward = function(matrix[double] dprobs, matrix[double] scores)
    return (matrix[double] dscores) {
  /*
   * Computes the backward pass for a softmax classifier.
   *
   * Note that dscores_ij has multiple source branches:
   *
   *   ```
   *   dprobs_ij/dscores_ij = probs_ij * (1 - probs_ij)
   *   dprobs_ik/dscores_ij = -probs_ik * probs_ij, for all k != j
   *
   *   dloss/dscores_ij =
   *      (dloss/dprobs_ij * dprobs_ij/dscores_ij)
   *      + sum_{k!=j}(dloss/dprobs_ik * dprobs_ik/dscores_ij)
   *   ```
   *
   * Inputs:
   *  - dprobs: Gradient wrt `probs` from upstream, of shape (N, D).
   *  - scores: Inputs, of shape (N, D).
   *
   * Outputs:
   *  - dscores: Gradient wrt `scores`, of shape (N, D).
   */
  scores = scores - rowMaxs(scores)  # numerical stability
  unnorm_probs = exp(scores)  # unnormalized probabilities
  probs = unnorm_probs / rowSums(unnorm_probs)  # normalized probabilities
  # After some cancellation:
  # dscores = dprobs*probs - probs*rowSums(dprobs*probs)
  dtemp = dprobs * probs
  dscores = dtemp - probs*rowSums(dtemp)
}