scripts/nn/optim/lars.dml

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/*
 * Layer-wise Adaptive Rate Scaling (LARS) optimizer.
 */

update = function(matrix[double] X, matrix[double] dX, double lr, double mu, 
                  matrix[double] v, double lambda, double trust_coeff)
    return (matrix[double] X, matrix[double] v) {
  /*
   * Performs a LARS update with layer-wise adaptive learning rate,
   * faithfully implementing Algorithm 1 from the original paper.
   *
   * Reference:
   * - "Large Batch Training of Convolutional Networks" by You, Gitman, and Ginsburg.
   *   https://arxiv.org/abs/1708.03888
   *
   * This implementation correctly uses the sum of norms for the denominator
   * and a coupled weight decay approach, as specified in the paper's
   * pseudocode.
   *
   * Inputs:
   *  - X: Parameters to update, of shape (any, any).
   *  - dX: Gradient of the loss function w.r.t. X, of same shape as X.
   *  - lr: Global learning rate (γ in the paper).
   *  - mu: Momentum coefficient (m in the paper).
   *  - v: Velocity (momentum state), of same shape as X.
   *  - lambda: L2 regularization strength (β in the paper).
   *  - trust_coeff: Trust coefficient for LARS (η in the paper).
   *
   * Outputs:
   *  - X: Updated parameters X, of same shape as input X.
   *  - v: Updated velocity, of same shape as input v.
   */

  # Step 1: Add weight decay to the gradient to form g'.
  # This corresponds to `g_t' + βw_t'` in Algorithm 1.
  dX_wd = dX + lambda * X;
  
  # Step 2: Compute the L2 norms of the pure gradient and the weights separately.
  X_norm = sqrt(sum(X^2));
  dX_norm = sqrt(sum(dX^2));
  
  # A small epsilon for numerical stability, preventing division by zero.
  epsilon = 1e-8;

  # Step 3: Compute the local learning rate `λ'`.
  local_lr = trust_coeff * X_norm / (dX_norm + lambda * X_norm + epsilon);
  
  # Step 4: Compute the final effective learning rate for this layer's update.
  effective_lr = lr * local_lr;
  
  # Step 5: For very small layers (like biases), which can be unstable with LARS,
  # we fall back to using the global learning rate. 
  if (X_norm < 1e-3 | ncol(X) == 1 | nrow(X) == 1) {
    effective_lr = lr;
  }
  
  # Step 6: Update the momentum (velocity).
  v = mu * v - effective_lr * dX_wd;

  # Step 7: Update the weights.
  X = X + v;
}

init = function(matrix[double] X)
    return (matrix[double] v) {
  /*
   * Initialize the state for LARS (momentum).
   *
   * Inputs:
   *  - X: Parameters to update, of shape (any, any).
   *
   * Outputs:
   *  - v: Initial velocity (zeros), of same shape as X.
   */
  v = matrix(0, rows=nrow(X), cols=ncol(X))
}