from builtins import range
import numpy as np

def softmax_loss_naive(W, X, y, reg):
    """
    Softmax loss function, naive implementation (with loops)

    Inputs have dimension D, there are C classes, and we operate on minibatches
    of N examples.

    Inputs:
    - W: A numpy array of shape (D, C) containing weights.
    - X: A numpy array of shape (N, D) containing a minibatch of data.
    - y: A numpy array of shape (N,) containing training labels; y[i] = c means
      that X[i] has label c, where 0 <= c < C.
    - reg: (float) regularization strength

    Returns a tuple of:
    - loss as single float
    - gradient with respect to weights W; an array of same shape as W
    """
    # Initialize the loss and gradient to zero.
    loss = 0.0
    dW = np.zeros_like(W)

    batch_size = X.shape[0]
    n_features = X.shape[1]
    n_classes  = W.shape[1]

    for i in range(batch_size):

      un_normalised_probs = X[i] @ W
      loss += -un_normalised_probs[y[i]] + np.log(np.sum(np.exp(un_normalised_probs))) 

      # Probabilities with the "log-sum-exp trick"
      un_normalised_probs -= np.max(un_normalised_probs)
      y_hat = np.exp(un_normalised_probs)/np.sum(np.exp(un_normalised_probs))
      
      for k in range(n_classes):
      	# Here we implement the gradient we derived above 
        dW[:, k] += X[i] * (y_hat[k] - (k == y[i]))
   
    loss /= batch_size 
    dW /= batch_size

    loss += reg * np.sum(W*W)   
    dW +=  reg * 2 * W

    return loss, dW

for k in range(n_classes):
	# Here we implement the gradient we derived above 
	dW[:, k] += X[i] * (y_hat[k] - (k == y[i]))

un_normalised_probs = X @ W
losses = np.log(np.sum(np.exp(un_normalised_probs), axis=1 , keepdims=True)) - np.take_along_axis(un_normalised_probs, np.reshape(y, (-1, 1)), axis=1)
loss   += np.sum(losses)/batch_size + reg *  np.sum(W*W) 

un_normalised_probs -=  np.amax(un_normalised_probs, axis=1, keepdims=True)
y_hats = np.exp(un_normalised_probs) / np.sum(np.exp(un_normalised_probs), axis=1, keepdims=True)
y_hats[np.arange(y.shape[0]), y] -= 1
dW = (X.T @ y_hats)/batch_size + reg * 2 * W