1. Logistic Loss Function

• The squared cost function for logistic regression shows many local minima, which is not ideal for gradient descent.

• The upper branch log function works for y = 1, because the cost decreases exponentially when expected value gets closer to 1.
• Closer to 0, cost goes to infinity.
  
• The opposite for y = 0.

2. Simplified Cost Function

• either one of the two terms gets eliminated because y is only either 1 or 0.

• average of all the losses