Sigmoid derivative in gradient descent

Question

This is a neural network written by James Loy.

The problem is that when adjusting the weights, the old weights are added to the gradient vector and not subtracted in: self.weights1 += d_weights1

In this post it suggests that the sigmoid derivative is missing a negative sign that will be compensated.

Where is the missing and what should be the sigmoid derivative if that sign was not missing ?

Sigmoid derivative implementation:

def sigmoid(x):
    return 1.0/(1+ np.exp(-x))

def sigmoid_derivative(x):
    return x * (1.0 - x)

Entire implementation:

import numpy as np

def sigmoid(x):
    return 1.0/(1+ np.exp(-x))

def sigmoid_derivative(x):
    return x * (1.0 - x)

class NeuralNetwork:
    def __init__(self, x, y):
        self.input      = x
        self.weights1   = np.random.rand(self.input.shape[1],4) 
        self.weights2   = np.random.rand(4,1)                 
        self.y          = y
        self.output     = np.zeros(self.y.shape)

    def feedforward(self):
        self.layer1 = sigmoid(np.dot(self.input, self.weights1))
        self.output = sigmoid(np.dot(self.layer1, self.weights2))

    def backprop(self):
        # application of the chain rule to find derivative of the loss function with respect to weights2 and weights1
        d_weights2 = np.dot(self.layer1.T, (2*(self.y - self.output) * sigmoid_derivative(self.output)))
        d_weights1 = np.dot(self.input.T,  (np.dot(2*(self.y - self.output) * sigmoid_derivative(self.output), self.weights2.T) * sigmoid_derivative(self.layer1)))

        # update the weights with the derivative (slope) of the loss function
        self.weights1 += d_weights1
        self.weights2 += d_weights2


if __name__ == "__main__":
    X = np.array([[0,0,1],
                  [0,1,1],
                  [1,0,1],
                  [1,1,1]])
    y = np.array([[0],[1],[1],[0]])
    nn = NeuralNetwork(X,y)

    for i in range(1500):
        nn.feedforward()
        nn.backprop()

    print(nn.output)

score 1 · Answer 1 · answered Jul 25 '20 at 19:15

def sigmoid_derivative(x):
    return x * (1.0 - x)

Should be changed to

def sigmoid_derivative(x):
    return sigmoid(x) * (1.0 - sigmoid(x))

Hope this solves your problem. The derived equation comes from simple differentiation. You can check the derivative here : Sigmoid

Sigmoid derivative in gradient descent

1 Answers1