Backpropagating bias in a neural network

Question

Following Andrew Traks's example, I want to implement a 3 layer neural network - 1 input, 1 hidden, 1 output - with a simple dropout, for binary classification.

If I include bias terms b1 and b2, then I would need to slightly modify Andrew's code as below.

X = np.array([ [0,0,1],[0,1,1],[1,0,1],[1,1,1] ])
y = np.array([[0,1,1,0]]).T
alpha,hidden_dim,dropout_percent = (0.5,4,0.2)
synapse_0 = 2*np.random.random((X.shape[1],hidden_dim)) - 1
synapse_1 = 2*np.random.random((hidden_dim,1)) - 1
b1 = np.zeros(hidden_dim)
b2 = np.zeros(1)
for j in range(60000):
    # sigmoid activation function
    layer_1 = (1/(1+np.exp(-(np.dot(X,synapse_0) + b1))))
    # dropout
    layer_1 *= np.random.binomial([np.ones((len(X),hidden_dim))],1-dropout_percent)[0] * (1.0/(1-dropout_percent))
    layer_2 = 1/(1+np.exp(-(np.dot(layer_1,synapse_1) + b2)))
    # sigmoid derivative = s(x)(1-s(x))
    layer_2_delta = (layer_2 - y)*(layer_2*(1-layer_2))
    layer_1_delta = layer_2_delta.dot(synapse_1.T) * (layer_1 * (1-layer_1))
    synapse_1 -= (alpha * layer_1.T.dot(layer_2_delta))
    synapse_0 -= (alpha * X.T.dot(layer_1_delta))
    b1 -= alpha*layer_1_delta
    b2 -= alpha*layer_2_delta

The problem is, of course, with the code above the dimensions of b1 dont match with the dimensions of layer_1_delta, similarly with b2 and layer_2_delta.

I don't understand how the delta is calculated to update b1 and b2 - according to Michael Nielsen's example, b1 and b2 should be updated by a delta which in my code I believe to be layer_1_delta and layer_2_delta respectively.

What am I doing wrong here? Have I messed up the dimensionality of the deltas or of the biases? I feel it is the latter, because if I remove the biases from this code it works fine. Thanks in advance

Answer 1

So first I would change X in bX to 0 and 1 to correspond to synapse_X, because this is where they belong and it makes it:

b1 -= alpha * 1.0 / m * np.sum(layer_2_delta)
b0 -= alpha * 1.0 / m * np.sum(layer_1_delta)

Where m is the number of examples in the training set. Also, the drop rate is stupidly high and actually hurts convergence. So in all considered the whole code:

import numpy as np

X = np.array([ [0,0,1],[0,1,1],[1,0,1],[1,1,1] ])
m = X.shape[0]
y = np.array([[0,1,1,0]]).T
alpha,hidden_dim,dropout_percent = (0.5,4,0.02)
synapse_0 = 2*np.random.random((X.shape[1],hidden_dim)) - 1
synapse_1 = 2*np.random.random((hidden_dim,1)) - 1
b0 = np.zeros(hidden_dim)
b1 = np.zeros(1)
for j in range(10000):
    # sigmoid activation function
    layer_1 = (1/(1+np.exp(-(np.dot(X,synapse_0) + b0))))
    # dropout
    layer_1 *= np.random.binomial([np.ones((len(X),hidden_dim))],1-dropout_percent)[0] * (1.0/(1-dropout_percent))
    layer_2 = 1/(1+np.exp(-(np.dot(layer_1,synapse_1) + b1)))
    # sigmoid derivative = s(x)(1-s(x))
    layer_2_delta = (layer_2 - y)*(layer_2*(1-layer_2))
    layer_1_delta = layer_2_delta.dot(synapse_1.T) * (layer_1 * (1-layer_1))
    synapse_1 -= (alpha * layer_1.T.dot(layer_2_delta))
    synapse_0 -= (alpha * X.T.dot(layer_1_delta))
    b1 -= alpha * 1.0 / m * np.sum(layer_2_delta)
    b0 -= alpha * 1.0 / m * np.sum(layer_1_delta)

print layer_2