def sqerror_gradients(network: List[List[Vector]],
input_vector: Vector,
target_vector: Vector) -> List[List[Vector]]:
"""
Given a neural network, an input vector, and a target vector,
make a prediction and compute the gradient of the squared error
loss with respect to the neuron weights.
"""
# forward pass
hidden_outputs, outputs = feed_forward(network, input_vector)
# gradients with respect to output neuron pre-activation outputs
output_deltas = [output * (1 - output) * (output - target)
for output, target in zip(outputs, target_vector)]
# gradients with respect to output neuron weights
output_grads = [[output_deltas[i] * hidden_output
for hidden_output in hidden_outputs + [1]]
for i, output_neuron in enumerate(network[-1])]
# gradients with respect to hidden neuron pre-activation outputs
hidden_deltas = [hidden_output * (1 - hidden_output) *
dot(output_deltas, [n[i] for n in network[-1]])
for i, hidden_output in enumerate(hidden_outputs)]
# gradients with respect to hidden neuron weights
hidden_grads = [[hidden_deltas[i] * input for input in input_vector + [1]]
for i, hidden_neuron in enumerate(network[0])]
return [hidden_grads, output_grads]
def neuron_output(weights: Vector, inputs: Vector) -> float:
'''weights includes the bias term, inputs includes a 1'''
return sigmoid(dot(weights, inputs))
def ridge_penalty(beta: Vector, alpha: float) -> float:
return alpha * dot(beta[1:], beta[1:])
def predict(x: Vector, beta: Vector) -> float:
'''assumes that the first element of x is 1'''
return dot(x, beta)