Gradients of a Single Neuron

To train a neuron we need to know how each weight affects the error. For one training example:

• z = inputs · weights + bias
• y = sigmoid(z) = 1 / (1 + e^(-z))
• loss L = (y - target)²

By the chain rule, with delta = dL/dz = 2 × (y - target) × y × (1 - y):

• dL/dw_i = delta × x_i
• dL/dbias = delta

Implement neuron_gradients(inputs, weights, bias, target).

Input

inputs, weights: Lists of the same length n.

bias: The neuron's bias.

target: The correct output for this example (between 0 and 1).

Output

Return a list of n + 1 numbers: the gradient for each weight in order, then the gradient for the bias.

Contraintes

1 ≤ n ≤ 100

-10 ≤ each input, weight and bias ≤ 10

0 ≤ target ≤ 1

Exemples

Entrée :
bias = 0.0
inputs = [1.0, 2.0]
target = 1.0
weights = [0.5, -0.25]

Sortie attendue :
expected = [-0.25, -0.5, -0.25]

Entrée :
double bias = 0.0;
std::vector<double> inputs = {1.0, 2.0};
double target = 1.0;
std::vector<double> weights = {0.5, -0.25};

Sortie attendue :
std::vector<double> expected = {-0.25, -0.5, -0.25};

Entrée :
double bias = 0.0;
double[] inputs = new double[] {1.0, 2.0};
double target = 1.0;
double[] weights = new double[] {0.5, -0.25};

Sortie attendue :
double[] expected = new double[] {-0.25, -0.5, -0.25};

Entrée :
bias = 0.0
inputs = [3.0]
target = 0.0
weights = [0.0]

Sortie attendue :
expected = [0.75, 0.25]

Entrée :
double bias = 0.0;
std::vector<double> inputs = {3.0};
double target = 0.0;
std::vector<double> weights = {0.0};

Sortie attendue :
std::vector<double> expected = {0.75, 0.25};

Entrée :
double bias = 0.0;
double[] inputs = new double[] {3.0};
double target = 0.0;
double[] weights = new double[] {0.0};

Sortie attendue :
double[] expected = new double[] {0.75, 0.25};

Références

Ce défi a été adapté pour CodeUtility. Suivez l'énoncé et les contraintes ci-dessus.

• Michael Nielsen, How the backpropagation algorithm works
• Goodfellow, Bengio and Courville, Deep Learning