Problema
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.
Restricciones
1 ≤ n ≤ 100
-10 ≤ each input, weight and bias ≤ 10
0 ≤ target ≤ 1
Ejemplos
Entrada:
bias = 0.0
inputs = [1.0, 2.0]
target = 1.0
weights = [0.5, -0.25]
Salida esperada:
expected = [-0.25, -0.5, -0.25]Entrada:
double bias = 0.0;
std::vector<double> inputs = {1.0, 2.0};
double target = 1.0;
std::vector<double> weights = {0.5, -0.25};
Salida esperada:
std::vector<double> expected = {-0.25, -0.5, -0.25};Entrada:
double bias = 0.0;
double[] inputs = new double[] {1.0, 2.0};
double target = 1.0;
double[] weights = new double[] {0.5, -0.25};
Salida esperada:
double[] expected = new double[] {-0.25, -0.5, -0.25};Explicación
z = 1×0.5 + 2×(-0.25) = 0, so y = 0.5. delta = 2 × (0.5 - 1) × 0.5 × 0.5 = -0.25. Gradients: [-0.25×1, -0.25×2, -0.25] = [-0.25, -0.5, -0.25].
Entrada:
bias = 0.0
inputs = [3.0]
target = 0.0
weights = [0.0]
Salida esperada:
expected = [0.75, 0.25]Entrada:
double bias = 0.0;
std::vector<double> inputs = {3.0};
double target = 0.0;
std::vector<double> weights = {0.0};
Salida esperada:
std::vector<double> expected = {0.75, 0.25};Entrada:
double bias = 0.0;
double[] inputs = new double[] {3.0};
double target = 0.0;
double[] weights = new double[] {0.0};
Salida esperada:
double[] expected = new double[] {0.75, 0.25};Explicación
z = 0 and y = 0.5. delta = 2 × (0.5 - 0) × 0.5 × 0.5 = 0.25, so the gradients are [0.25×3, 0.25] = [0.75, 0.25]. Positive gradients mean the weight should go down.
Cómo abordarlo
Inténtalo primero por tu cuenta y luego abre las pistas una a una.
Pista 1
Start by computing the neuron's sigmoid output and the loss derivative with respect to its pre-activation value.
Pista 2
The gradient for each weight is the same delta scaled by that input, and the bias uses the delta alone.
Pista 3
Keep the weight gradients in input order, then append the bias gradient at the end.
Complejidad esperada
O(n) time, O(n) extra space
Referencias
Este reto está adaptado para CodeUtility. Sigue el enunciado y las restricciones indicadas arriba.