Problem
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.
Constraints
1 ≤ n ≤ 100
-10 ≤ each input, weight and bias ≤ 10
0 ≤ target ≤ 1
Examples
Input:
bias = 0.0
inputs = [1.0, 2.0]
target = 1.0
weights = [0.5, -0.25]
Expected output:
expected = [-0.25, -0.5, -0.25]Input:
double bias = 0.0;
std::vector<double> inputs = {1.0, 2.0};
double target = 1.0;
std::vector<double> weights = {0.5, -0.25};
Expected output:
std::vector<double> expected = {-0.25, -0.5, -0.25};Input:
double bias = 0.0;
double[] inputs = new double[] {1.0, 2.0};
double target = 1.0;
double[] weights = new double[] {0.5, -0.25};
Expected output:
double[] expected = new double[] {-0.25, -0.5, -0.25};Explanation
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].
Input:
bias = 0.0
inputs = [3.0]
target = 0.0
weights = [0.0]
Expected output:
expected = [0.75, 0.25]Input:
double bias = 0.0;
std::vector<double> inputs = {3.0};
double target = 0.0;
std::vector<double> weights = {0.0};
Expected output:
std::vector<double> expected = {0.75, 0.25};Input:
double bias = 0.0;
double[] inputs = new double[] {3.0};
double target = 0.0;
double[] weights = new double[] {0.0};
Expected output:
double[] expected = new double[] {0.75, 0.25};Explanation
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.
How to approach it
Try on your own first, then open one hint at a time.
Hint 1
Start by computing the neuron's sigmoid output and the loss derivative with respect to its pre-activation value.
Hint 2
The gradient for each weight is the same delta scaled by that input, and the bias uses the delta alone.
Hint 3
Keep the weight gradients in input order, then append the bias gradient at the end.
Expected complexity
O(n) time, O(n) extra space
References
This challenge is adapted for CodeUtility. Follow the statement and constraints above.