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.
Einschränkungen
1 ≤ n ≤ 100
-10 ≤ each input, weight and bias ≤ 10
0 ≤ target ≤ 1
Beispiele
Eingabe:
bias = 0.0
inputs = [1.0, 2.0]
target = 1.0
weights = [0.5, -0.25]
Erwartete Ausgabe:
expected = [-0.25, -0.5, -0.25]Eingabe:
double bias = 0.0;
std::vector<double> inputs = {1.0, 2.0};
double target = 1.0;
std::vector<double> weights = {0.5, -0.25};
Erwartete Ausgabe:
std::vector<double> expected = {-0.25, -0.5, -0.25};Eingabe:
double bias = 0.0;
double[] inputs = new double[] {1.0, 2.0};
double target = 1.0;
double[] weights = new double[] {0.5, -0.25};
Erwartete Ausgabe:
double[] expected = new double[] {-0.25, -0.5, -0.25};Erläuterung
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].
Eingabe:
bias = 0.0
inputs = [3.0]
target = 0.0
weights = [0.0]
Erwartete Ausgabe:
expected = [0.75, 0.25]Eingabe:
double bias = 0.0;
std::vector<double> inputs = {3.0};
double target = 0.0;
std::vector<double> weights = {0.0};
Erwartete Ausgabe:
std::vector<double> expected = {0.75, 0.25};Eingabe:
double bias = 0.0;
double[] inputs = new double[] {3.0};
double target = 0.0;
double[] weights = new double[] {0.0};
Erwartete Ausgabe:
double[] expected = new double[] {0.75, 0.25};Erläuterung
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.
So gehst du vor
Versuch es zuerst allein und öffne dann einen Hinweis nach dem anderen.
Hinweis 1
Start by computing the neuron's sigmoid output and the loss derivative with respect to its pre-activation value.
Hinweis 2
The gradient for each weight is the same delta scaled by that input, and the bias uses the delta alone.
Hinweis 3
Keep the weight gradients in input order, then append the bias gradient at the end.
Erwartete Komplexität
O(n) time, O(n) extra space
Referenzen
Diese Aufgabe wurde für CodeUtility angepasst. Beachte die oben genannten Vorgaben und Einschränkungen.