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.
Vincoli
1 ≤ n ≤ 100
-10 ≤ each input, weight and bias ≤ 10
0 ≤ target ≤ 1
Esempi
Input:
bias = 0.0
inputs = [1.0, 2.0]
target = 1.0
weights = [0.5, -0.25]
Output atteso:
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};
Output atteso:
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};
Output atteso:
double[] expected = new double[] {-0.25, -0.5, -0.25};Spiegazione
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]
Output atteso:
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};
Output atteso:
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};
Output atteso:
double[] expected = new double[] {0.75, 0.25};Spiegazione
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.
Come affrontarlo
Prova prima da solo, poi apri un suggerimento alla volta.
Suggerimento 1
Start by computing the neuron's sigmoid output and the loss derivative with respect to its pre-activation value.
Suggerimento 2
The gradient for each weight is the same delta scaled by that input, and the bias uses the delta alone.
Suggerimento 3
Keep the weight gradients in input order, then append the bias gradient at the end.
Complessità attesa
O(n) time, O(n) extra space
Riferimenti
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