Условие
Backpropagation applies the chain rule layer by layer, from the output back to the inputs. The network has n inputs, h hidden neurons and one output, all with sigmoid activation (σ(z) = 1 / (1 + e^(-z))):
- hidden: a_j = σ(inputs · hidden_weights[j] + hidden_biases[j])
- output: y = σ(a · output_weights + output_bias)
- loss: L = (y - target)²
Gradients:
- delta_out = 2 × (y - target) × y × (1 - y)
- Output layer: dL/doutput_weights[j] = delta_out × a_j and dL/doutput_bias = delta_out
- delta_j = delta_out × output_weights[j] × a_j × (1 - a_j)
- Hidden layer: dL/dhidden_weights[j][i] = delta_j × inputs[i] and dL/dhidden_biases[j] = delta_j
Implement backprop_gradients(inputs, hidden_weights, hidden_biases, output_weights, output_bias, target).
Output
Return a matrix with h + 1 rows:
- Row j (for each hidden neuron j, in order): the n gradients of
hidden_weights[j], then the gradient ofhidden_biases[j]. - Last row: the h gradients of
output_weights, then the gradient ofoutput_bias.
Ограничения
1 ≤ n, h ≤ 20
-5 ≤ each input, weight and bias ≤ 5
0 ≤ target ≤ 1
Примеры
Ввод:
inputs = [1.0]
target = 1.0
output_bias = 0.0
hidden_biases = [0.0]
hidden_weights = [
[0.0]
]
output_weights = [0.0]
Ожидаемый вывод:
expected = [
[0.0, 0.0],
[-0.125, -0.25]
]Ввод:
std::vector<double> inputs = {1.0};
double target = 1.0;
double output_bias = 0.0;
std::vector<double> hidden_biases = {0.0};
std::vector<std::vector<double>> hidden_weights = {
{0.0}
};
std::vector<double> output_weights = {0.0};
Ожидаемый вывод:
std::vector<std::vector<double>> expected = {
{0.0, 0.0},
{-0.125, -0.25}
};Ввод:
double[] inputs = new double[] {1.0};
double target = 1.0;
double output_bias = 0.0;
double[] hidden_biases = new double[] {0.0};
double[][] hidden_weights = new double[][] {
new double[] {0.0}
};
double[] output_weights = new double[] {0.0};
Ожидаемый вывод:
double[][] expected = new double[][] {
new double[] {0.0, 0.0},
new double[] {-0.125, -0.25}
};Пояснение
a = σ(0) = 0.5 and y = σ(0) = 0.5. delta_out = 2 × (0.5 - 1) × 0.25 = -0.25. Output row: [-0.25 × 0.5, -0.25] = [-0.125, -0.25]. delta_1 = -0.25 × 0 × 0.25 = 0, so the hidden row is [0, 0]: a zero output weight blocks the gradient.
Ввод:
inputs = [1.0, 0.0]
target = 0.0
output_bias = -1.0
hidden_biases = [0.0, 0.0]
hidden_weights = [
[0.0, 1.0],
[0.0, -1.0]
]
output_weights = [1.0, 1.0]
Ожидаемый вывод:
expected = [
[0.0625, 0.0, 0.0625],
[0.0625, 0.0, 0.0625],
[0.125, 0.125, 0.25]
]Ввод:
std::vector<double> inputs = {1.0, 0.0};
double target = 0.0;
double output_bias = -1.0;
std::vector<double> hidden_biases = {0.0, 0.0};
std::vector<std::vector<double>> hidden_weights = {
{0.0, 1.0},
{0.0, -1.0}
};
std::vector<double> output_weights = {1.0, 1.0};
Ожидаемый вывод:
std::vector<std::vector<double>> expected = {
{0.0625, 0.0, 0.0625},
{0.0625, 0.0, 0.0625},
{0.125, 0.125, 0.25}
};Ввод:
double[] inputs = new double[] {1.0, 0.0};
double target = 0.0;
double output_bias = -1.0;
double[] hidden_biases = new double[] {0.0, 0.0};
double[][] hidden_weights = new double[][] {
new double[] {0.0, 1.0},
new double[] {0.0, -1.0}
};
double[] output_weights = new double[] {1.0, 1.0};
Ожидаемый вывод:
double[][] expected = new double[][] {
new double[] {0.0625, 0.0, 0.0625},
new double[] {0.0625, 0.0, 0.0625},
new double[] {0.125, 0.125, 0.25}
};Пояснение
Both hidden values are σ(0) = 0.5, so y = σ(0.5 + 0.5 - 1) = 0.5 and delta_out = 2 × 0.5 × 0.25 = 0.25. Output row: [0.125, 0.125, 0.25]. Each delta_j = 0.25 × 1 × 0.25 = 0.0625, giving hidden rows [0.0625, 0, 0.0625] (the second input is 0).
Как подступиться
Сначала попробуйте сами, затем открывайте подсказки по одной.
Подсказка 1
Work backward from the output neuron first, then reuse that signal to compute each hidden neuron's contribution.
Подсказка 2
You need one delta for the output layer and one delta per hidden neuron, both based on sigmoid derivatives.
Подсказка 3
Be careful to use each hidden neuron's own output weight when propagating error back to that neuron.
Ожидаемая сложность
O(hn) time, O(hn) extra space
Ссылки
Это задание адаптировано для CodeUtility. Следуйте условию и ограничениям выше.