समस्या
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.
सीमाएँ
1 ≤ n, h ≤ 20
-5 ≤ each input, weight and bias ≤ 5
0 ≤ target ≤ 1
उदाहरण
इनपुट:
inputs = [1.0]
target = 1.0
output_bias = 0.0
hidden_biases = [0.0]
hidden_weights = [
[0.0]
]
output_weights = [0.0]
अपेक्षित आउटपुट:
expected = [
[0.0, 0.0],
[-0.125, -0.25]
]इनपुट:
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};
अपेक्षित आउटपुट:
std::vector<std::vector<double>> expected = {
{0.0, 0.0},
{-0.125, -0.25}
};इनपुट:
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};
अपेक्षित आउटपुट:
double[][] expected = new double[][] {
new double[] {0.0, 0.0},
new double[] {-0.125, -0.25}
};व्याख्या
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.
इनपुट:
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]
अपेक्षित आउटपुट:
expected = [
[0.0625, 0.0, 0.0625],
[0.0625, 0.0, 0.0625],
[0.125, 0.125, 0.25]
]इनपुट:
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};
अपेक्षित आउटपुट:
std::vector<std::vector<double>> expected = {
{0.0625, 0.0, 0.0625},
{0.0625, 0.0, 0.0625},
{0.125, 0.125, 0.25}
};इनपुट:
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};
अपेक्षित आउटपुट:
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}
};व्याख्या
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).
कैसे हल करें
पहले खुद कोशिश करें, फिर एक-एक करके संकेत खोलें।
संकेत 1
Work backward from the output neuron first, then reuse that signal to compute each hidden neuron's contribution.
संकेत 2
You need one delta for the output layer and one delta per hidden neuron, both based on sigmoid derivatives.
संकेत 3
Be careful to use each hidden neuron's own output weight when propagating error back to that neuron.
अपेक्षित जटिलता
O(hn) time, O(hn) extra space
संदर्भ
यह चुनौती CodeUtility के लिए अनुकूलित की गई है। ऊपर दिए गए स्टेटमेंट और सीमाओं का पालन करें।