Problem
XOR (exclusive or) is the classic problem a single neuron cannot solve, but a network with one hidden layer can. Train a network with 2 inputs, 2 hidden neurons and 1 output, all sigmoid (σ(z) = 1 / (1 + e^(-z))), using the loss L = (y - target)².
The training data, always used in this order in every epoch:
| input | target |
|---|---|
| [0, 0] | 0 |
| [0, 1] | 1 |
| [1, 0] | 1 |
| [1, 1] | 0 |
For each example:
- Forward: a_j = σ(x₁ × hidden_weights[j][0] + x₂ × hidden_weights[j][1] + hidden_biases[j]), y = σ(a₁ × output_weights[0] + a₂ × output_weights[1] + output_bias)
- delta_out = 2 × (y - target) × y × (1 - y), and delta_j = delta_out × output_weights[j] × a_j × (1 - a_j), using the weights before this example's update
- Update every parameter with learning rate lr: output_weights[j] -= lr × delta_out × a_j, output_bias -= lr × delta_out, hidden_weights[j][i] -= lr × delta_j × x_i, hidden_biases[j] -= lr × delta_j
Implement train_xor(hidden_weights, hidden_biases, output_weights, output_bias, learning_rate, epochs).
Output
After training, return the network's predictions y for the inputs [0, 0], [0, 1], [1, 0] and [1, 1], in that order.
Einschränkungen
0 ≤ epochs ≤ 5,000
0 < learning_rate ≤ 5
-2 ≤ each starting weight and bias ≤ 2
Beispiele
Eingabe:
epochs = 0
output_bias = 0.0
hidden_biases = [0.0, 0.3]
learning_rate = 1.5
hidden_weights = [
[0.2, 0.8],
[0.9, -0.7]
]
output_weights = [-0.6, 0.4]
Erwartete Ausgabe:
expected = [0.482451, 0.436972, 0.494378, 0.452729]Eingabe:
long long epochs = 0;
double output_bias = 0.0;
std::vector<double> hidden_biases = {0.0, 0.3};
double learning_rate = 1.5;
std::vector<std::vector<double>> hidden_weights = {
{0.2, 0.8},
{0.9, -0.7}
};
std::vector<double> output_weights = {-0.6, 0.4};
Erwartete Ausgabe:
std::vector<double> expected = {0.482451, 0.436972, 0.494378, 0.452729};Eingabe:
long epochs = 0;
double output_bias = 0.0;
double[] hidden_biases = new double[] {0.0, 0.3};
double learning_rate = 1.5;
double[][] hidden_weights = new double[][] {
new double[] {0.2, 0.8},
new double[] {0.9, -0.7}
};
double[] output_weights = new double[] {-0.6, 0.4};
Erwartete Ausgabe:
double[] expected = new double[] {0.482451, 0.436972, 0.494378, 0.452729};Erläuterung
With 0 epochs nothing is trained, so the result is just the forward pass with the starting weights: about [0.482, 0.437, 0.494, 0.453]. The untrained network gives every input nearly the same answer.
Eingabe:
epochs = 2000
output_bias = 0.0
hidden_biases = [0.0, 0.3]
learning_rate = 1.5
hidden_weights = [
[0.2, 0.8],
[0.9, -0.7]
]
output_weights = [-0.6, 0.4]
Erwartete Ausgabe:
expected = [0.015727, 0.982119, 0.985515, 0.013682]Eingabe:
long long epochs = 2000;
double output_bias = 0.0;
std::vector<double> hidden_biases = {0.0, 0.3};
double learning_rate = 1.5;
std::vector<std::vector<double>> hidden_weights = {
{0.2, 0.8},
{0.9, -0.7}
};
std::vector<double> output_weights = {-0.6, 0.4};
Erwartete Ausgabe:
std::vector<double> expected = {0.015727, 0.982119, 0.985515, 0.013682};Eingabe:
long epochs = 2000;
double output_bias = 0.0;
double[] hidden_biases = new double[] {0.0, 0.3};
double learning_rate = 1.5;
double[][] hidden_weights = new double[][] {
new double[] {0.2, 0.8},
new double[] {0.9, -0.7}
};
double[] output_weights = new double[] {-0.6, 0.4};
Erwartete Ausgabe:
double[] expected = new double[] {0.015727, 0.982119, 0.985515, 0.013682};Erläuterung
After 2,000 epochs the predictions are about [0.016, 0.982, 0.986, 0.014]: low for [0, 0] and [1, 1], high for [0, 1] and [1, 0]. The network has learned XOR.
So gehst du vor
Versuch es zuerst allein und öffne dann einen Hinweis nach dem anderen.
Hinweis 1
Process the four XOR samples in the same fixed order for every epoch, and update parameters immediately after each one.
Hinweis 2
Use forward propagation, then backpropagate the output error into both hidden neurons with the current weights before applying updates.
Hinweis 3
Watch for the sigmoid derivative and the fact that hidden-layer updates depend on the input values, so zero inputs leave some weights unchanged.
Erwartete Komplexität
O(epochs) time, O(1) extra space
Referenzen
Diese Aufgabe wurde für CodeUtility angepasst. Beachte die oben genannten Vorgaben und Einschränkungen.