Problema
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.
Restricciones
0 ≤ epochs ≤ 5,000
0 < learning_rate ≤ 5
-2 ≤ each starting weight and bias ≤ 2
Ejemplos
Entrada:
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]
Salida esperada:
expected = [0.482451, 0.436972, 0.494378, 0.452729]Entrada:
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};
Salida esperada:
std::vector<double> expected = {0.482451, 0.436972, 0.494378, 0.452729};Entrada:
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};
Salida esperada:
double[] expected = new double[] {0.482451, 0.436972, 0.494378, 0.452729};Explicación
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.
Entrada:
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]
Salida esperada:
expected = [0.015727, 0.982119, 0.985515, 0.013682]Entrada:
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};
Salida esperada:
std::vector<double> expected = {0.015727, 0.982119, 0.985515, 0.013682};Entrada:
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};
Salida esperada:
double[] expected = new double[] {0.015727, 0.982119, 0.985515, 0.013682};Explicación
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.
Cómo abordarlo
Inténtalo primero por tu cuenta y luego abre las pistas una a una.
Pista 1
Process the four XOR samples in the same fixed order for every epoch, and update parameters immediately after each one.
Pista 2
Use forward propagation, then backpropagate the output error into both hidden neurons with the current weights before applying updates.
Pista 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.
Complejidad esperada
O(epochs) time, O(1) extra space
Referencias
Este reto está adaptado para CodeUtility. Sigue el enunciado y las restricciones indicadas arriba.