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.
Restrições
0 ≤ epochs ≤ 5,000
0 < learning_rate ≤ 5
-2 ≤ each starting weight and bias ≤ 2
Exemplos
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]
Saída 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};
Saída 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};
Saída esperada:
double[] expected = new double[] {0.482451, 0.436972, 0.494378, 0.452729};Explicação
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]
Saída 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};
Saída 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};
Saída esperada:
double[] expected = new double[] {0.015727, 0.982119, 0.985515, 0.013682};Explicação
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.
Como abordar
Tente sozinho primeiro e depois abra uma dica de cada vez.
Dica 1
Process the four XOR samples in the same fixed order for every epoch, and update parameters immediately after each one.
Dica 2
Use forward propagation, then backpropagate the output error into both hidden neurons with the current weights before applying updates.
Dica 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.
Complexidade esperada
O(epochs) time, O(1) extra space
Referências
Este desafio foi adaptado para o CodeUtility. Siga o enunciado e as restrições acima.