समस्या
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.
सीमाएँ
0 ≤ epochs ≤ 5,000
0 < learning_rate ≤ 5
-2 ≤ each starting weight and bias ≤ 2
उदाहरण
इनपुट:
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]
अपेक्षित आउटपुट:
expected = [0.482451, 0.436972, 0.494378, 0.452729]इनपुट:
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};
अपेक्षित आउटपुट:
std::vector<double> expected = {0.482451, 0.436972, 0.494378, 0.452729};इनपुट:
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};
अपेक्षित आउटपुट:
double[] expected = new double[] {0.482451, 0.436972, 0.494378, 0.452729};व्याख्या
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.
इनपुट:
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]
अपेक्षित आउटपुट:
expected = [0.015727, 0.982119, 0.985515, 0.013682]इनपुट:
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};
अपेक्षित आउटपुट:
std::vector<double> expected = {0.015727, 0.982119, 0.985515, 0.013682};इनपुट:
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};
अपेक्षित आउटपुट:
double[] expected = new double[] {0.015727, 0.982119, 0.985515, 0.013682};व्याख्या
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.
कैसे हल करें
पहले खुद कोशिश करें, फिर एक-एक करके संकेत खोलें।
संकेत 1
Process the four XOR samples in the same fixed order for every epoch, and update parameters immediately after each one.
संकेत 2
Use forward propagation, then backpropagate the output error into both hidden neurons with the current weights before applying updates.
संकेत 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.
अपेक्षित जटिलता
O(epochs) time, O(1) extra space
संदर्भ
यह चुनौती CodeUtility के लिए अनुकूलित की गई है। ऊपर दिए गए स्टेटमेंट और सीमाओं का पालन करें।