โจทย์
Gradient descent improves the weights a little at a time by moving against the gradient:
w ← w - learning_rate × dL/dw
Train a sigmoid neuron with stochastic gradient descent. For each epoch, go through the samples in the given order, and for each sample:
- z = sample · weights + bias, y = 1 / (1 + e^(-z))
- delta = 2 × (y - target) × y × (1 - y)
- weights[i] ← weights[i] - learning_rate × delta × sample[i] (for every i), then bias ← bias - learning_rate × delta
Implement train_neuron(samples, targets, weights, bias, learning_rate, epochs).
Input
samples: m training examples, each with n values.
targets: The m correct outputs (between 0 and 1).
weights, bias: The starting weights (n values) and bias.
learning_rate, epochs: The step size and how many passes over the data.
Output
Return the trained weights followed by the trained bias (n + 1 numbers).
เงื่อนไข
1 ≤ m ≤ 50, 1 ≤ n ≤ 10, 1 ≤ epochs ≤ 1,000
0 < learning_rate ≤ 5
-5 ≤ each value ≤ 5
ตัวอย่าง
อินพุต:
bias = 0.0
epochs = 1
samples = [
[1.0]
]
targets = [1.0]
weights = [0.0]
learning_rate = 1.0
ผลลัพธ์ที่คาดหวัง:
expected = [0.25, 0.25]อินพุต:
double bias = 0.0;
long long epochs = 1;
std::vector<std::vector<double>> samples = {
{1.0}
};
std::vector<double> targets = {1.0};
std::vector<double> weights = {0.0};
double learning_rate = 1.0;
ผลลัพธ์ที่คาดหวัง:
std::vector<double> expected = {0.25, 0.25};อินพุต:
double bias = 0.0;
long epochs = 1;
double[][] samples = new double[][] {
new double[] {1.0}
};
double[] targets = new double[] {1.0};
double[] weights = new double[] {0.0};
double learning_rate = 1.0;
ผลลัพธ์ที่คาดหวัง:
double[] expected = new double[] {0.25, 0.25};คำอธิบาย
y = σ(0) = 0.5, delta = 2 × (0.5 - 1) × 0.25 = -0.25. Weight: 0 - 1 × (-0.25) × 1 = 0.25. Bias: 0 - 1 × (-0.25) = 0.25. Result: [0.25, 0.25].
อินพุต:
bias = 0.0
epochs = 500
samples = [
[0.0, 0.0],
[0.0, 1.0],
[1.0, 0.0],
[1.0, 1.0]
]
targets = [0.0, 0.0, 0.0, 1.0]
weights = [0.0, 0.0]
learning_rate = 2.0
ผลลัพธ์ที่คาดหวัง:
expected = [6.346558, 6.339863, -9.592092]อินพุต:
double bias = 0.0;
long long epochs = 500;
std::vector<std::vector<double>> samples = {
{0.0, 0.0},
{0.0, 1.0},
{1.0, 0.0},
{1.0, 1.0}
};
std::vector<double> targets = {0.0, 0.0, 0.0, 1.0};
std::vector<double> weights = {0.0, 0.0};
double learning_rate = 2.0;
ผลลัพธ์ที่คาดหวัง:
std::vector<double> expected = {6.346558, 6.339863, -9.592092};อินพุต:
double bias = 0.0;
long epochs = 500;
double[][] samples = new double[][] {
new double[] {0.0, 0.0},
new double[] {0.0, 1.0},
new double[] {1.0, 0.0},
new double[] {1.0, 1.0}
};
double[] targets = new double[] {0.0, 0.0, 0.0, 1.0};
double[] weights = new double[] {0.0, 0.0};
double learning_rate = 2.0;
ผลลัพธ์ที่คาดหวัง:
double[] expected = new double[] {6.346558, 6.339863, -9.592092};คำอธิบาย
This is the logical AND gate. After 500 epochs both weights are large and positive and the bias is strongly negative, so only the input [1, 1] gives an output above 0.5.
แนวทางการแก้
ลองทำเองก่อน แล้วค่อยเปิดคำใบ้ทีละข้อ
คำใบ้ 1
Process the training samples in order for each epoch, updating weights immediately after each example.
คำใบ้ 2
For each sample, compute the sigmoid output, then use its gradient to move weights and bias downhill.
คำใบ้ 3
Because updates are online, later samples in the same epoch see earlier updates.
ความซับซ้อนที่คาดหวัง
O(epochs × m × n) time, O(1) extra space
แหล่งอ้างอิง
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