문제
Mini-batch gradient descent is how almost every neural network is trained: split the data into small batches and update the weights after each one.
Train y = w × x + b on the mean squared error, starting from w = 0 and b = 0. In each epoch, split the samples in order into batches of batch_size (the last batch may be smaller). For each batch of size n:
- errorᵢ = w × xᵢ + b - yᵢ for each sample in the batch
- dw = (2 / n) × Σ errorᵢ × xᵢ and db = (2 / n) × Σ errorᵢ
- w ← w - learning_rate × dw and b ← b - learning_rate × db
Implement train_mini_batch(xs, ys, batch_size, learning_rate, epochs).
Input
xs, ys: The data points (same length).
batch_size: Samples per batch.
learning_rate, epochs: The step size and the number of passes over the data.
Output
Return [w, b] after training.
제약 조건
1 ≤ number of points ≤ 1,000, 1 ≤ batch_size ≤ number of points, 0 ≤ epochs ≤ 1,000
0 < learning_rate ≤ 1
-100 ≤ each value ≤ 100
예시
입력:
xs = [1.0, 2.0, 3.0]
ys = [2.0, 4.0, 6.0]
epochs = 1
batch_size = 2
learning_rate = 0.1
예상 출력:
expected = [2.44, 1.08]입력:
std::vector<double> xs = {1.0, 2.0, 3.0};
std::vector<double> ys = {2.0, 4.0, 6.0};
long long epochs = 1;
long long batch_size = 2;
double learning_rate = 0.1;
예상 출력:
std::vector<double> expected = {2.44, 1.08};입력:
double[] xs = new double[] {1.0, 2.0, 3.0};
double[] ys = new double[] {2.0, 4.0, 6.0};
long epochs = 1;
long batch_size = 2;
double learning_rate = 0.1;
예상 출력:
double[] expected = new double[] {2.44, 1.08};설명
Batch 1 is x = [1, 2]: errors -2 and -4, dw = -10, db = -6, so w = 1 and b = 0.6. Batch 2 is x = [3] alone: error 1×3 + 0.6 - 6 = -2.4, dw = 2 × (-2.4) × 3 = -14.4 and db = -4.8, so w = 2.44 and b = 1.08. Result: [2.44, 1.08].
입력:
xs = [1.0, 2.0, 3.0, 4.0]
ys = [3.0, 5.0, 7.0, 9.0]
epochs = 1
batch_size = 4
learning_rate = 0.05
예상 출력:
expected = [1.75, 0.6]입력:
std::vector<double> xs = {1.0, 2.0, 3.0, 4.0};
std::vector<double> ys = {3.0, 5.0, 7.0, 9.0};
long long epochs = 1;
long long batch_size = 4;
double learning_rate = 0.05;
예상 출력:
std::vector<double> expected = {1.75, 0.6};입력:
double[] xs = new double[] {1.0, 2.0, 3.0, 4.0};
double[] ys = new double[] {3.0, 5.0, 7.0, 9.0};
long epochs = 1;
long batch_size = 4;
double learning_rate = 0.05;
예상 출력:
double[] expected = new double[] {1.75, 0.6};설명
A batch as big as the dataset is ordinary batch gradient descent: one update per epoch. Result: [1.75, 0.6].
접근 방법
먼저 스스로 풀어 본 뒤 힌트를 하나씩 열어 보세요.
힌트 1
Process the samples in the given order, updating the parameters after every batch instead of after every single point.
힌트 2
For each batch, compute prediction errors first, then use their averages with x values to get the two gradients.
힌트 3
The last batch may be smaller than the others, so its gradient scaling must use that batch's size.
예상 복잡도
O(epochs × n) time, O(1) extra space
참고 자료
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