問題
Adam keeps two moving averages per weight, of the gradient (m) and of the squared gradient (v), and uses them to give each weight its own step size.
Start with m = 0 and v = 0 for every weight. For each step t = 1, 2, 3, … with gradient g:
- m ← beta1 × m + (1 - beta1) × g
- v ← beta2 × v + (1 - beta2) × g²
- m̂ = m / (1 - beta1ᵗ) and v̂ = v / (1 - beta2ᵗ) (bias correction)
- w ← w - learning_rate × m̂ / (√v̂ + epsilon)
Implement adam_steps(gradients, weights, learning_rate, beta1, beta2, epsilon), applying one update per row of gradients, in order.
Input
gradients: One row per step; each row has one gradient per weight.
weights: The starting weights.
learning_rate, beta1, beta2, epsilon: Adam's hyperparameters (commonly 0.001, 0.9, 0.999 and 0.00000001).
Output
Return the weights after all steps.
制約
0 ≤ steps ≤ 1,000, 1 ≤ weights ≤ 100
0 < learning_rate ≤ 1, 0 ≤ beta1, beta2 < 1, 0 < epsilon ≤ 0.001
-100 ≤ each value ≤ 100
例
入力：
beta1 = 0.9
beta2 = 0.999
epsilon = 1.0e-08
weights = [1.0]
gradients = [
[2.0]
]
learning_rate = 0.1
期待される出力：
expected = [0.9]入力：
double beta1 = 0.9;
double beta2 = 0.999;
double epsilon = 1.0e-08;
std::vector<double> weights = {1.0};
std::vector<std::vector<double>> gradients = {
{2.0}
};
double learning_rate = 0.1;
期待される出力：
std::vector<double> expected = {0.9};入力：
double beta1 = 0.9;
double beta2 = 0.999;
double epsilon = 1.0e-08;
double[] weights = new double[] {1.0};
double[][] gradients = new double[][] {
new double[] {2.0}
};
double learning_rate = 0.1;
期待される出力：
double[] expected = new double[] {0.9};解説
m = 0.1 × 2 = 0.2 and v = 0.001 × 4 = 0.004. Bias correction gives m̂ = 0.2 / 0.1 = 2 and v̂ = 0.004 / 0.001 = 4, so the step is 0.1 × 2 / √4 = 0.1: w = 0.9. On its first step Adam moves each weight by about the learning rate, whatever the gradient's size.
入力：
beta1 = 0.9
beta2 = 0.999
epsilon = 1.0e-08
weights = [0.0, 0.0]
gradients = [
[100.0, 0.01]
]
learning_rate = 0.01
期待される出力：
expected = [-0.01, -0.01]入力：
double beta1 = 0.9;
double beta2 = 0.999;
double epsilon = 1.0e-08;
std::vector<double> weights = {0.0, 0.0};
std::vector<std::vector<double>> gradients = {
{100.0, 0.01}
};
double learning_rate = 0.01;
期待される出力：
std::vector<double> expected = {-0.01, -0.01};入力：
double beta1 = 0.9;
double beta2 = 0.999;
double epsilon = 1.0e-08;
double[] weights = new double[] {0.0, 0.0};
double[][] gradients = new double[][] {
new double[] {100.0, 0.01}
};
double learning_rate = 0.01;
期待される出力：
double[] expected = new double[] {-0.01, -0.01};解説
The gradients differ by a factor of 10,000, yet both weights move by about 0.01, to [-0.01, -0.01]: Adam scales each step by the gradient's own size.
解き方のヒント
まずは自分で考えてから、ヒントを 1 つずつ開いてみましょう。
ヒント 1
Track two moving averages per weight: one for the gradient direction and one for its squared size.
ヒント 2
Apply bias correction using the current step number before dividing by the square root term.
ヒント 3
Pay attention to epsilon in the denominator, because it prevents division by zero and changes the final update slightly.
目安の計算量
O(s × w) time, O(w) extra space
参考資料
この課題はCodeUtility用に調整されています。上記の説明と制約に従ってください。