Gradient Descent with Momentum

Momentum makes gradient descent behave like a ball rolling downhill: each step keeps part of the previous one.

Start with a velocity of 0 for every weight. For each step t, with gradient vector g (one value per weight):

v ← beta × v + g
w ← w - learning_rate × v

Implement momentum_steps(gradients, weights, learning_rate, beta), 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: The step size.

beta: The momentum coefficient (0 means plain gradient descent).

Output

Return the weights after all steps.

Constraints

0 ≤ steps ≤ 1,000, 1 ≤ weights ≤ 100

0 < learning_rate ≤ 1, 0 ≤ beta < 1

-100 ≤ each value ≤ 100

Examples

Example 1

Input:
gradients = [
    [1.0],
    [1.0],
    [1.0]
]
weights = [0.0]
learning_rate = 0.1
beta = 0.9

Expected output:
[-0.561]

Explanation

The velocity grows 1 → 1.9 → 2.71 because the gradient keeps pointing the same way, so the steps are 0.1, 0.19 and 0.271: w = -0.561.

Example 2

Input:
gradients = [
    [2.0, -4.0]
]
weights = [1.0, 1.0]
learning_rate = 0.5
beta = 0.0

Expected output:
[0.0, 3.0]

Explanation

With beta = 0 this is plain gradient descent: w = [1 - 0.5 × 2, 1 - 0.5 × (-4)] = [0, 3].

References

Run the sample tests to check your code without submitting.

Example 1 Not run

Input:

gradients = [
    [1.0],
    [1.0],
    [1.0]
]
weights = [0.0]
learning_rate = 0.1
beta = 0.9

Expected output:

[-0.561]

Your output

Example 2 Not run

Input:

gradients = [
    [2.0, -4.0]
]
weights = [1.0, 1.0]
learning_rate = 0.5
beta = 0.0

Expected output:

[0.0, 3.0]

Your output

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package main

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const Pi = 3.14

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data, _ := ioutil.ReadFile("file.txt")
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println(strings.TrimSpace(text))
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words := strings.Split(text, " ")
fmt.Println(words)

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type Person struct {
  Name string
}

func (p Person) Greet() string {
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}

p := Person{Name: "Alice"}
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x := 10
ptr := &x
*ptr = 20
println(x) // 20