Backpropagation in a Two-Layer Network

Backpropagation applies the chain rule layer by layer, from the output back to the inputs. The network has n inputs, h hidden neurons and one output, all with sigmoid activation (σ(z) = 1 / (1 + e^(-z))):

hidden: a_j = σ(inputs · hidden_weights[j] + hidden_biases[j])
output: y = σ(a · output_weights + output_bias)
loss: L = (y - target)²

Gradients:

delta_out = 2 × (y - target) × y × (1 - y)
Output layer: dL/doutput_weights[j] = delta_out × a_j and dL/doutput_bias = delta_out
delta_j = delta_out × output_weights[j] × a_j × (1 - a_j)
Hidden layer: dL/dhidden_weights[j][i] = delta_j × inputs[i] and dL/dhidden_biases[j] = delta_j

Implement backprop_gradients(inputs, hidden_weights, hidden_biases, output_weights, output_bias, target).

Output

Return a matrix with h + 1 rows:

Row j (for each hidden neuron j, in order): the n gradients of hidden_weights[j], then the gradient of hidden_biases[j].
Last row: the h gradients of output_weights, then the gradient of output_bias.

Constraints

1 ≤ n, h ≤ 20

-5 ≤ each input, weight and bias ≤ 5

0 ≤ target ≤ 1

Examples

Example 1

Input:
inputs = [1.0]
hidden_weights = [
    [0.0]
]
hidden_biases = [0.0]
output_weights = [0.0]
output_bias = 0.0
target = 1.0

Expected output:
[
    [0.0, 0.0],
    [-0.125, -0.25]
]

Explanation

a = σ(0) = 0.5 and y = σ(0) = 0.5. delta_out = 2 × (0.5 - 1) × 0.25 = -0.25. Output row: [-0.25 × 0.5, -0.25] = [-0.125, -0.25]. delta_1 = -0.25 × 0 × 0.25 = 0, so the hidden row is [0, 0]: a zero output weight blocks the gradient.

Example 2

Input:
inputs = [1.0, 0.0]
hidden_weights = [
    [0.0, 1.0],
    [0.0, -1.0]
]
hidden_biases = [0.0, 0.0]
output_weights = [1.0, 1.0]
output_bias = -1.0
target = 0.0

Expected output:
[
    [0.0625, 0.0, 0.0625],
    [0.0625, 0.0, 0.0625],
    [0.125, 0.125, 0.25]
]

Explanation

Both hidden values are σ(0) = 0.5, so y = σ(0.5 + 0.5 - 1) = 0.5 and delta_out = 2 × 0.5 × 0.25 = 0.25. Output row: [0.125, 0.125, 0.25]. Each delta_j = 0.25 × 1 × 0.25 = 0.0625, giving hidden rows [0.0625, 0, 0.0625] (the second input is 0).

References

Run the sample tests to check your code without submitting.

Example 1 Not run

Input:

inputs = [1.0]
hidden_weights = [
    [0.0]
]
hidden_biases = [0.0]
output_weights = [0.0]
output_bias = 0.0
target = 1.0

Expected output:

[
    [0.0, 0.0],
    [-0.125, -0.25]
]

Your output

Example 2 Not run

Input:

inputs = [1.0, 0.0]
hidden_weights = [
    [0.0, 1.0],
    [0.0, -1.0]
]
hidden_biases = [0.0, 0.0]
output_weights = [1.0, 1.0]
output_bias = -1.0
target = 0.0

Expected output:

[
    [0.0625, 0.0, 0.0625],
    [0.0625, 0.0, 0.0625],
    [0.125, 0.125, 0.25]
]

Your output

4 hidden tests also run when you submit. If your code fails one, you'll see that test. Solve the challenge to see them all.

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💡 Kotlin Basics Guide for Beginners

1. Declaring Variables and Constants

Kotlin uses val for immutable variables and var for mutable ones.

val name: String = "Alice"
var age: Int = 30
val pi = 3.14  // type inferred

// Constants
const val MAX_USERS = 100

2. Conditionals (if / when)

Use if expressions or when for multiple branches.

val x = 2
if (x == 1) {
    println("One")
} else if (x == 2) {
    println("Two")
} else {
    println("Other")
}

when (x) {
    1 -> println("One")
    2 -> println("Two")
    else -> println("Other")
}

3. Loops

Use for, while, and do-while for iteration.

for (i in 0..2) {
    println(i)
}

var n = 3
while (n > 0) {
    println(n)
    n--
}

4. Arrays

Kotlin supports arrays using the arrayOf() function.

val numbers = arrayOf(10, 20, 30)
println(numbers[1])

5. List Manipulation

Use mutableListOf for dynamic lists.

val nums = mutableListOf(1, 2, 3)
nums.add(4)
nums.removeAt(0)

for (n in nums) {
    print("$n ")
}

6. Console Input/Output

Use readLine() for input and println() for output.

print("Enter your name: ")
val name = readLine()
println("Hello, $name")

7. Functions

Functions use fun keyword with optional return type.

fun add(a: Int, b: Int): Int {
    return a + b
}

println(add(3, 4))

8. Maps

mutableMapOf stores key-value pairs.

val ages = mutableMapOf("Alice" to 30)
println(ages["Alice"])

9. Exception Handling

Use try, catch, and finally to handle errors.

try {
    val result = 10 / 0
} catch (e: ArithmeticException) {
    println("Error: ${e.message}")
}

10. File I/O

Use File from java.io for file operations.

import java.io.File

File("file.txt").writeText("Hello File")
val text = File("file.txt").readText()
println(text)

11. String Manipulation

Kotlin strings support many methods and interpolation.

val text = "Hello World"
println(text.length)
println(text.substring(0, 5))
println(text.contains("World"))

12. Classes & Objects

Kotlin classes are concise and support default constructors.

class Person(val name: String) {
    fun greet() = println("Hi, I'm $name")
}

val p = Person("Alice")
p.greet()

13. Null Safety

Kotlin distinguishes nullable and non-nullable types. Use ? and ?: for safety.

val name: String? = null
println(name?.length ?: 0)