THE FIRST FRAUD SCORE

Difficulty: Easy Tags: Loss Functions

Story

PayPal's very first fraud model outputs a raw probability per transaction. Before anyone trusts the model, its loss on a labeled calibration batch needs to be computed exactly, matching the framework reference to the last digit.

The Math

L = -(1/n) * sum_i [ y_i * log(yhat_i) + (1 - y_i) * log(1 - yhat_i) ]

Input Format

n
y_1 yhat_1
...
y_n yhat_n

Output Format

Scalar loss, 6 decimals.

Constraints

• 1 <= n <= 10^5, 10^-7 <= yhat_i <= 1 - 10^-7
• Time limit: 1.0 second.

Example

Input

4
1 0.9
0 0.2
1 0.6
0 0.3

Output

0.299001