THE FEED RANKER'S LAYER NORM

Difficulty: Medium Tags: Deep Learning, Layer Normalization, Numerical Stability

Story

A feed-ranking transformer processes user and item features with wildly different scales. Its layer-normalization kernel must normalize each token's feature vector independently, using population variance and a small epsilon.

The Math

For a feature vector x of length d:

mu = mean(x)
v  = mean((x - mu)^2)       (population variance, divide by d not d-1)
xhat_i = (x_i - mu) / sqrt(v + eps)

Use the supplied eps > 0. There are no learnable scale or bias parameters.

Input Format

x (length d), eps.

Output Format

The d normalized values, 6 decimal places.

Constraints

• 1 <= d <= 10^5, 10^-12 <= eps <= 1, |x_i| <= 10^6.
• Time limit: 1.0 second. Memory: 64 MB.

Example

Input

x = [1, 2, 3], eps = 0.000001

Output

-1.224744 0.000000 1.224744

Explanation: The mean is 2, population variance is 2/3, and the epsilon-adjusted standard deviation is sqrt(2/3 + 1e-6).