THE GRADIENT SEATBELT

Difficulty: Easy Tags: Optimization

Story

PayPal's fraud model occasionally sees gradient spikes from rare, extreme-value transactions. Global-norm gradient clipping is the seatbelt that keeps one outlier batch from blowing up training.

The Math

if ||g||_2 > tau:
    g <- g * (tau / ||g||_2)
else:
    g <- g

Input Format

d tau
g_1 ... g_d

Output Format

Clipped gradient, d values, 6 decimals.

Constraints

• 1 <= d <= 10^4, tau > 0
• Time limit: 1.0 second.

Example

Input

2 2.0
3.0 4.0

Output

1.200000 1.600000

Explanation: ||g||_2 = 5.0 > tau = 2.0, so scale by 2.0/5.0 = 0.4: (3.0, 4.0) * 0.4 = (1.2, 1.6).