THE PERPLEXITY DASHBOARD

Difficulty: Medium Tags: Language Models, Cross-Entropy, Perplexity

Story

A language-model evaluation dashboard reports perplexity from token-level negative log likelihoods. It accidentally exponentiates the sum rather than the mean, causing scores to depend exponentially on evaluation-set length.

The Math

Given token negative log-likelihoods l_1, ..., l_N (already token-level negative log likelihoods):

H   = mean(l)
PPL = e^H

Input Format

losses: a list of N non-negative loss values.

Output Format

(mean_loss, perplexity), 6 decimal places.

Constraints

• 1 <= N <= 10^5, 0 <= l_i <= 20.
• Time limit: 1.0 second.

Example

Input

losses = [0.5, 1.0, 1.5]

Output

1.000000 2.718282

Explanation: The mean loss is 1, so perplexity is e^1.