What is perplexity?

Perplexity is a commonly used evaluation metric in natural language processing (NLP) that measures how well a language model predicts a sample of text.

Perplexity is a measurement of how well a probability distribution or a probability model predicts a sample.

In the context of language modeling, perplexity measures how well a language model predicts the next word in a sequence based on the words that come before it.

Mathematically, perplexity is defined as the exponential of the average negative log-likelihood of a sequence of words. The formula for perplexity is:

Perplexity = $exp(-1/N * sum(log(P(w_i|w_1, w_2, ..., w_{i-1}))))$

where:

• N is the total number of words in the sequence

• $P(w_i|w_1, w_2, ..., w_{i-1})$ is the probability of the word $w_i$ given the preceding words $w_1, w_2, ..., w_{i-1}$

• log is the natural logarithm

Intuitive Understanding

Perplexity can be thought of as a measure of how "surprised" or "confused" the language model is when predicting the next word.

A lower perplexity indicates that the model is less surprised and can predict the next word more accurately, while a higher perplexity suggests that the model is more uncertain or confused.

For example, if a language model has a perplexity of 10 on a given text dataset, it means that, on average, the model is as confused as if it had to choose uniformly and independently from 10 possibilities for each word.

Technical Explanation

To calculate perplexity, you first need to compute the cross-entropy loss between the predicted word probabilities and the actual word probabilities. Cross-entropy loss measures the difference between two probability distributions.

In the context of language modeling, the model predicts the probability distribution over the vocabulary for the next word, given the preceding words. The actual word distribution is represented as a one-hot vector, where the correct word has a probability of 1, and all other words have a probability of 0.

The cross-entropy loss for a single word is calculated as:

Loss = $-log(P(w_i|w_1, w_2, ..., w_{i-1}))$

To get the average cross-entropy loss for the entire sequence, you sum up the individual word losses and divide by the total number of words:

Average Loss = $-1/N * sum(log(P(w_i|w_1, w_2, ..., w_{i-1})))$

Finally, perplexity is obtained by exponentiating the average cross-entropy loss:

Perplexity = $exp(Average Loss)$

The perplexity score is often used to compare different language models or to evaluate the improvement of a model during training.

A lower perplexity indicates better language modeling performance.

It's important to note that while perplexity is a useful metric, it has some limitations.

It doesn't directly measure the quality or coherence of the generated text, and it can be sensitive to the choice of vocabulary and the specifics of the training data.

Therefore, perplexity should be used in conjunction with other evaluation metrics and human judgment to assess the overall performance of a language model.