Perplexity-based Performance (PP) Measure
A Perplexity-based Performance (PP) Measure is an intrinsic performance measure that is based on a perplexity function.
- Context:
- It can (typically) quantifies how well a probability model predicts a sample, often used in the context of Language Model Evaluation.
- It can reflects the level of uncertainty a model has when predicting subsequent elements in a sequence.
- It can (typically) measure the likelihood of a Language Model generating a given text using probabilities assigned to sequences of words, which is quantified using a Perplexity Score.
- It can (often) be calculated as the exponential of the entropy or the average negative log-likelihood of a probabilistic model.
- It can serve as a benchmark to compare different Statistical Models or Machine Learning Algorithms in terms of their efficiency in handling and predicting language data.
- It can range from being very high, indicating a model with poor predictive performance, to being very low, suggesting a model that effectively predicts text sequences with high accuracy.
- It can be influenced by the size and diversity of the dataset used to train the Statistical Model.
- It can be an input to a Perplexity Measuring Task, and its calculation is often represented by the mathematical expression: [math]\displaystyle{ 2^{\it{Entropy}} = 2^{-\Sigma \ p \log p} }[/math].
- ...
- Example(s):
- a Language Model such as GPT-3 which achieves low perplexity scores on a wide range of text corpora, indicating strong predictive performance.
- a Unigram Model that demonstrates higher perplexity values, showcasing its limited capability in capturing word dependencies.
- a Language Model Perplexity Measure using standard datasets like WikiText-103 corpus to illustrate significant reductions in perplexity, as seen in advanced models like TransformerXL.
- ...
- Counter-Example(s):
- Generalized Perplexity (Derived from Rényi Entropy): The exponential of Rényi entropy leads to a generalized form of perplexity, which serves as a performance measure by indicating how well a model with the entropy characteristics defined by \( \alpha \) can predict new data.
- Cross-Entropy and Its Exponential**: Cross-entropy measures the difference between two probability distributions. The exponential of the negative cross-entropy is a performance measure that evaluates how similar one probability distribution is to another, akin to how perplexity quantifies the surprise of a model in predictive scenarios.
- q-Exponential (Derived from Tsallis Entropy): In statistical mechanics, the q-exponential function related to Tsallis entropy helps define ensembles that describe systems' behaviors, serving as a tool to measure how systems deviate from the expected normative behaviors based on classical statistical mechanics.
- Exponential of Topological Entropy: This measures the "chaoticness" of a system by quantifying the exponential growth rate of distinguishable orbits, thus providing a performance measure of the system's dynamical complexity and unpredictability.
- Exponential of Von Neumann Entropy: This can be interpreted as the effective number of quantum states that contribute to the state of the system, offering a performance measure of the quantum system's complexity and state diversity.
- Accuracy and Precision, which are metrics used in classification tasks, not suitable for measuring uncertainty or randomness in sequence prediction.
- Extrinsic Performance Measures such as Word Error Rate for Automatic Speech Recognition or BLEU Score for Automated Machine Translation, which focus on external validation rather than model's intrinsic capabilities.
- ...
- See: Entropy, Cross-Entropy, Information Theory, Predictive Modeling, Entropy Measure, Empirical Analysis.
References
2024
- GPT-4
- As an intrinsic performance measure, perplexity evaluates the effectiveness of a probabilistic model in language processing and other statistical applications. It reflects how well a model predicts a sample and is particularly useful in models where predictions involve likelihood estimations of sequential data, such as in language modeling.
From this perspective, perplexity quantifies how "surprised" a model is when encountering new data; a lower perplexity indicates that the model is less surprised by the new data, implying better predictive performance. In practical terms, perplexity measures the weighted average branching factor of a language model. A lower branching factor (or lower perplexity) means that the model has a more confident prediction of the next item in the sequence.
- For language models, the perplexity is formally defined as: [math]\displaystyle{ \text{PP}(P) = 2^{-\frac{1}{N} \sum_{i=1}^{N} \log_2 p(x_i)} }[/math] where \( N \) is the number of tokens in the test corpus, and \( p(x_i) \) is the probability the model assigns to the \( i \)-th token in the sequence. This definition showcases perplexity as a measure of prediction power, balancing between model simplicity (avoiding overfitting) and complexity (capturing the nuances in the data).
In essence, perplexity as an intrinsic performance measure assesses a model’s ability to efficiently use the information it has learned to make accurate predictions about unseen data, which is crucial for determining the effectiveness of models in real-world tasks.
- As an intrinsic performance measure, perplexity evaluates the effectiveness of a probabilistic model in language processing and other statistical applications. It reflects how well a model predicts a sample and is particularly useful in models where predictions involve likelihood estimations of sequential data, such as in language modeling.
2020
- https://thegradient.pub/understanding-evaluation-metrics-for-language-models/
- QUOTE: ... Intuitively, perplexity can be understood as a measure of uncertainty. The perplexity of a language model can be seen as the level of perplexity when predicting the following symbol. Consider a language model with an entropy of three bits, in which each bit encodes two possible outcomes of equal probability. This means that when predicting the next symbol, that language model has to choose among possible options. Thus, we can argue that this language model has a perplexity of 8.
Mathematically, the perplexity of a language model is defined as: ...
- QUOTE: ... Intuitively, perplexity can be understood as a measure of uncertainty. The perplexity of a language model can be seen as the level of perplexity when predicting the following symbol. Consider a language model with an entropy of three bits, in which each bit encodes two possible outcomes of equal probability. This means that when predicting the next symbol, that language model has to choose among possible options. Thus, we can argue that this language model has a perplexity of 8.
2019
- https://openreview.net/forum?id=HJePno0cYm¬eId=Hkla0-dp27
- QUOTE: This paper proposes a variant of transformer to train language model, ... Extensive experiments in terms of perplexity results are reported, specially on WikiText-103 corpus, significant perplexity reduction has been achieved.
Perplexity is not a gold standard for language model, the authors are encouraged to report experimental results on real world applications such as word error rate reduction on ASR or BLEU score improvement on machine translation.
- QUOTE: This paper proposes a variant of transformer to train language model, ... Extensive experiments in terms of perplexity results are reported, specially on WikiText-103 corpus, significant perplexity reduction has been achieved.
2018
- (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Perplexity Retrieved:2018-3-7.
- In information theory, perplexity is a measurement of how well a probability distribution or probability model predicts a sample. It may be used to compare probability models. A low perplexity indicates the probability distribution is good at predicting the sample.
2016
- https://www.slideshare.net/alopezfoo/edinburgh-mt-lecture-11-neural-language-models
- QUOTE: Given: [math]\displaystyle{ \bf{w}, \it{p}_{\text{LM}} }[/math]; [math]\displaystyle{ \text{PPL} = 2 \frac{1}{\bf{w}} \log_w \it{p}_{\text{LM}} (\bf{w}) }[/math]; [math]\displaystyle{ 0 \le \text{PPL} \le \infty }[/math]
- Perplexity is a generation of the notion of branching factor: How many choices to I have at each position?
- State-of-the-art English LMs have a PPL of ~100 word choices per position
- A uniform LM has a perplexity of [math]\displaystyle{ |\Sigma| }[/math]
- Humans do much better … and bad models can do even worse than uniform!
2017
- https://web.stanford.edu/class/cs124/lec/languagemodeling2017.pdf
- QUOTE: The best language model is one that best predicts an unseen test setGives the highest P(sentence).
- Perplexity is the inverse probability of the test set, normalized by the number of words:
[math]\displaystyle{ \text{PP}(\bf{w}) = \it{p} (w_1,w_2, ..., w_n)^{-\frac{1}{N}} = \sqrt[N]{\frac{1}{ \it{p}(w_1,w_2, ..., w_n)}} }[/math] - Chain rule: [math]\displaystyle{ \text{PP}(\bf{w}) = \sqrt[N]{ \Pi^{N}_{i=1} \frac{1}{ \it{p}(w_i | w_1,w_2, ..., w_n)}} }[/math]
- For bigrams: [math]\displaystyle{ \text{PP}(\bf{w}) = \sqrt[N]{ \Pi^{N}_{i=1} \frac{1}{ \it{p}(w_i | w_{i-1})}} }[/math]
- Perplexity is the inverse probability of the test set, normalized by the number of words:
- Minimizing perplexity is the same as maximizing probability
- Lower perplexity = better model
Training 38 million words, test 1.5 million words, WSJ: Unigram=162 ; Bigram=170 ; Trigram = 109.
- QUOTE: The best language model is one that best predicts an unseen test setGives the highest P(sentence).
2009
- (Jurafsky & Martin, 2009) ⇒ Daniel Jurafsky, and James H. Martin. (2009). “Speech and Language Processing, 2nd edition." Pearson Education. ISBN:0131873210
- Perplexity is the most common intrinsic evaluation metric for N-gram language models.
1977
- (Jelinek et al., 1977) ⇒ Fred Jelinek, Robert L. Mercer, Lalit R. Bahl, and James K. Baker. (1977). “Perplexity — a Measure of the Difficulty of Speech Recognition Tasks.” The Journal of the Acoustical Society of America 62, no. S1