This blog post by Nikki Nikkhoui delves into the concept of entropy as applied to the output of Large Language Models (LLMs). It meticulously explores how entropy can be used as a metric to quantify the uncertainty or randomness inherent in the text generated by these models. The author begins by establishing a foundational understanding of entropy itself, drawing parallels to its use in information theory as a measure of information content. They explain how higher entropy corresponds to greater uncertainty and a wider range of possible outcomes, while lower entropy signifies more predictability and a narrower range of potential outputs.
Nikkhoui then proceeds to connect this theoretical framework to the practical realm of LLMs. They describe how the probability distribution over the vocabulary of an LLM, which essentially represents the likelihood of each word being chosen at each step in the generation process, can be used to calculate the entropy of the model's output. Specifically, they elucidate the process of calculating the cross-entropy and then using it to approximate the true entropy of the generated text. The author provides a detailed breakdown of the formula for calculating cross-entropy, emphasizing the role of the log probabilities assigned to each token by the LLM.
The blog post further illustrates this concept with a concrete example involving a fictional LLM generating a simple sentence. By showcasing the calculation of cross-entropy step-by-step, the author clarifies how the probabilities assigned to different words contribute to the overall entropy of the generated sequence. This practical example reinforces the connection between the theoretical underpinnings of entropy and its application in evaluating LLM output.
Beyond the basic calculation of entropy, Nikkhoui also discusses the potential applications of this metric. They suggest that entropy can be used as a tool for evaluating the performance of LLMs, arguing that higher entropy might indicate greater creativity or diversity in the generated text, while lower entropy could suggest more predictable or repetitive outputs. The author also touches upon the possibility of using entropy to control the level of randomness in LLM generations, potentially allowing users to fine-tune the balance between predictable and surprising outputs. Finally, the post briefly considers the limitations of using entropy as the sole metric for evaluating LLM performance, acknowledging that other factors, such as coherence and relevance, also play crucial roles.
In essence, the blog post provides a comprehensive overview of entropy in the context of LLMs, bridging the gap between abstract information theory and the practical analysis of LLM-generated text. It explains how entropy can be calculated, interpreted, and potentially utilized to understand and control the characteristics of LLM outputs.
This blog post presents a different perspective on deriving Shannon entropy, distinct from the traditional axiomatic approach. Instead of starting with desired properties and deducing the entropy formula, it begins with a fundamental problem: quantifying the average number of bits needed to optimally represent outcomes from a probabilistic source. The author argues this approach provides a more intuitive and grounded understanding of why the entropy formula takes the shape it does.
The post meticulously constructs this derivation. It starts by considering a source emitting symbols from a finite alphabet, each with an associated probability. The core idea is to group these symbols into sets based on their probabilities, specifically targeting sets where the cumulative probability is a power of two. This allows for efficient representation using binary codes, as each set can be uniquely identified by a binary prefix.
The process begins with the most probable symbol and continues iteratively, grouping less probable symbols into progressively larger sets until all symbols are assigned. The author demonstrates how this grouping mirrors the process of building a Huffman code, a well-known algorithm for creating optimal prefix-free codes.
The post then carefully analyzes the expected number of bits required to encode a symbol using this method. This expectation involves summing the product of the number of bits assigned to a set (which relates to the negative logarithm of the cumulative probability of that set) and the cumulative probability of the symbols within that set.
Through a series of mathematical manipulations and approximations, leveraging the properties of logarithms and the behavior of probabilities as the number of samples increases, the author shows that this expected number of bits converges to the familiar Shannon entropy formula: the negative sum of each symbol's probability multiplied by the logarithm base 2 of that probability.
Crucially, the derivation highlights the relationship between optimal coding and entropy. It demonstrates that Shannon entropy represents the theoretical lower bound on the average number of bits needed to encode messages from a given source, achievable through optimal coding schemes like Huffman coding. This construction emphasizes that entropy is not just a measure of uncertainty or information content, but intrinsically linked to efficient data compression and representation. The post concludes by suggesting this alternative construction offers a more concrete and less abstract understanding of Shannon entropy's significance in information theory.
The Hacker News post titled "An alternative construction of Shannon entropy," linking to an article exploring a different way to derive Shannon entropy, has generated a moderate discussion with several interesting comments.
One commenter highlights the pedagogical value of the approach presented in the article. They appreciate how it starts with desirable properties for a measure of information and derives the entropy formula from those, contrasting this with the more common axiomatic approach where the formula is presented and then shown to satisfy the properties. They believe this method makes the concept of entropy more intuitive.
Another commenter focuses on the historical context, mentioning that Shannon's original derivation was indeed based on desired properties. They point out that the article's approach is similar to the one Shannon employed, further reinforcing the pedagogical benefit of seeing the formula emerge from its intended properties rather than the other way around. They link to a relevant page within a book on information theory which seemingly discusses Shannon's original derivation.
A third commenter questions the novelty of the approach, suggesting that it seems similar to standard treatments of the topic. They wonder if the author might be overselling the "alternative construction" aspect. This sparks a brief exchange with another user who defends the article, arguing that while the fundamental ideas are indeed standard, the specific presentation and the emphasis on the grouping property could offer a fresh perspective, especially for educational purposes.
Another commenter delves into more technical details, discussing the concept of entropy as a measure of average code length and relating it to Kraft's inequality. They connect this idea to the article's approach, demonstrating how the desired properties lead to a formula that aligns with the coding interpretation of entropy.
Finally, a few comments touch upon related concepts like cross-entropy and Kullback-Leibler divergence, briefly extending the discussion beyond the scope of the original article. One commenter mentions an example of how entropy is useful, by stating how optimizing for log-loss in a neural network can be interpreted as an attempt to make the predicted distribution very similar to the true distribution.
Overall, the comments section provides a valuable supplement to the article, offering different perspectives on its significance, clarifying some technical points, and connecting it to broader concepts in information theory. While not groundbreaking, the discussion reinforces the importance of pedagogical approaches that derive fundamental formulas from their intended properties.
Summary of Comments ( 15 )
https://news.ycombinator.com/item?id=42649315
Hacker News users discussed the relationship between LLM output entropy and interestingness/creativity, generally agreeing with the article's premise. Some debated the best metrics for measuring "interestingness," suggesting alternatives like perplexity or considering audience-specific novelty. Others pointed out the limitations of entropy alone, highlighting the importance of semantic coherence and relevance. Several commenters offered practical applications, like using entropy for prompt engineering and filtering outputs, or combining it with other metrics for better evaluation. There was also discussion on the potential for LLMs to maximize entropy for "clickbait" generation and the ethical implications of manipulating these metrics.
The Hacker News post titled "Entropy of a Large Language Model output," linking to an article on llm-entropy.html, has generated a moderate amount of discussion. Several commenters engage with the core concept of using entropy to measure the predictability or "surprise" of LLM output.
One commenter questions the practical utility of entropy calculations, especially given that perplexity, a related metric, is already commonly used. They suggest that while intellectually interesting, the entropy analysis might not offer significant new insights for LLM development or evaluation.
Another commenter builds upon this by suggesting that the focus should shift towards the change in entropy over the course of a conversation. They hypothesize that a decreasing entropy could indicate the LLM getting "stuck" in a repetitive loop or predictable pattern, a phenomenon often observed in practice. This suggests a potential application for entropy analysis in detecting and mitigating such issues.
A different thread of discussion arises around the interpretation of high vs. low entropy. One commenter points out that high entropy doesn't necessarily equate to "good" output. A randomly generated string of characters would have high entropy but be nonsensical. They argue that optimal LLM output likely lies within a "goldilocks zone" of moderate entropy – structured enough to be coherent but unpredictable enough to be interesting and informative.
Another commenter introduces the concept of "cross-entropy" and its potential relevance to evaluating LLM output against a reference text. While not fully explored, this suggestion hints at a possible avenue for using entropy-based metrics to assess the faithfulness or accuracy of LLM-generated summaries or translations.
Finally, there's a brief exchange regarding the computational cost of calculating entropy, with one commenter noting that efficient libraries exist to make this calculation manageable even for large texts.
Overall, the comments reflect a cautious but intrigued reception to the idea of using entropy to analyze LLM output. While some question its practical value compared to existing metrics, others identify potential applications in areas like detecting repetitive behavior or evaluating against reference texts. The discussion highlights the ongoing exploration of novel methods for understanding and improving LLM performance.