Cross-entropy and KL divergence are closely related measures of difference between probability distributions. While cross-entropy quantifies the average number of bits needed to encode events drawn from a true distribution p using a coding scheme optimized for a predicted distribution q, KL divergence measures how much more information is needed on average when using q instead of p. Specifically, KL divergence is the difference between cross-entropy and the entropy of the true distribution p. Therefore, minimizing cross-entropy with respect to q is equivalent to minimizing the KL divergence, as the entropy of p is constant. While both can measure the dissimilarity between distributions, KL divergence is a true "distance" metric (though asymmetric), whereas cross-entropy is not. The post illustrates these concepts with detailed numerical examples and explains their significance in machine learning, particularly for tasks like classification where the goal is to match a predicted distribution to the true data distribution.
The blog post "Entropy Attacks" argues against blindly trusting entropy sources, particularly in cryptographic contexts. It emphasizes that measuring entropy based solely on observed outputs, like those from /dev/random, is insufficient for security. An attacker might manipulate or partially control the supposedly random source, leading to predictable outputs despite seemingly high entropy. The post uses the example of an attacker influencing the timing of network packets to illustrate how seemingly unpredictable data can still be exploited. It concludes by advocating for robust key-derivation functions and avoiding reliance on potentially compromised entropy sources, suggesting deterministic random bit generators (DRBGs) seeded with a high-quality initial seed as a preferable alternative.
The Hacker News comments discuss the practicality and effectiveness of entropy-reduction attacks, particularly in the context of Bernstein's blog post. Some users debate the real-world impact, pointing out that while theoretically interesting, such attacks often rely on unrealistic assumptions like attackers having precise timing information or access to specific hardware. Others highlight the importance of considering these attacks when designing security systems, emphasizing defense-in-depth strategies. Several comments delve into the technical details of entropy estimation and the challenges of accurately measuring it. A few users also mention specific examples of vulnerabilities related to insufficient entropy, like Debian's OpenSSL bug. The overall sentiment suggests that while these attacks aren't always easily exploitable, understanding and mitigating them is crucial for robust security.
Succinct data structures represent data in space close to the information-theoretic lower bound, while still allowing efficient queries. The blog post explores several examples, starting with representing a bit vector using only one extra bit beyond the raw data, while still supporting constant-time rank and select operations. It then extends this to compressed bit vectors using Elias-Fano encoding and explains how to represent arbitrary sets and sparse arrays succinctly. Finally, it touches on representing trees succinctly, demonstrating how to support various navigation operations efficiently despite the compact representation. Overall, the post emphasizes the power of succinct data structures to achieve substantial space savings without significant performance degradation.
Hacker News users discussed the practicality and performance trade-offs of succinct data structures. Some questioned the real-world benefits given the complexity and potential performance hits compared to simpler, less space-efficient solutions, especially with the abundance of cheap memory. Others highlighted the value in specific niches like bioinformatics and embedded systems where memory is constrained. The discussion also touched on the difficulty of implementing and debugging these structures and the lack of mature libraries in common languages. A compelling comment highlighted the use case of storing large language models efficiently, where succinct data structures can significantly reduce storage requirements and memory access times, potentially enabling new applications on resource-constrained devices. Others noted the theoretical elegance of the approach, even if practical applications remain somewhat niche.
The blog post "On Zero Sum Games (The Informational Meta-Game)" argues that while many real-world interactions appear zero-sum, they often contain hidden non-zero-sum elements, especially concerning information. The author uses poker as an analogy: while the chips exchanged represent a zero-sum component, the information revealed through betting, bluffing, and tells creates a meta-game that isn't zero-sum. This meta-game involves learning about opponents and improving one's own strategies, generating future value even within apparently zero-sum situations like negotiations or competitions. The core idea is that leveraging information asymmetry can transform seemingly zero-sum interactions into opportunities for mutual gain by increasing overall understanding and skill, thus expanding the "pie" over time.
HN commenters generally appreciated the post's clear explanation of zero-sum games and its application to informational meta-games. Several praised the analogy to poker, finding it illuminating. Some extended the discussion by exploring how this framework applies to areas like politics and social dynamics, where manipulating information can create perceived zero-sum scenarios even when underlying resources aren't truly limited. One commenter pointed out potential flaws in assuming perfect rationality and complete information, suggesting the model's applicability is limited in real-world situations. Another highlighted the importance of trust and reputation in navigating these information games, emphasizing the long-term cost of deceptive tactics. A few users also questioned the clarity of certain examples, requesting further elaboration from the author.
Bell Labs, celebrating its centennial, represents a century of groundbreaking innovation. From its origins as a research arm of AT&T, it pioneered advancements in telecommunications, including the transistor, laser, solar cell, information theory, and the Unix operating system and C programming language. This prolific era fostered a collaborative environment where scientific exploration thrived, leading to numerous Nobel Prizes and shaping the modern technological landscape. However, the breakup of AT&T and subsequent shifts in corporate focus impacted Bell Labs' trajectory, leading to a diminished research scope and a transition towards more commercially driven objectives. Despite this evolution, Bell Labs' legacy of fundamental scientific discovery and engineering prowess remains a benchmark for industrial research.
HN commenters largely praised the linked PDF documenting Bell Labs' history, calling it well-written, informative, and a good overview of a critical institution. Several pointed out specific areas they found interesting, like the discussion of "directed basic research," the balance between pure research and product development, and the evolution of corporate research labs in general. Some lamented the decline of similar research-focused environments today, contrasting Bell Labs' heyday with the current focus on short-term profits. A few commenters added further historical details or pointed to related resources like the book Idea Factory. One commenter questioned the framing of Bell Labs as primarily an American institution given its reliance on global talent.
The blog post explores using entropy as a measure of the predictability and "surprise" of Large Language Model (LLM) outputs. It explains how to calculate entropy character-by-character and demonstrates that higher entropy generally corresponds to more creative or unexpected text. The author argues that while tools like perplexity exist, entropy offers a more granular and interpretable way to analyze LLM behavior, potentially revealing insights into the model's internal workings and helping identify areas for improvement, such as reducing repetitive or predictable outputs. They provide Python code examples for calculating entropy and showcase its application in evaluating different LLM prompts and outputs.
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.
This blog post presents a different way to derive Shannon entropy, focusing on its property as a unique measure of information content. Instead of starting with desired properties like additivity and then finding a formula that satisfies them, the author begins with a core idea: measuring the average number of binary questions needed to pinpoint a specific outcome from a probability distribution. By formalizing this concept using a binary tree representation of the questioning process and leveraging Kraft's inequality, they demonstrate that -∑pᵢlog₂(pᵢ) emerges naturally as the optimal average question length, thus establishing it as the entropy. This construction emphasizes the intuitive link between entropy and the efficient encoding of information.
Hacker News users discuss the alternative construction of Shannon entropy presented in the linked article. Some express appreciation for the clear explanation and visualizations, finding the geometric approach insightful and offering a fresh perspective on a familiar concept. Others debate the pedagogical value of the approach, questioning whether it truly simplifies understanding for those unfamiliar with entropy, or merely offers a different lens for those already versed in the subject. A few commenters note the connection to cross-entropy and Kullback-Leibler divergence, suggesting the geometric interpretation could be extended to these related concepts. There's also a brief discussion on the practical implications and potential applications of this alternative construction, although no concrete examples are provided. Overall, the comments reflect a mix of appreciation for the novel approach and a pragmatic assessment of its usefulness in teaching and application.
Summary of Comments ( 4 )
https://news.ycombinator.com/item?id=43670171
Hacker News users generally praised the clarity and helpfulness of the article explaining cross-entropy and KL divergence. Several commenters pointed out the value of the concrete code examples and visualizations provided. One user appreciated the explanation of the difference between minimizing cross-entropy and maximizing likelihood, while another highlighted the article's effective use of simple language to explain complex concepts. A few comments focused on practical applications, including how cross-entropy helps in model selection and its relation to log loss. Some users shared additional resources and alternative explanations, further enriching the discussion.
The Hacker News post titled "Cross-Entropy and KL Divergence," linking to an article explaining these concepts, has generated several comments. Many commenters appreciate the clarity and helpfulness of the article.
One commenter points out a potential area of confusion in the article regarding the base of the logarithm used in the calculations. They explain that while the article uses base 2 for its examples, other bases like e (natural logarithm) are common, and the choice affects the units (bits vs. nats) of the result. This commenter emphasizes the importance of understanding the relationship between these different units and how the chosen base impacts the interpretation of the calculated values.
Another commenter expresses gratitude for the clear and concise explanation, stating that they've often seen these terms used without proper definition. They specifically praise the article's use of concrete examples and its intuitive approach to explaining complex mathematical concepts.
Another comment focuses on the practical implications of cross-entropy, particularly its use in machine learning as a loss function. They discuss how minimizing cross-entropy leads to improved model performance and how it relates to maximizing the likelihood of the observed data. This comment connects the theoretical concepts to real-world applications, enhancing the practical understanding of the topic.
One user provides a link to another resource, a blog post by Tim Vieira, which offers further explanation and builds upon the original article's content. This contribution extends the discussion by providing additional avenues for learning and exploring related concepts.
A few other commenters express their agreement with the positive sentiment towards the article, confirming its usefulness and clarity. They appreciate the article's straightforward approach and the way it demystifies these often-confusing concepts.
In summary, the comments on the Hacker News post overwhelmingly praise the linked article for its clear and accessible explanation of cross-entropy and KL divergence. They delve into specific aspects like the importance of the logarithm base, the practical applications in machine learning, and provide additional resources for further learning. The comments contribute to a deeper understanding and appreciation of the article's subject matter.