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.
Researchers at the University of Surrey have theoretically demonstrated that two opposing arrows of time can emerge within specific quantum systems. By examining the evolution of entanglement within these systems, they found that while one subsystem experiences time flowing forward as entropy increases, another subsystem can simultaneously experience time flowing backward, with entropy decreasing. This doesn't violate the second law of thermodynamics, as the overall combined system still sees entropy increase. This discovery offers new insights into the foundations of quantum mechanics and its relationship with thermodynamics, particularly in understanding the flow of time at the quantum level.
HN users express skepticism about the press release's interpretation of the research, questioning whether the "two arrows of time" are a genuine phenomenon or simply an artifact of the chosen model. Some suggest the description is sensationalized and oversimplifies complex quantum behavior. Several commenters call for access to the actual paper rather than relying on the university's press release, emphasizing the need to examine the methodology and mathematical framework to understand the true implications of the findings. A few commenters delve into the specifics of microscopic reversibility and entropy, highlighting the challenges in reconciling these concepts with the claims made in the article. There's a general consensus that the headline is attention-grabbing but potentially misleading without deeper analysis of the underlying research.
The blog post "Fat Rand: How Many Lines Do You Need to Generate a Random Number?" explores the surprising complexity hidden within seemingly simple random number generation. It dissects the code behind Python's random.randint() function, revealing a multi-layered process involving system-level entropy sources, hashing, and bit manipulation to ultimately produce a seemingly simple random integer. The post highlights the extensive effort required to achieve statistically sound randomness, demonstrating that generating even a single random number relies on a significant amount of code and underlying system functionality. This complexity is necessary to ensure unpredictability and avoid biases, which are crucial for security, simulations, and various other applications.
Hacker News users discussed the surprising complexity of generating truly random numbers, agreeing with the article's premise. Some commenters highlighted the difficulty in seeding pseudo-random number generators (PRNGs) effectively, with suggestions like using /dev/random, hardware sources, or even mixing multiple sources. Others pointed out that the article focuses on uniformly distributed random numbers, and that generating other distributions introduces additional complexity. A few users mentioned specific use cases where simple PRNGs are sufficient, like games or simulations, while others emphasized the critical importance of robust randomness in cryptography and security. The discussion also touched upon the trade-offs between performance and security when choosing a random number generation method, and the value of having different "grades" of randomness for various applications.
Klarity is an open-source Python library designed to analyze uncertainty and entropy in large language model (LLM) outputs. It provides various metrics and visualization tools to help users understand how confident an LLM is in its generated text. This can be used to identify potential errors, biases, or areas where the model is struggling, ultimately enabling better prompt engineering and more reliable LLM application development. Klarity supports different uncertainty estimation methods and integrates with popular LLM frameworks like Hugging Face Transformers.
Hacker News users discussed Klarity's potential usefulness, but also expressed skepticism and pointed out limitations. Some questioned the practical applications, wondering if uncertainty analysis is truly valuable for most LLM use cases. Others noted that Klarity focuses primarily on token-level entropy, which may not accurately reflect higher-level semantic uncertainty. The reliance on temperature scaling as the primary uncertainty control mechanism was also criticized. Some commenters suggested alternative approaches to uncertainty quantification, such as Bayesian methods or ensembles, might be more informative. There was interest in seeing Klarity applied to different models and tasks to better understand its capabilities and limitations. Finally, the need for better visualization and integration with existing LLM workflows was highlighted.
This post provides a high-level overview of compression algorithms, categorizing them into lossless and lossy methods. Lossless compression, suitable for text and code, reconstructs the original data perfectly using techniques like Huffman coding and LZ77. Lossy compression, often used for multimedia like images and audio, achieves higher compression ratios by discarding less perceptible data, employing methods such as discrete cosine transform (DCT) and quantization. The post briefly explains the core concepts behind these techniques and illustrates how they reduce data size by exploiting redundancy and irrelevancy. It emphasizes the trade-off between compression ratio and data fidelity, with lossy compression prioritizing smaller file sizes at the expense of some information loss.
Hacker News users discussed various aspects of compression, prompted by a blog post overviewing different algorithms. Several commenters highlighted the importance of understanding data characteristics when choosing a compression method, emphasizing that no single algorithm is universally superior. Some pointed out the trade-offs between compression ratio, speed, and memory usage, with specific examples like LZ77 being fast for decompression but slower for compression. Others discussed more niche compression techniques like ANS and its use in modern codecs, as well as the role of entropy coding. A few users mentioned practical applications and tools, like using zstd for backups and mentioning the utility of brotli. The complexities of lossy compression, particularly for images, were also touched upon.
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.