toplogo
Sign In

A Mathematical Theory for Learning Semantic Capabilities in Abstract Language Models


Core Concepts
A mathematical theory is developed to explain the emergence of learned skills in large language models when the number of system parameters and the size of training data surpass certain thresholds.
Abstract
The paper presents a mathematical theory for learning semantic languages using abstract learners. Key highlights: Semantic languages are defined using a skill-text bipartite graph, where skills represent the latent capabilities required to understand texts. Two types of abstract learners are introduced - 1-skill learners and Poisson learners. These learners can learn skills by repeatedly being presented with training texts. Density evolution analysis is used to show the emergence of learned skills when the ratio of the number of training texts to the number of skills exceeds a certain threshold. This threshold corresponds to a sharp drop in the testing error, defined as the probability that a randomly selected text can be understood by the learner. The analysis also provides a scaling law for the testing error with respect to the ratio of training texts to skills. The trained learners can be used for semantic compression, where texts are encoded using the indices of the learned skills required to understand them. This enables more efficient compression compared to traditional lossless compression. The paper discusses the application of the trained learners in a semantic communication system, where the semantic encoder/decoder is separate from the physical channel encoder/decoder. The key contribution is the development of a mathematical framework to explain the emergence of learned skills in large language models, which is an active area of research.
Stats
None
Quotes
None

Deeper Inquiries

How can the proposed theory be extended to model the hierarchical structure of skills in real-world languages

To extend the proposed theory to model the hierarchical structure of skills in real-world languages, we can introduce a multi-level abstraction in the skill-text bipartite graph model. Instead of treating all skills as equal entities, we can categorize them into different levels based on their complexity or dependency. For instance, basic skills can be at the lower levels, while more advanced or specialized skills can be at higher levels. This hierarchical structure can be represented in the graph by connecting skills at different levels based on their relationships and dependencies. By incorporating this hierarchical organization of skills, the learning process can be more reflective of how skills are acquired in real-world scenarios, where foundational skills are mastered before progressing to more advanced ones.

How can the scaling laws derived in this paper be reconciled with the observed scaling behavior of the number of parameters in large language models like GPT-3 and GPT-4

The scaling laws derived in the paper can be reconciled with the observed scaling behavior of the number of parameters in large language models like GPT-3 and GPT-4 by considering the capacity of the models to store and learn patterns or skills. The scaling law derived in the paper focuses on the relationship between the size of training texts, the number of skills, and the emergence of learned skills. In the context of large language models, the number of parameters directly influences the capacity of the model to learn and store patterns or skills. Therefore, the scaling behavior of the number of parameters in these models can be seen as a mechanism to increase the capacity for learning a larger set of skills. The scaling laws derived in the paper can provide insights into how the performance of these models improves as the number of parameters increases, leading to the emergence of new learned skills.

What are the potential connections between the learned skill representations in the proposed theory and the internal representations learned by transformer-based language models

The learned skill representations in the proposed theory and the internal representations learned by transformer-based language models share commonalities in terms of capturing semantic relationships and patterns in language. In the proposed theory, learned skills are acquired through the iterative decoding process, where skills are identified and understood based on the presented texts. Similarly, in transformer-based models, the attention mechanism allows the model to capture dependencies between tokens in a text, effectively learning patterns and semantic relationships. The representations of learned skills in the theory can be likened to the learned embeddings or representations of tokens in transformer models, where each skill or token representation encodes specific semantic information. By aligning the representations learned in the theory with the internal representations of transformer models, we can draw parallels between the theoretical framework and the practical implementation of language understanding and generation.
0