Physics in Next-token Prediction (2411.00660v2)

Abstract: We discovered the underlying physics in Next-token Prediction (NTP). We identified the law of information conservation within NTP and proposed the First Law of Information Capacity (IC-1), demonstrating that the essence of intelligence emergence in auto-regressive models is fundamentally a process of information transfer. We also introduced Landauer's Principle into NTP, formulating the Second Law of Information Capacity (IC-2), which establishes the relationship between auto-regressive model training and energy consumption. Additionally, we presented several corollaries, which hold practical significance for production practices. Finally, we demonstrate the consistency between the Law of Information Capacity and the Scaling Law for Neural LLMs, the Knowledge Capacity Scaling Laws, and the Scaling Laws for Precision.

• The paper introduces a framework with two foundational laws—IC-1 for information conservation and IC-2 linking energy consumption to model capacity.
• The paper applies Landauer’s Principle to connect minimal energy requirements with training efficiency in auto-regressive models.
• The paper presents dynamic corollaries that guide improvements in dataset quality, model scaling, and energy-efficient AI design.

An Expert Overview of "Physics in Next-token Prediction"

The paper "Physics in Next-token Prediction" by Hongjun An, Yiliang Song, and Xuelong Li presents a novel theoretical framework addressing the underlying principles governing Next-token Prediction (NTP) in auto-regressive models. The authors propose two foundational laws named the First and Second Laws of Information Capacity: the law of information conservation within NTP and the introduction of Landauer's Principle into this domain.

Key Contributions and Theorems

1. Law of Information Conservation (IC-1): The authors derive and propose the First Law of Information Capacity (IC-1), asserting that the emergence of intelligence in auto-regressive models fundamentally represents a process of information transfer. They describe this process using the equation ηN = D(H - L), where η denotes the information capacity of the model, N is the parameter size (in bits), D the number of trained tokens, H the entropy of the dataset, and L the average cross-entropy training loss. This equation signifies that model training corresponds to an efficient compression of the dataset.
2. Landauer’s Principle and Energy Relationship (IC-2): By incorporating Landauer's Principle, which relates the erasure of information to energy consumption, the authors formulate the Second Law of Information Capacity (IC-2). This law posits that the minimum energy required to train a model is linked to the information capacity, forming an energy-information relationship in the NTP context: E₀ = ηN(k_B T ln 2), where E₀ is the minimal energy, k_B is the Boltzmann constant, and T the temperature of the heat reservoir in Kelvin.
3. Dynamic Perspectives and Practical Corollaries: The paper further introduces dynamic interpretations of these laws throughout the training process, from initial to terminal states, illustrating that as training progresses, the model's information capacity increases dynamically. Several corollaries derived from these laws can guide practical applications, such as estimating data entropy, optimizing dataset quality, matching model size to dataset size, and identifying energy limitations during model training.

Implications and Theoretical Consistency

The implications of these findings are significant for both theoretical understanding and practical applications in AI model development. The laws offer a compelling framework to evaluate the efficiency of information transfer in NTP, facilitating more effective allocation of computational resources. The introduction of energy considerations through Landauer’s Principle illuminates the inevitable physical constraints in AI model training, suggesting avenues for more energy-efficient algorithmic designs and hardware advancements.

The theoretical framework proposed in this paper is consistent with existing empirical scaling laws, such as those outlined by Kaplan et al. (2020). The analysis demonstrates compatibility with practical observations of model training scales, thus supporting the broader applicability of the Information Capacity laws across various auto-regressive model architectures.

Future Directions

This research opens several potential pathways for future investigation. Further empirical validation across diverse model types and architectures could solidify the axes of theoretical and practical insights offered by IC-1 and IC-2. Moreover, real-world applications could benefit from optimizing models for minimal information capacity and energy consumption, particularly as computational demands continue to escalate.

The exploration of information and energy interplay within AI systems might also spur advances in quantum computing, possibly redefining limits currently imposed by classical computational paradigms. As such, the foundational principles articulated in this paper could well underpin the next wave of scalable and sustainable AI technologies.