Unifying Learning Dynamics and Generalization in Transformers Scaling Law

Abstract: The scaling law, a cornerstone of LLM development, predicts improvements in model performance with increasing computational resources. Yet, while empirically validated, its theoretical underpinnings remain poorly understood. This work formalizes the learning dynamics of transformer-based LLMs as an ordinary differential equation (ODE) system, then approximates this process to kernel behaviors. Departing from prior toy-model analyses, we rigorously analyze stochastic gradient descent (SGD) training for multi-layer transformers on sequence-to-sequence data with arbitrary data distribution, closely mirroring real-world conditions. Our analysis characterizes the convergence of generalization error to the irreducible risk as computational resources scale with data, especially during the optimization process. We establish a theoretical upper bound on excess risk characterized by a distinct phase transition. In the initial optimization phase, the excess risk decays exponentially relative to the computational cost ${\sf C}$. However, once a specific resource allocation threshold is crossed, the system enters a statistical phase, where the generalization error follows a power-law decay of $Θ(\mathsf{C}^{-1/6})$. Beyond this unified framework, our theory derives isolated scaling laws for model size, training time, and dataset size, elucidating how each variable independently governs the upper bounds of generalization.

• The paper introduces an ODE-based framework to capture transformer learning dynamics and their impact on scaling behavior.
• The analysis simplifies training into a kernel regime, linking model size, training time, and dataset volume to generalization performance.
• Results highlight a dual-phase error decay with an exponential drop transitioning to a power-law decline, pinpointing resource thresholds for enhanced risk bounds.

Summary of "Unifying Learning Dynamics and Generalization in Transformers Scaling Law" (2512.22088)

This paper investigates the theoretical foundations of the scaling law for transformer-based LLMs, focusing on their learning dynamics and generalization performance. By modeling the training process as an ODE system and analyzing the use of stochastic gradient descent (SGD) in multi-layer transformers, the paper derives a comprehensive theoretical framework that connects computational resources, training dynamics, and generalization.

Theoretical Foundations of Scaling Law

The paper addresses a significant gap in understanding the theoretical underpinnings of scaling laws in LLMs. Previous studies primarily focused on simpler statistical models, while this work extends the analysis to complex, contemporary transformer architectures. The authors achieve this by representing transformer learning dynamics through kernel behavior and analyzing the impact of scaling computational resources, model size, training time, and dataset size.

Key Contributions

• ODE-Based Learning Dynamics: The authors formalize the learning process of LLMs using an ODE system, capturing the interaction between model parameters and computational resources.
• Kernel Regime Simplification: Leveraging the over-parameterization of LLMs, the study adopts a kernel-based approach to describe LLMs' behavior during training, akin to "Lazy Learning."
• Two-Stage Upper Boundary on Expected Risk: The study identifies a phase transition in the scaling process characterized by an exponential drop in excess risk, which turns into a power-law decay as resources increase beyond a critical threshold.
• Isolated Scaling Laws: The paper derives individual scaling laws for various factors—model size, training duration, and dataset size—highlighting each factor's role in determining generalization performance.

Training Convergence and Generalization Bounds

Training convergence is guaranteed under specific conditions using the Neural Tangent Kernel (NTK) framework. The NTK setup provides insights into the generalization performance of LLMs:

• Convergence in Kernel Regimes: By bounding kernel perturbation and employing a lazy learning assumption, the paper ensures provable convergence of training with arbitrary error constraints.
• Empirical Risk Minimization: The bounds on generalization error incorporate approximation and estimation errors, connected through the scaling laws.

Scaling Law Dynamics and Patterns

The paper presents a unified scaling law framework, including:

• Compute-Starved and Data-Limited Stages: The analysis distinguishes between initial compute-starved conditions and subsequent data-limited phases, revealing distinct decay patterns in generalization error.
• Single-Law Effects: Insight into how individual scaling variables (model size, training time, dataset size) influence LLM performance, with the paper quantifying diminishing returns and potential failure points in scaling efforts.

Implications and Future Directions

This work establishes a robust theoretical foundation for analyzing the effects of scaling in transformer LLMs. It underscores the need for proportional scaling of datasets with model and compute increases to optimize generalization. Future developments might focus on exploring the interplay between scaling and emerging transformer architectures, potentially enhancing the guidelines for designing next-generation AI systems.

Conclusion

The paper "Unifying Learning Dynamics and Generalization in Transformers Scaling Law" (2512.22088) provides a comprehensive theoretical analysis of transformer scaling laws, offering clarity on the connection between learning dynamics and generalization in LLMs. It introduces a dual-phase framework for understanding the implications of computational scaling and offers analytical tools for predicting and optimizing LLM performance across various dimensions of scaling.