aiqbench analysis
This research provides a rigorous theoretical framework for transformer scaling laws by modeling training dynamics as ordinary differential equations, bridging the gap between empirical observations and formal proof. The study identifies a distinct phase transition where generalization error shifts from exponential decay to a power-law regime ($\Theta(\mathsf{C}^{-1/7})$) once a specific resource threshold is crossed, offering precise bounds on how model size, time, and data size independently govern performance limits. For practitioners and researchers, this work validates the tightness of scaling laws up to logarithmic factors and condition-number gaps, moving beyond toy models to analyze real-world sequence-to-sequence training conditions.
Unifying Learning Dynamics and Generalization in Transformers Scaling Law
arXiv:2512.22088v3 Announce Type: replace Abstract: The scaling law, a cornerstone of Large Language Model (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 language models 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 matching upper and lower bounds on the 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 $\Theta(\mathsf{C}^{-1/7})$. These rates are certified by complementary lower bounds -- statistical, via an information-theoretic two-point reduction, and optimization-side, via a first-order oracle argument -- rendering the two-stage law tight up to constants, logarithmic factors, and a condition-number gap. 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 bounds of generalization.
arXiv:2512.22088 (cs)
Abstract:The scaling law, a cornerstone of Large Language Model (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 language models 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 matching upper and lower bounds on the 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 $\Theta(\mathsf{C}^{-1/7})$. These rates are certified by complementary lower bounds -- statistical, via an information-theoretic two-point reduction, and optimization-side, via a first-order oracle argument -- rendering the two-stage law tight up to constants, logarithmic factors, and a condition-number gap. 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 bounds of generalization.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computation and Language (cs.CL)
arXiv:2512.22088 [cs.LG]
(or arXiv:2512.22088v3 [cs.LG] for this version)
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