The two clocks and the innovation window: When and how generative models learn rules

2026-05-11

AI Summary

This work addresses the challenge that generative models, when trained on limited data, often converge to the empirical distribution without capturing the underlying generative rules. To dissect this behavior, the authors propose a “dual-clock” mechanism that tracks two critical timepoints during training: τ_rule, when the model first generates rule-compliant samples, and τ_mem, when it begins memorizing training instances. The interval between these timestamps defines an “innovation window,” which characterizes the model’s genuine generalization capacity. Experiments on DiT diffusion and GPT autoregressive architectures reveal that τ_rule increases with rule complexity but decreases with model capacity, whereas τ_mem is largely rule-agnostic and scales linearly with dataset size. Consequently, the innovation window widens with more data yet narrows—or even vanishes—as rule complexity rises, offering a unified framework to analyze rule-learning dynamics across distinct generative paradigms.

Abstract

Generative models trained on finite data face a fundamental tension: their score-matching or next-token objective converges to the empirical training distribution rather than the population distribution we seek to learn. Using rule-valid synthetic tasks, we trace this tension across two training timescales: $τ_{\mathrm{rule}}$, the step at which generations first become rule-valid, and $τ_{\mathrm{mem}}$, the step at which models begin reproducing training samples. Focusing on parity and extending to other binary rules and combinatorial puzzles, we characterize how these two clocks, $τ_{\mathrm{rule}}$ and $τ_{\mathrm{mem}}$, depend on key aspects of the learning setup. Specifically, we show that $τ_{\mathrm{rule}}$ increases with rule complexity and decreases with model capacity, while $τ_{\mathrm{mem}}$ is approximately invariant to the rule and scales nearly linearly with dataset size $N$. We define the \emph{innovation window} as the interval $[τ_{\mathrm{rule}}, τ_{\mathrm{mem}}]$. This window widens with increasing $N$ and narrows with rule complexity, and may vanish entirely when $τ_{\mathrm{rule}} \geq τ_{\mathrm{mem}}$. The same two-clock structure arises in both diffusion (DiT) and autoregressive (GPT) models, with architecture-dependent offsets. Dissecting the learned score of DiT models reveals a corresponding evolution of the optimization landscapes, where rule-valid samples' basins expand substantially around $τ_{\mathrm{rule}}$, while training samples' basins begin to dominate around $τ_{\mathrm{mem}}$. Together, these results yield a unified and predictive account of when and how generative models exhibit genuine innovation.

Problem

• generative models
• rule learning
• training dynamics
• innovation window
• distribution learning

Innovation

• innovation window
• rule learning
• generative models
• score matching
• training dynamics