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

arXiv: https://arxiv.org/abs/2605.10019 · HTML: https://mlsanity.com/paper/2605.10019/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 49 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 30 of 45)

- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [T2I-Adapter: Learning Adapters to Dig out More Controllable Ability for Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.08453/) (2023)
- [Vision Transformers Need Registers](https://mlsanity.com/paper/2309.16588/) (2023)
- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Theoretical Limitations of Self-Attention in Neural Sequence Models](https://mlsanity.com/paper/1906.06755/) (2019)
- [Transformers Learn Shortcuts to Automata](https://mlsanity.com/paper/2210.10749/) (2022)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [Overcoming a Theoretical Limitation of Self-Attention](https://mlsanity.com/paper/2202.12172/) (2022)
- [Self-Attention Networks Can Process Bounded Hierarchical Languages](https://mlsanity.com/paper/2105.11115/) (2021)
- [An analytic theory of creativity in convolutional diffusion models](https://mlsanity.com/paper/2412.20292/) (2024)
- [Why Diffusion Models Don't Memorize: The Role of Implicit Dynamical Regularization in Training](https://mlsanity.com/paper/2505.17638/) (2025)
- [Why are Sensitive Functions Hard for Transformers?](https://mlsanity.com/paper/2402.09963/) (2024)
- [Generalization on the Unseen, Logic Reasoning and Degree Curriculum](https://mlsanity.com/paper/2301.13105/) (2023)
- [Simplicity Bias in Transformers and their Ability to Learn Sparse Boolean Functions](https://mlsanity.com/paper/2211.12316/) (2022)
- [Diffusion Beats Autoregressive in Data-Constrained Settings](https://mlsanity.com/paper/2507.15857/) (2025)
- [Transformers Provably Solve Parity Efficiently with Chain of Thought](https://mlsanity.com/paper/2410.08633/) (2024)
- [Diffusion Language Models are Super Data Learners](https://mlsanity.com/paper/2511.03276/) (2025)
- [Theoretical limitations of multi-layer Transformer](https://mlsanity.com/paper/2412.02975/) (2024)
- [In search of dispersed memories: Generative diffusion models are associative memory networks](https://mlsanity.com/paper/2309.17290/) (2023)
- [SGD Finds then Tunes Features in Two-Layer Neural Networks with near-Optimal Sample Complexity: A Case Study in the XOR problem](https://mlsanity.com/paper/2309.15111/) (2023)
- [Understanding Generalizability of Diffusion Models Requires Rethinking the Hidden Gaussian Structure](https://mlsanity.com/paper/2410.24060/) (2024)
- [A Good Score Does not Lead to A Good Generative Model](https://mlsanity.com/paper/2401.04856/) (2024)
- [How Far Can Transformers Reason? The Globality Barrier and Inductive Scratchpad](https://mlsanity.com/paper/2406.06467/) (2024)
- [The Unreasonable Effectiveness of Gaussian Score Approximation for Diffusion Models and its Applications](https://mlsanity.com/paper/2412.09726/) (2024)

## Built on by (most cited first, 1 of 1)

- [Evaluating the Representation Space of Diffusion Models via Self-Supervised Principles](https://mlsanity.com/paper/2606.09718/) (2026)

## Abstract (arXiv)

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: τrule, the step at which generations first become rule-valid, and τmem, 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, τrule and τmem, depend on key aspects of the learning setup. Specifically, we show that τrule increases with rule complexity and decreases with model capacity, while τmem is approximately invariant to the rule and scales nearly linearly with dataset size N. We define the innovation window as the interval [τrule, τmem]. This window widens with increasing N and narrows with rule complexity, and may vanish entirely when τrule ≥ τmem. 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 τrule, while training samples' basins begin to dominate around τmem. Together, these results yield a unified and predictive account of when and how generative models exhibit genuine innovation.