The Coverage Principle: How Pre-Training Enables Post-Training
The Coverage Principle: How Pre-Training Enables Post-Training
Citation
- Authors: Fan Chen, Audrey Huang, Noah Golowich, Sadhika Malladi, Adam Block, Jordan T. Ash, Akshay Krishnamurthy, and Dylan J. Foster
- Year: 2025
- Venue: arXiv preprint
- arXiv: arXiv:2510.15020
- URL: https://arxiv.org/abs/2510.15020
One-Line Takeaway
The paper argues that downstream success is governed less by pre-training cross-entropy than by coverage: how much probability mass the pretrained model already assigns to high-quality responses that post-training can later amplify.
Technical Core
- The central object is the coverage profile
Cov_N, which measures whether a pretrained model places enough mass on target responses for best-of-Nstyle post-training or decoding to recover them. - Coverage is presented as necessary and sufficient for best-of-
Nsuccess, making it a more relevant statistic than raw cross-entropy when the downstream task depends on rare but high-quality traces. - A key theoretical point is that next-token pre-training can implicitly optimize coverage even when cross-entropy appears to be a poor predictor of downstream performance.
Theoretical Result
- Coverage can generalize faster than cross-entropy and need not inherit spurious dependence on raw sequence length.
- The paper analyzes SGD through this lens and shows that gradient normalization can improve coverage.
- It also proposes model-selection and test-time procedures that target coverage more directly than naive validation loss.
Assumptions And Limits
The paper studies post-training largely through best-of-N and related test-time scaling proxies. That makes it very informative about support accessibility, but it does not by itself answer how to create entirely new strategies outside the pretrained support.
Relevance To This Project
This note matters because it identifies the pre-training side of the repo's exploration problem. If downstream RL or search can only amplify what pre-training already covers, then the central question becomes: can a latent-strategy factorization expose useful structure that ordinary token-level coverage hides, and can interventions enlarge effective coverage over strategies even when token-level support is weak?
Comparison Notes
The Sharpening Mechanism is the closest companion paper: both are about how pretraining support makes later self-improvement possible. The Invisible Leash explains what goes wrong later in RLVR when post-training contracts empirical support instead of preserving it. Zhang et al. (2025) adds a controlled synthetic analysis of when pretraining and RL align well enough to get real capability gains.
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