# The Coverage Principle: How Pre-Training Enables Post-Training

arXiv: https://arxiv.org/abs/2510.15020 · HTML: https://mlsanity.com/paper/2510.15020/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2025.

## Built on by (most cited first, 30 of 30)

- [From Prior to Pro: Efficient Skill Mastery via Distribution Contractive RL Finetuning](https://mlsanity.com/paper/2603.10263/) (2026)
- [Posterior Behavioral Cloning: Pretraining BC Policies for Efficient RL Finetuning](https://mlsanity.com/paper/2512.16911/) (2025)
- [From Reasoning Traces to Reusable Modules: Understanding Compositional Generalization in Language Model Reasoning](https://mlsanity.com/paper/2606.18089/) (2026)
- [Understanding Reasoning from Pretraining to Post-Training](https://mlsanity.com/paper/2607.16097/) (2026)
- [Learning to Reason with Curriculum I: Provable Benefits of Autocurriculum](https://mlsanity.com/paper/2603.18325/) (2026)
- [Revisiting the (Sub)Optimality of Best-of-N for Inference-Time Alignment](https://mlsanity.com/paper/2603.05739/) (2026)
- [Post-Training with Policy Gradients: Optimality and the Base Model Barrier](https://mlsanity.com/paper/2603.06957/) (2026)
- [Reject, Resample, Repeat: Understanding Parallel Reasoning in Language Model Inference](https://mlsanity.com/paper/2603.07887/) (2026)
- [TMRL: Diffusion Timestep-Modulated Pretraining Enables Exploration for Efficient Policy Finetuning](https://mlsanity.com/paper/2605.12236/) (2026)
- [Learning to Reason with Curriculum II: Compositional Generalization](https://mlsanity.com/paper/2606.27721/) (2026)
- [On Advantage Estimates for Max@K Policy Gradients](https://mlsanity.com/paper/2606.06080/) (2026)
- [OrderGrad: Optimizing Beyond the Mean with Order-Statistic Policy Gradient Estimation](https://mlsanity.com/paper/2606.06096/) (2026)
- [Anatomy of Post-Training: Using Interpretability to Characterize Data and Shape the Learning Signal](https://mlsanity.com/paper/2606.12360/) (2026)
- [Learning Generalizable Behaviors for Terminal Agents](https://mlsanity.com/paper/2608.22631/) (2026)
- [Demystifying Reinforcement Learning Post-Training of Language Models](https://mlsanity.com/paper/2608.24949/) (2026)
- [TailSFT: Filtered Fine-Tuning Improves Post-Training Performance](https://mlsanity.com/paper/2608.25756/) (2026)
- [Controlling Multimodal Conversational Agents with Coverage-Enhanced Latent Actions](https://mlsanity.com/paper/2601.07516/) (2026)
- [Characterizing Online and Private Learnability under Distributional Constraints via Generalized Smoothness](https://mlsanity.com/paper/2602.20585/) (2026)
- [A Theoretical Framework for Statistical Evaluability of Generative Models](https://mlsanity.com/paper/2604.05324/) (2026)
- [Peer-Predictive Self-Training for Language Model Reasoning](https://mlsanity.com/paper/2604.13356/) (2026)
- [RL Excursions during Pre-Training: Re-examining Policy Optimization for LLM training](https://mlsanity.com/paper/2606.04272/) (2026)
- [How Post-Training Shapes Biological Reasoning Models](https://mlsanity.com/paper/2606.16517/) (2026)
- [Recipes for Steering and Scaling LLMs via Sampling](https://mlsanity.com/paper/2608.26120/) (2026)
- [Partition Function Estimation under Bounded f-Divergence](https://mlsanity.com/paper/2602.23535/) (2026)
- [The Role of Generator Access in Autoregressive Post-Training](https://mlsanity.com/paper/2604.04855/) (2026)
- [Mirror Horizon: Viable Path Entropy as a Measure of Bounded Reflection](https://mlsanity.com/paper/2607.11937/) (2026)
- [From Imitation to Reward Discovery: On-Policy Warmup for Agentic RL](https://mlsanity.com/paper/2609.39436/) (2026)
- [How RL Reshapes LLM Reasoning: Transferability, Coverage, and Scaling Laws](https://mlsanity.com/paper/2610.04158/) (2026)
- [Understanding the Weight Averaging Mechanism in LLM Training for Post-Training Quantization](https://mlsanity.com/paper/2610.05329/) (2026)
- [Do Small Language Models Learn to Negotiate? A Controlled Scaling Study of RL-Trained Sellers](https://mlsanity.com/paper/2610.06204/) (2026)

## Abstract (arXiv)

Language models demonstrate remarkable abilities when pre-trained on large text corpora and fine-tuned for specific tasks, but how and why pre-training shapes the success of the final model remains poorly understood. Notably, although pre-training success is often quantified by cross-entropy loss, cross-entropy can be a poor predictor of downstream performance. Instead, we provide a theoretical perspective on this relationship through the lens of coverage, which quantifies the probability mass the pre-trained model places on high-quality responses and which is necessary and sufficient for post-training and test-time scaling methods such as Best-of-N to succeed. Our main results develop an understanding of the coverage principle, a phenomenon whereby next-token prediction (more generally, maximum likelihood) implicitly optimizes toward a model with good coverage. In particular, we uncover a mechanism that explains the power of coverage in predicting downstream performance: coverage generalizes faster than cross-entropy, avoiding spurious dependence on problem-dependent parameters such as the sequence length. We also study practical algorithmic interventions with provable benefits for improving coverage, including (i) model/checkpoint selection procedures, (ii) gradient normalization schemes, and (iii) test-time decoding strategies.