# Towards Efficient LLMs Annealing with Principled Sample Selection

arXiv: https://arxiv.org/abs/2605.31175 · HTML: https://mlsanity.com/paper/2605.31175/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 40 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 30 of 31)

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- [InfoBatch: Lossless Training Speed Up by Unbiased Dynamic Data Pruning](https://mlsanity.com/paper/2303.04947/) (2023)
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- [Velocitune: A Velocity-based Dynamic Domain Reweighting Method for Continual Pre-training](https://mlsanity.com/paper/2411.14318/) (2024)
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- [A Survey on Efficient Large Language Model Training: From Data-centric Perspectives](https://mlsanity.com/paper/2510.25817/) (2025)
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## Abstract (arXiv)

The annealing phase is a pivotal convergence stage in LLM pre-training that ultimately determines final model quality. However, effectively selecting training data during this phase remains a key challenge. Current strategies rely on empirical heuristics, such as domain filtering or context extension, which lack a principled grounding in optimization theory. In this work, we characterize the annealing phase through the lens of the loss landscape's spectral geometry. We argue that optimal convergence requires gradient updates to satisfy heterogeneous constraints across different eigen-directions. Building on this insight, we formulate data selection as a problem of satisfying these directional constraints. To this end, we propose DiReCT (Directionally-Restrained Constrained Training), a novel framework that reformulates sample selection in the annealing stage as a constrained optimization problem. By imposing explicit directional constraints on per-sample gradients based on the spectral properties of the Hessian, DiReCT identifies samples that align with the optimal curvature-aware descent path. Extensive experiments across various model scales demonstrate that DiReCT consistently achieves state-of-the-art performance. For future research, code is available at https://github.com/xuyj233/Direct.