AdaKP: Online Adaptive Knowledge-Point Selection for Reasoning-Oriented Reinforcement Learning

Reinforcement learning (RL) finetuning significantly enhances the reasoning capabilities of large language models (LLMs), yet its effectiveness critically depends on selecting prompts of appropriate difficulty for the current policy. This is challenging because prompt difficulty evolves throughout training. Existing online methods therefore face a trade-off: evaluation-based approaches are accurate but expensive, while prediction-based approaches are efficient but typically assume stationary difficulty, making them ill-suited to RL's non-stationary training dynamics. To address these issues, we propose a Kalman-Guided Prompt Selection method (KGPS), which reformulates prompt selection as a dynamic state estimation problem rather than static difficulty prediction. KGPS models each prompt's latent success rate in logit space using a linear-Gaussian state-space model, with process noise coupled to the magnitude of policy updates so that uncertainty increases when the policy changes more substantially. A Kalman filter then maintains a calibrated Gaussian posterior over prompt difficulty, and prompts are selected by maximizing a posterior-expected training utility that favors intermediate-difficulty prompts while naturally revisiting uncertain ones. The resulting procedure is adaptive to policy drift and requires no additional rollouts beyond standard policy training. Extensive experiments across mathematics, planning, and geometry reasoning benchmarks, as well as multiple RL algorithms, show that KGPS consistently improves both final accuracy and rollout efficiency over strong baselines, establishing state-of-the-art performance among online prompt selection methods. For example, on DeepSeek-R1-Distill-7B, KGPS uses 83% fewer rollouts than DS while even improving the average performance by 0.12 point across six math reasoning benchmarks.

Context engineering can improve large language models without updating their weights, but mathematical reasoning exposes a key limitation: feedback accumulated in one growing prompt causes context bloat and limits the amount of learned guidance that can be used. Existing methods often conflate storage, what is learned across runs, with usage, what is included for a particular problem, and therefore inherit this prompt-size ceiling. We introduce Knowledge-Adaptive Context Engineering (KACE), which separates storage from usage through difficulty- and domain-based organization. Offline, a self-reflective learning loop distills training traces into an epistemic tree: a knowledge base of typed cards stratified by problem difficulty and epistemic domain. Each card is assigned to the difficulty-domain node corresponding to the failure from which it originated. At evaluation time, tiered self-consistency with per-tier agreement gates dynamically classifies each problem as easy, medium, or hard. Easy problems exit without retrieved cards, while harder problems retrieve only the matching branch of the tree. This tiered scheme matches or exceeds Best-of-N while using comparable compute, and it classifies problem difficulty with 78 percent pairwise concordance. The main empirical contribution is the construction and use of a difficulty- and domain-stratified knowledge base enabled by tiered self-consistency. On AIME 2025, KACE achieves 62.2 percent accuracy, a 10.4-point absolute gain over fixed Best-of-5 self-consistency at a comparable solver-call budget and a 5.6-point gain over the strongest learned-context baseline, Tiered + GEPA. We also observe consistent gains on MATH-HARD and the verifiable subset of OlymMATH.

Reinforcement learning has become the standard for improving reasoning in large language models, yet evidence increasingly suggests that RL does not teach new strategies; it redistributes probability mass over solutions the base model already contains. In this work, we ask: if RL merely steers the model toward paths it already knows, is the RL optimization loop itself necessary? Through token-level analysis across multiple model families and RL algorithms, we find that RL's beneficial footprint is a sparse, predictable correction concentrated at high-entropy decision points where the model is uncertain which branch to take. Only 1--3% of token positions are affected, the promoted token always lies within the base model's top-5 alternatives, and targeted corrections at those few positions causally recover a large fraction of RL's accuracy gain, while random corrections fail. The base model's own entropy identifies these positions without any RL-trained model, and the entire correction is low-dimensional, representable in a tiny fraction of model parameters. These findings reframe reasoning improvement as sparse policy selection, not capability acquisition. We translate this insight into ReasonMaxxer, a minimal RL-free method that applies contrastive loss only at entropy-gated decision points, using a few hundred base-model rollouts and no online generation. Across three model families, six scales, and six math reasoning benchmarks, ReasonMaxxer matches or exceeds full RL performance while requiring only tens of problems and minutes of single-GPU training, a reduction in training cost of roughly three orders of magnitude.