FADE: Factored Ability--Difficulty Filtering for Online RLVR Scheduling

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Group-based reinforcement learning with verifiable rewards (RLVR) learns only from prompts whose sampled completions disagree: all-pass and all-fail groups produce zero relative advantage and burn generation budget without moving the policy. Adaptive schedulers try to steer rollouts toward informative prompts, yet they rely on per-prompt pass-rate trackers that entangle two quantities evolving at very different speeds---the policy's shifting competence across reasoning skills and each prompt's intrinsic difficulty. When a single exponential average or Beta counter tracks both at once, short memory scrambles the relative ordering of prompts whereas long memory reports stale pass rates. We introduce FADE (Factored Ability--Difficulty Estimation), a bilinear state-space filter that separates the two timescales. FADE represents each prompt's success logit as the inner product of a whitened $K$-dimensional skill loading with a fast-drifting policy ability vector, minus a slow-shrinking scalar item intercept. Using Pólya--Gamma augmentation, each minibatch of binary rollout outcomes updates the $K$-dimensional ability posterior in closed form via a rank-$K$ Woodbury step, instantly transferring competence shifts to all unobserved prompts that share those skills while leaving item intercepts stable. From the resulting posterior mean and variance we derive a closed-form utility for the probability of drawing a non-degenerate rollout group. Across five reasoning benchmarks, two base models, and four scheduling paradigms under a matched rollout budget, FADE consistently improves downstream accuracy and calibration while adding less than $0.6$\,ms per step.

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