ThinkPrior: Zero-Rollout Difficulty Priors for Cold-Start Prompt Selection in RLVR

Reinforcement learning with verifiable rewards (RLVR) has emerged as a central paradigm for improving the reasoning capabilities of large language models. Group-based policy optimization methods, such as GRPO, typically allocate a fixed number of rollouts to every prompt. This uniform allocation can be inefficient: it over-allocates compute to prompts whose sampled groups are already saturated while under-exploring prompts for which additional samples may reveal useful correct trajectories. To address this limitation, we introduce hit utility, the posterior probability that at least one rollout in a proposed additional allocation for a prompt will be correct. Building on this notion, we propose Hit-Utility Optimal Rollout Allocation (HORA), a learning-free rollout allocation policy that maximizes total posterior hit utility within each allocation batch. HORA adaptively reallocates rollout budgets while leaving the downstream reward evaluation and group-based advantage estimator unchanged. Across four mathematical reasoning benchmarks and three model scales, HORA preserves comparable Pass@1 and improves Pass@K over compute-matched GRPO in ten of twelve model--benchmark configurations, with one tie and one saturated exception. It is also drop-in compatible with other group-based estimators such as RLOO. Ablation studies indicate that the uniform prior used by HORA is competitive with five prompt-conditioned learned-prior alternatives.

Reinforcement learning with verifiable rewards (RLVR) is bottlenecked by rollout generation, yet many sampled prompts produce saturated groups (all responses correct or all incorrect) whose zero reward variance yields no policy-gradient signal. Existing remedies either oversample a larger candidate pool and discard saturated prompts (dynamic sampling), paying heavy extra rollouts, or predict prompt difficulty before sampling, which is fragile under a shifting policy. We observe that a group's effectiveness is usually decided early, within the first few of its rollouts, so spending a full group on an already-decided prompt is wasteful. We cast per-step rollout collection as a budget-constrained sequential allocation (optimal stopping) problem and introduce SARA (Sequential Adaptive Rollout Allocation). SARA maintains a Beta posterior over each prompt's success rate, evaluates a closed-form predictor of group effectiveness, and applies a two-threshold, SPRT-style rule that commits effective groups, abandons saturated ones after a short probe, and reallocates the freed budget to fresh prompts, without any extra prediction rollouts. We prove abandonment reliability, expected rollout savings, fixed-budget yield dominance, and a link between effective-group yield and the GRPO gradient norm. On mathematical reasoning and planning with 1.5B/3B models on a single GPU, SARA matches DPS (both below the DS oracle) while using 22% fewer rollouts than DS; composing SARA with DPS yields the best accuracy, slightly above DS, at 67% fewer rollouts (near-uniform cost).

Reinforcement Learning with Verifiable Rewards (RLVR) trains reasoning models without labeled trajectories, relying on grouped rollouts to expose the policy to alternative reasoning paths and a verifier to score them. Rollout diversity has accordingly emerged as a central bottleneck in RLVR, with most existing methods broadening exploration through temperature, prefix, or rollout-selection adjustments. We identify a structurally distinguished but overlooked position for broadening this diversity: the first token after the reasoning marker. The policy's first-token distribution exhibits a sharply peaked yet correctness-decoupled phenomenon, and this first token position can broaden the regions a rollout group covers without altering the correctness signal. We introduce REFT (Rollout Exploration with First-Token Diversification), a light addition to the RLVR pipeline that samples first tokens uniformly from the policy's own top-$N$ candidates and allocates rollouts evenly, leaving every other component unchanged. Trained on the resulting diversified rollouts, REFT improves aggregate Pass@1, Pass@8, and Pass@64 over DAPO and GRPO baselines across four base models (0.5B-7B) and three difficulty regimes.