Hindsight-Privileged RLVR: Single-Rollout Token Credit Without Group Sampling

• GRPO requires multiple attempts per prompt, but homogeneous outcomes can leave every attempt with zero reward advantage.Additional sampling therefore incurs cost without necessarily supplying an informative comparison.Source: §1 · p. 1
• Outcome-based GRPO broadcasts a trajectory-level advantage across tokens, while intermediate decisions can play different roles in reasoning and interaction.The introduction contrasts this coarse credit with approaches that use intermediate labels, learned process rewards, or additional continuations.Source: §1 · p. 1
• HP-RLVR uses the generating policy to compare token distributions with and without the realized future and verifier feedback, weighting historically centered reward with bounded, nonnegative normalized JSD.It uses one rollout per prompt and needs neither a reference solution nor a learned critic; failures can receive negative credit.Source: §1 · p. 2, Figure 2 · p. 2
• The evaluation covers Qwen3 models at 1.7B, 4B, and 8B on mathematics, code generation, and ALFWorld, assessing accuracy and training-signal diagnostics.Figure 1 summarizes Qwen3-8B cost at a shared prompt and step budget using one eighth as many rollouts per prompt.Source: Figure 1 · p. 1, §1 · p. 2
• At a shared step schedule, HP-RLVR reports lower training cost; Figure 1 compares Qwen3-8B with G=1 against GRPO with G=8.The evaluation also considers accuracy and training-signal diagnostics.Source: Figure 1 · p. 1, §1 · p. 2

2.1 Related work

• Hindsight credit assignment revisits past decisions using observed future information; HP-RLVR constructs weights from full-vocabulary divergence between two views of the generating snapshot.The discussion distinguishes this from value estimation, reward redistribution, process supervision, and methods using additional continuations or repeated states.Source: §2.1 · pp. 2–3
• HP-RLVR uses unsigned divergence to weight reward-based updates on failures without a corrective reference, unlike H2SD's use of reference hints to correct failures.The paper also notes that update dynamics depend on weighting, loss normalization, and length dynamics.Source: §2.1 · p. 3

2.2 Problem formulation

• RLVR models a verifier-assigned reward for a policy-generated response or interactive trajectory, with policy scores contributed only by generated tokens.For a parameter-independent verifier and environment, the stated score-function gradient is E[R∇θ log πθ(y | x)].Source: §2.2 · p. 3
• GRPO assigns a group-relative advantage to each trajectory and applies that same advantage to every generated token; homogeneous groups yield zero advantages.The formulation defines σR as the group reward standard deviation and εstab as a stabilizer for the denominator.Source: §2.2 · p. 3

3.1 From trajectory rewards to token credit

• A historical baseline lets a single on-policy trajectory provide an unbiased estimator of the unweighted reward gradient under the stated assumptions.The baseline is the mean reward of earlier trajectories; the current batch is added only after its advantages are constructed.Source: §3.1 · p. 3

3.2 Hindsight-conditioned scoring

• The generating policy scores each token under its causal history and under hindsight context from the same completed trajectory, without sampling extra continuations.The hindsight scorer receives the allowed future and verifier report; the causal view does not.Source: §3.2 · pp. 3–4

3.3 Outcome-centered token credit

• Normalized Jensen–Shannon divergence weights each token’s centered reward, producing bounded credit whose sign follows the reward relative to baseline.For scored positions, credit is zero when reward equals baseline or the causal and hindsight distributions agree; otherwise divergence sets its relative magnitude within a trajectory.Source: §3.3 · p. 4

3.4 Training objective and reduction

• The PPO-style surrogate uses eligible generated tokens and averages valid tokens within each trajectory before averaging across trajectories.Only the current policy receives gradients, and training uses one optimization epoch with ϵ = 0.2.Source: §3.4 · pp. 4–5

4.1 Experimental setup

• Paired Qwen3-1.7B, 4B, and 8B runs use seed 42, shared training settings, 100 rollout-and-training steps, task-specific verifiers, and one training seed per method.Mathematics and code use sequence-level verification; ALFWorld uses terminal environment success.Source: §4.1 · pp. 5–6

4.2 Cost of replacing group sampling

• At a shared four-prompt budget and 100 rollout steps, Qwen3-8B HP-RLVR reduced end-to-end training time across three task families, with GRPO-to-HP-RLVR ratios of 1.21–2.30, using four versus 32 trajectories per step.The comparison measures cost at a fixed prompt and step budget, not equal trajectory counts or matched accuracy.Source: §4.2 · p. 6

4.3 Task accuracy under single-rollout training

• Accuracy varies by task: HP-RLVR has higher mathematics Avg@12 at all sizes, while GRPO leads code Pass@1 and Pass@8 and ALFWorld success at all sizes.Mathematics training responses are capped at 1,024 tokens, and approximately 42% of the 4B GRPO responses reach that cap.Source: §4.1 · p. 5, §4.3 · p. 6, Table 3 · p. 8

4.4 Training-signal availability and controls

• In ALFWorld, zero-gradient records are 43.2% (4B) and 49.0% (8B) for GRPO, versus none for HP-RLVR; 47.7% of 4B mathematics reward groups are homogeneous.Homogeneous reward groups measure uninformative comparisons, whereas logged gradient records measure current-gradient availability.Source: §4.4 · p. 8, Figure 4 · p. 8

5 Conclusion

• 1.21×, 1.46×, and 2.30× faster training than G=8 GRPO on mathematics, code, and ALFWorld, respectively, uses one eighth as many rollouts per prompt.The runs share a 100-step budget; accuracy is comparable overall.Source: §5 · p. 8
• HP-RLVR is ahead on mathematics at all three model sizes, while GRPO is ahead on code and ALFWorld.These task-specific accuracy results qualify the overall comparable-accuracy finding.Source: §5 · p. 8
• 43.2% (4B) and 49.0% (8B) of GRPO’s ALFWorld gradient-norm records are zero, versus none for HP-RLVR.HP-RLVR’s historical reward centering sets update direction, while bounded, nonnegative JSD weights allocate token-credit magnitude.Source: §5 · p. 8

Limitations

• Experiments cover Qwen3 models and text tasks with terminal verification.Other model families, multimodal feedback, and longer training contexts are natural extensions.Source: Limitations · p. 9

A Notation and Preliminaries

• Table 4 summarizes HP-RLVR and GRPO notation, while symbols used only in proofs are defined where introduced.The table is titled “Core notation.”Source: Table 4 · p. 11, A Notation and Preliminaries · p. 11

B Proofs

• The proof sequence establishes historical-baseline centering before the divergence properties used for local allocation.The first result is an expectation identity for the unweighted trajectory score; the second applies to each realized pair of scoring distributions.Source: B Proofs · p. 11

B.1 Historical-baseline identity

• The historical-baseline identity assumes a fixed completed history and a matched on-policy rollout, with parameter-independent task and verifier factors and stated regularity conditions.The current baseline is fixed before the rollout; under these assumptions, a single sampled estimator is conditionally unbiased for the unweighted reward gradient.Source: B Proofs · p. 11, §B.1 · p. 12

B.2 Zero divergence

• JSD is zero exactly when the two scoring distributions are equal, including distributions with zero probabilities.The proof applies nonnegativity to the two divergences from their mixture.Source: §B.2 · p. 12

B.3 Bounded credit

• |Aj| is bounded by 1, with equality only when reward and baseline are opposite extremes and token-distribution supports are disjoint.Overlapping supports imply a divergence weight below 1.Source: §B.3 · p. 12
• Aj is zero exactly when reward equals the historical baseline or the causal and hindsight distributions match.Source: §B.3 · p. 13
• A zero Aj makes that position’s credit-weighted objective term and gradient zero, but the converse need not hold.Clipping or a zero policy score can also yield a zero gradient when Aj is nonzero.Source: §B.3 · p. 13
• The historical-centering identity does not establish unbiasedness of the final surrogate, and divergence measures predictive disagreement rather than causal responsibility.Detaching the weight does not remove its statistical dependence on the rollout; masking, length normalization, and clipping also modify the update.Source: §B.3 · p. 13

C.1 Sampling and scoring details

• For each prompt, the old policy snapshot samples one trajectory (G = 1), which receives its verifier reward.Source: §C.1 · p. 13
• Section 4.1 reports generation-time recording, temperature convention, replay check, and memory footprint of the exact divergence; evaluation follows each benchmark’s specified sampling protocol.Source: §C.1 · p. 13
• Both scoring views use the same detached generating snapshot; the outcome descriptor O and eligibility rule for F_j are specified in Section 3.2.Source: §C.1 · p. 13

D.1 Setup Details

• Qwen3-4B GRPO mathematics records 16 prompts per step and 128 generated sequences, with global batch 128; Qwen3-8B timing runs use four-prompt accounting.Source: §D.1 · p. 13
• ALFWorld uses a global batch setting of 32 turn rows, which is not an environment-trajectory budget.Source: §D.1 · p. 13
• Per-step trajectory counts differ between the timing runs and the longer evaluation runs.Source: §D.1 · p. 13

D.2 Paired comparisons at Qwen3-1.7B

• Code uses eight completions per problem on the full 1,055-problem LiveCodeBench set, with Pass@1 and Pass@8 computed from the same samples.Source: §D.2 · p. 13
• At 1.7B, HP-RLVR leads mathematics best observed three-benchmark score (38.90% vs 37.80%) and Pass@12 (66.67% vs 65.56%), while GRPO leads code Pass@1 and Pass@8 at all three sizes.The contrast limits the accuracy claim to the tested task and model setting; both metrics use temperature 0.2.Source: §D.2 · p. 13

E.1 Current run timings and signal records

• Completed Qwen3-8B timings appear in main-text Table 1, but the 4B timing profile is not a completed paired 100-step run.Source: §E.1 · p. 13
• Table 5 preserves Fig. 4’s logged-record denominators, and its fractions count all logged gradient records rather than distinct optimizer steps.Source: Table 5 · p. 14

F LLM Usage Disclosure

• The authors used an LLM for writing support, related-work discovery, plotting code for Fig. 1, and consistency checks; they report that research decisions and numerical results were their own and verified against run artifacts.Source: F LLM Usage Disclosure · p. 14