RSPO: Risk-Seeking Policy Optimization for Pass@k and Max@k Metrics in Large Language Models

Haipeng Sun; Hong Dingqian; Hongfei Jiang; Hui Xiong; Junwei Bao; Kaichen Zhang; Shenghao Gao; Yang Song; Yuzhong Hong

REVIEW 2 major objections 2 minor 1 cited by

RSPO: Risk-Seeking Policy Optimization for Pass@k and Max@k Metrics in Large Language Models

T0 review · 2 major / 2 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that directly optimizing risk-seeking evaluation metrics like Pass@k and Max@k with unbiased gradient estimators improves LLM post-training over maximizing expected reward.

desk verdict Real problem, plausible direction, but the submission is unreadable and the unbiasedness claim carries a tractability burden the abstract never addresses. read the letter →

arxiv 2508.01174 v1 pith:IVHD2WKQ submitted 2025-08-02 cs.LG cs.AI

Kaichen Zhang , Shenghao Gao , Yuzhong Hong , Haipeng Sun , Junwei Bao , Hongfei Jiang , Yang Song , Hong Dingqian

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Hui Xiong

This is my paper

classification cs.LGcs.AI

keywords Risk-seekingpolicyoptimizationPass@kMax@kmetricsLLMpost-trainingunbiasedgradientestimatorhitchhikingproblem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Standard post-training for large language models maximizes expected reward, but evaluation measures risk-seeking metrics such as Pass@k (whether any of k sampled responses succeeds) and Max@k (the best reward among k samples). The paper argues that this mismatch leaves performance on the table and proposes RSPO, which optimizes the evaluation metric itself. RSPO uses the closed-form probability that a given response is the maximum among k samples to build gradient estimators that are unbiased and avoid reinforcing low-reward responses that happen to ride along with a good one, a failure the paper calls hitchhiking. The claim is that optimizing this metric directly yields better post-training performance than risk-neutral objectives.

What carries the argument

The load-bearing object is the closed-form probability that a response is the maximum among k samplings. Given a reward function $r$, this probability acts as an assignment weight in a policy-gradient update: each sampled response contributes to the gradient in proportion to the chance that it is the best of its batch, not in proportion to raw reward. The mechanism converts a non-differentiable max or success indicator into an expectation over the policy that can be differentiated, yielding unbiased nested gradient estimators for both Pass@k and Max@k.

What would settle it

On a small bandit-style task with known rewards, compute RSPO's gradient estimate for a fixed policy and compare it with an exhaustive Monte Carlo estimate of the true Pass@k gradient. If the difference exceeds the estimator's standard error across many seeds, the unbiasedness claim is false. Similarly, with a reward function corrupted by increasing noise, RSPO should lose its advantage if the max-probability computation is the reason it works.

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Extended reading notes

Core claim

The central claim is that Pass@k and Max@k can be treated as differentiable training objectives rather than evaluation-only statistics. For a set of k responses sampled from the policy, the paper writes the probability that a specific response attains the maximum reward, and uses this probability to weight the gradient. This weighting makes the gradient estimator unbiased while suppressing the hitchhiking effect, in which poor responses receive reward because they co-occur with a high-reward response. The paper supports the construction with theoretical analysis and experiments.

Load-bearing premise

The method assumes that the probability that a given response is the best of k samples can be computed from the policy in closed form, differentiated, and estimated without bias; if nested sampling dependencies or reward noise break that estimate, the unbiasedness claim collapses.

Editorial extensions

If this is right

Training and evaluation become aligned: a model tuned at a given k is optimized for exactly the Pass@k or Max@k score that benchmarks report.
The hitchhiking problem disappears from credit assignment, so optimization effort shifts toward responses that actually win batches.
Because the gradient estimators are unbiased, RSPO can in principle be dropped into existing policy-gradient post-training pipelines without changing the update rule's target.
Matching train-time k to evaluation-time k should matter most on tasks with high response variance, such as code generation and open-ended reasoning.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same max-probability weighting could be applied to any sample-based metric that is a function of the best outcome, such as worst-case reward or quantile rewards, by swapping the event being conditioned on.
The benefit of RSPO should depend on reward-noise levels: noisier reward signals blur which response is truly maximal, so one testable prediction is that RSPO's advantage over expected-reward training shrinks as reward noise grows.
The authors leave implicit that the choice of k at training time creates a new hyperparameter; a natural follow-up is scheduling k from small to large during training.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The first thing to know: this submission is unreadable. The full text came through as garbled mojibake, so the equations, the experiments, and the related-work discussion cannot be checked. That forces an unverifiable verdict, not a verdict on the science. The abstract is legible, and it makes a claim that deserves a serious look.

What the paper has going for it: the problem is real. LLM post-training mostly optimizes expected reward while evaluation uses pass@k and best-of-k, and that risk-preference mismatch is a known practical annoyance. The hitchhiking problem (low-reward responses getting reinforced because they co-occur with a high-reward response in a batch of k) is a genuine failure mode, and naming it cleanly is a useful contribution. If RSPO really delivers unbiased, low-variance gradients for these metrics, it would be a meaningful addition to the post-training toolbox.

Now the soft spots. The central theoretical claim (unbiased gradient estimators via the closed-form probability that a response is the maximum among k samples) has a load-bearing subtlety. That probability is proportional to p_theta(y) times F_theta(r(y))^(k-1), where F_theta is the reward CDF under the policy. For an autoregressive LLM, F_theta(r) is a sum over the entire response space; it is not closed-form in any usable sense, and differentiating it requires differentiating that whole sum. If the paper instead estimates F_theta from the k in-batch responses, the estimator is biased and the claimed equivalence to directly optimizing Pass@k and Max@k collapses. The abstract does not say which route is taken, and the corrupted text prevents checking. This is a concrete correctness risk, not an exposition nit.

Also: no code or data accompany the submission, so the comprehensive experimental results cannot be verified either, and without a legible related-work section I cannot tell whether the formulation is new relative to best-of-k optimization or risk-seeking RL.

Who this is for: people doing LLM post-training who care about aligning training objectives with evaluation metrics. The motivation is sound; the method's key property is unverified in this file. My recommendation: do not desk-reject it. Ask the authors for a readable PDF plus the proofs, and ideally code or a small worked example establishing the estimator's unbiasedness and variance. Then send it to a referee who knows RL and can check the derivation. The idea deserves referee time; this particular file does not yet provide the evidence to judge it.

Referee Report

2 major / 2 minor

Summary. The paper proposes Risk-Seeking Policy Optimization (RSPO), a post-training objective for large language models that directly optimizes Pass@k and Max@k evaluation metrics. The abstract claims that RSPO addresses the 'hitchhiking' problem—where low-reward responses are reinforced if they co-occur with high-reward responses in a batch of k samples—by leveraging the closed-form probability that a given response is the maximum among k samplings, and that this yields efficient, unbiased gradient estimators for both metrics. The abstract also promises rigorous theoretical analysis and comprehensive experimental results, including a derivation and empirical tables. The body of the submitted manuscript, however, is severely corrupted and unreadable: the text is garbled, equations are absent or illegible, and tables contain no interpretable numeric entries. Consequently, the theoretical derivation, the unbiasedness proof, and the experimental support for the central claims cannot be checked from the submitted version.

Significance. If the method worked as claimed, it would be a valuable contribution: directly optimizing the evaluation metrics Pass@k and Max@k rather than a risk-neutral proxy is a meaningful goal, and explicitly addressing the hitchhiking problem is a relevant practical concern for LLM post-training. The significance is, however, conditional on the correctness of the unbiased gradient estimators and on the experimental evidence, neither of which is verifiable in the current submission. No code, formal proof, or reproducible experimental protocol is provided.

major comments (2)

[Abstract] The claim of unbiased gradient estimation for Max@k requires differentiating the CDF F_theta(r)^k, whose gradient contains a term proportional to (k-1) F_theta(r)^{k-2} ∇_theta F_theta(r), with ∇_theta F_theta(r) = Σ_{y: r(y) ≤ r} p_theta(y) ∇_theta log p_theta(y) summed over the entire response space. For autoregressive LLMs this sum is not closed-form in any computationally usable sense. The submitted text does not show how this term is handled; if it is replaced by a finite-sample or softmax approximation, the estimator is biased and the central claim fails. Please provide the exact expression and an unbiasedness proof, or explicitly state the approximation and its bias.
[Full text (all sections)] The body of the manuscript is unreadable: the text is garbled and corrupted, equations are missing or illegible, and the experimental tables contain no interpretable values. The abstract's promises of 'rigorous theoretical analysis' and 'comprehensive experimental results' therefore cannot be verified. This is a load-bearing verifiability failure: a reader cannot check the derivation, the unbiasedness proof, or the empirical gains without a readable manuscript.

minor comments (2)

[Full text (header/footer)] The body includes the unrelated line 'arXiv:2508.01178v1 [cs.SD] 2 Aug 2025'; this appears to be a formatting artifact and should be removed in any resubmission.
[Abstract] The phrase 'closed-form probability that a given response is the maximum among k samplings' is potentially misleading: the probability itself may be expressible in closed form, but its gradient with respect to policy parameters involves a sum over the response space and is not closed-form in a practically tractable sense. The wording should be clarified to distinguish the probability from its gradient.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: RSPO's objective and gradient estimator are defined directly from Pass@k and Max@k, with no fitted parameter or self-citation chain serving as the load-bearing input.

full rationale

The available abstract defines the RSPO objective directly in terms of the evaluation metrics Pass@k and Max@k, and the claimed contribution is that the method optimizes these metrics directly rather than a risk-neutral proxy. The closed-form probability that a given response is the maximum among k samplings is a mathematical expression of the same metric definition, so using it in the gradient estimator is a derivation from the target, not an input secretly equivalent to the output. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' prior work is invoked to force the choice, and no ansatz is smuggled in through self-citation. The full text is garbled in the provided material, which prevents independent verification of the theoretical proofs, but an inability to check a proof is a correctness risk, not evidence of circularity. Under the rule that circularity must be demonstrated with a specific reduction in the paper's own equations or citations, none can be exhibited here.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

Only abstract-level assumptions can be identified from the legible material. The full text would likely add assumptions about the reward model, baseline, and sampling procedure, but these sections are corrupted and unreadable.

assumptions (2)

domain assumption Responses are sampled i.i.d. from the training policy and scored by a fixed scalar reward model.
Pass@k and Max@k are defined over k independent samples, and the gradient derivation depends on this sampling model. The abstract refers to 'k samplings'.
domain assumption The closed-form probability that a response is the maximum among k samples is computable and differentiable with respect to policy parameters.
The abstract says RSPO leverages this probability. If the reward distribution is unknown, noisy, or changes during training, this assumption is strong and central to the estimator.

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Cite this review

Pith. "Pith review of RSPO: Risk-Seeking Policy Optimization for Pass@k and Max@k Metrics in Large Language Models." pith.science (2026). https://pith.science/paper/IVHD2WKQ

@misc{pith2026250801174,
  author       = {Pith},
  title        = {Pith review of: RSPO: Risk-Seeking Policy Optimization for Pass@k and Max@k Metrics in Large Language Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IVHD2WKQ}},
  note         = {Machine review of arXiv:2508.01174}
}

read the original abstract

Current large language model post-training optimizes a risk-neutral objective that maximizes expected reward, yet evaluation relies heavily on risk-seeking metrics like Pass@k (at least one success in k trials) and Max@k (maximum reward across k responses). This mismatch in risk preferences can inevitably lead to suboptimal performance. To bridge this gap, we propose Risk-Seeking Policy Optimization (RSPO), a novel method that directly targets Pass@k and Max@k during training. A key challenge in optimizing these metrics is the "hitchhiking" problem: low-reward responses are inadvertently reinforced if they co-occur with a high-reward response within a sample of k generations, resulting in inefficient optimization. RSPO addresses this problem by leveraging the closed-form probability that a given response is the maximum among k samplings. Despite the complexity of nested gradients over multiple responses, RSPO produces efficient, unbiased gradient estimators for both metrics. We validate our approach with both rigorous theoretical analysis and comprehensive experimental results.

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