More Bang for the Buck: Improving the Inference of Large Language Models at a Fixed Budget using Reset and Discard (ReD)

Barak Hirshberg; Noam Levi; Sagi Meir; Shlomi Reuveni; Tommer D. Keidar

REVIEW 3 major objections 5 minor 1 cited by

A simple query policy—reset after each failed attempt, discard solved questions—provably increases the number of distinct tasks an LLM solves for any fixed inference budget, regardless of the shape of pass@k.

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2026-08-03 06:55 UTC pith:465NOZE3

load-bearing objection Good idea, real theory, overbroad claim: ReD's 'any budget' guarantee fails on a simple finite-pool counterexample, but the renewal mapping is worth publishing after a careful revision. the 3 major comments →

arxiv 2601.21522 v2 pith:465NOZE3 submitted 2026-01-29 cs.LG cond-mat.dis-nncs.AIstat.ML

More Bang for the Buck: Improving the Inference of Large Language Models at a Fixed Budget using Reset and Discard (ReD)

Sagi Meir , Tommer D. Keidar , Noam Levi , Shlomi Reuveni , Barak Hirshberg This is my paper

classification cs.LG cond-mat.dis-nncs.AIstat.ML MSC 60K0560K10

keywords coverage@costpass@kreset and discardinference budget allocationrenewal theorypower-law scalingLLM inferenceverifiable tasks

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that a simple query policy—after a fixed number of failed attempts at a question, move on to the next question, and drop questions as soon as they are solved—strictly increases the number of distinct LLM-solved problems that a fixed inference budget buys, for any difficulty mix and regardless of the shape of pass@k. It proves that resetting every attempt is the optimal reset interval, derives the exact mapping from pass@k to coverage@cost using renewal theory, and shows that the empirically common power-law decay of 1−pass@k turns solve-to-completion coverage growth sublinear, while ReD restores linear growth. It also shows how to predict the attempt, token, and USD savings from pass@k alone and how to estimate the power-law exponent from ReD runs without expensive large-k evaluation. If correct, this gives practitioners a provable, deployable allocation rule that converts more of an inference budget into solved problems.

Core claim

The central claim is that Reset-and-Discard (ReD) improves coverage@cost—the expected number of unique problems solved after a fixed total number of attempts—for any budget and any underlying difficulty distribution, and that the optimal resetting protocol is to reset after every attempt (τ=1). Under the empirically observed power-law behavior 1−pass@k ∝ k^{-α} with 0<α<1, the standard solve-to-completion policy has divergent mean time per question and yields sublinear coverage growth; ReD makes the mean time finite and coverage grow linearly. Given pass@k, the paper provides an exact closure (Eq. 6 plus Eq. 10) predicting ReD's coverage@cost and the resulting attempt savings; without pass@k

What carries the argument

A renewal-process representation of coverage@cost: solving questions is a renewal process whose inter-arrival distribution is T, the attempts needed for one question, whose CDF equals pass@k. The paper leverages the Z-transform and Tauberian theorems to extract asymptotic coverage growth, and the resetting CDF F_τ(t) = 1−(1−F(τ))^n (1−F(u)) to re-derive coverage under deterministic resetting every τ attempts. The proof of optimality of τ=1 reduces to a covariance inequality: E[T_{τ+1}]−E[T_τ] is non-negative because p(1−p)^τ/(1−(1−p)^τ) is strictly decreasing in p. For exponent inference, the surviving-difficulty distribution after n ReD rounds converges to Beta(α, β+n), giving the ratio ide

Load-bearing premise

The proofs assume every question has a fixed, independent success probability and a perfect verifier; if repeated attempts on a question are correlated, or if verification can be wrong, the predicted coverage gains and the optimality of resetting every attempt do not follow.

What would settle it

Run ReD (τ=1) versus solve-to-completion on a task where the model retains memory of previous attempts or uses self-correction, so attempts are correlated; if coverage@cost no longer improves, the i.i.d.-attempt assumption is the load-bearing premise. Alternatively, inject controlled verifier error rates into a code benchmark and check where the predicted attempt savings stop matching the measured curve.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

If 0<α<1, standard solve-to-completion delivers sublinear coverage growth; ReD converts it to linear growth, a qualitative improvement at any finite budget.
Given an empirical pass@k curve, the paper's formulas predict exact attempt/token/USD savings from ReD before running it, enabling budget planning.
Resetting every attempt (τ=1) is strictly optimal for any difficulty distribution, so practitioners need not tune the reset interval.
ReD runs provide a statistically efficient estimator of the inference power-law exponent α, avoiding costly direct measurement of pass@k at large k.
Empirically, on coding benchmarks, ReD reaches a target coverage with fewer attempts, tokens, and dollars than solve-to-completion across all tested models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if attempts on the same question are correlated (self-correction, backtracking, memory), the independence-based renewal analysis may break; a natural extension is to test whether some reset interval other than 1 becomes optimal under such memory.
Editorial inference: ReD's difficulty-flattening dynamics double as an active-learning filter for self-improvement pipelines; it could preferentially harvest solvable trajectories, but this would need verification beyond the paper's coverage experiments.
Editorial inference: the paper's finite-dataset approximation (replacing t(n) by its average) predicts ReD's gains will hold for larger pools; this is testable on bigger benchmarks than HumanEval.
Editorial inference: the reported USD advantage conflates model price and verbosity; a practitioner could combine ReD with output-length control to amplify the savings further, but that is outside the paper's controlled setup.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Good idea, real theory, overbroad claim: ReD's 'any budget' guarantee fails on a simple finite-pool counterexample, but the renewal mapping is worth publishing after a careful revision.

read the letter

The thing to know: Reset-and-Discard is a simple, practical allocation policy with a genuine theoretical backbone, but the flagship claim — resetting strictly improves coverage@cost at any budget for any difficulty distribution — is not what the proofs show, and it is false for finite question pools. The paper deserves a serious referee, but it needs major revision before it can be accepted.

What is actually new: they map pass@k to coverage@cost via renewal theory, which is a legitimate and useful connection. The renewal model gives a clean explanation of why solve-to-completion has diminishing returns when the per-question success distribution has a power-law tail with alpha < 1. ReD, especially with tau = 1, makes the mean renewal time finite and converts sublinear coverage growth to linear growth asymptotically. That is a real result. The extension to inferring the power-law exponent from ReD runs is useful, and the HumanEval experiments show the advantage holds on average for three models.

Now the soft spots. The stress-test counterexample is on target: with two questions, p = (0.6, 0.4), and a budget of four attempts, solve-to-completion gives expected coverage about 1.58 while ReD with tau = 1 gives 1.48. So the 'any budget' pointwise claim is simply false in the finite-pool setting, which is the setting of the paper's own experiments. Theorem 4.1 only proves E[T_tau] is minimized at tau = 1 — that is a statement about the asymptotic slope of coverage@cost, not about dominance at every finite budget. The finite-size equations (13)--(17) are approximations that average over fluctuations, and the paper acknowledges this in Sec. 4.3 but continues to use the unqualified language. This needs to be fixed, not softened in a footnote.

Also, the abstract as submitted claims experiments on GSM8K and MMLU-Pro, but the paper only reports HumanEval. That is a straightforward inconsistency. If those results exist, report them; if not, remove them from the abstract.

The review should also push for code and data release. The paper says the code will be posted on GitHub, but for an empirical paper of this kind, the reviewers should see the actual pass@k matrices and the generation protocol.

Overall: the renewal mapping and the ReD policy are worth building on. The theory as stated is overbroad, and the empirical base is a single benchmark. With a corrected theorem statement, an honest finite-size discussion, and the missing benchmarks either removed or added, this could be a solid contribution to the inference-scaling literature.

Recommendation: send it to peer review, but with a clear request from the editor to address the finite-pool counterexample and the abstract mismatch. It is not a desk reject, and it is not an accept as is.

Referee Report

3 major / 5 minor

Summary. The paper proposes Reset-and-Discard (ReD), a query-allocation policy for LLM inference under a fixed budget. ReD cycles through questions, giving each at most τ attempts, discarding solved questions, and resetting unsolved ones to the back of the queue. The authors connect pass@k to coverage@cost using renewal theory, claim that resetting strictly improves coverage@cost for any budget and any difficulty distribution, and that τ=1 is optimal. They derive asymptotic growth laws, give an approximate finite-pool prediction, and validate the method on HumanEval with three LLMs, also proposing a method to infer the pass@k power-law exponent from ReD runs. The central theoretical result is that resetting minimizes the mean number of attempts per solved question; the empirical results show ReD improves coverage for the tested models and budgets.

Significance. If the claims were fully established, the paper would provide a simple, practical scheduling rule for LLM inference with quantifiable savings, plus a method to estimate inference scaling exponents from cheaper ReD runs. The renewal-theoretic setup is standard but well applied, and the connection between pass@k and coverage@cost is a useful framing. The paper responsibly states its perfect-verifier assumption and acknowledges the finite-pool approximation. However, the headline universal claim—ReD improves coverage@cost for any budget and any difficulty distribution—is false as stated for finite pools, which is the setting of the paper's own experiments. The proof in App. B only establishes a mean-completion-time inequality, not pointwise dominance of coverage@cost at every finite budget. The empirical demonstration on HumanEval is suggestive but does not rescue the unqualified theoretical claim. The paper's contribution is therefore real but needs substantial revision in its claims and scope.

major comments (3)

[§1, §4.2, Theorem 4.1, App. B] The abstract and §1 claim ReD 'increases coverage@cost for a given budget' and 'strictly improves coverage@cost for any budget and any underlying difficulty distribution.' This is false for finite question pools. Counterexample: N=2, p1=0.6, p2=0.4, τ=1, budget t=4. For solve-to-completion with random order, expected coverage is (1−0.4^4)+[0.6(1−0.6^3)+0.24(1−0.6^2)+0.096·0.4] = 1.6368 if q1 is first, and 1.5328 if q2 is first; averaging over orders gives 1.5848. ReD with τ=1 gives each question exactly two attempts, so expected coverage is (1−0.4^2)+(1−0.6^2)=1.48. Thus solve-to-completion beats ReD at this budget. App. B proves only E[Tτ] ≤ E[T] (mean completion time), which is an asymptotic-slope statement, not pointwise dominance of coverage@cost. The unqualified finite-budget claim must be removed or replaced by a precisely stated condition.
[§4.3, Eqs. (13)–(17)] The finite-pool prediction is an approximation that replaces the random total attempt count t(n) by its mean and neglects fluctuations in Rn and tn. The paper acknowledges this, but the counterexample above shows the discrepancy is structural, not merely a fluctuation: the finite-pool process changes the pool composition after discarding, so the renewal model with i.i.d. questions does not apply. The paper should state explicitly that the 'any budget' improvement is an infinite-pool/large-budget asymptotic result, and that the finite-pool version is only an approximation validated empirically on specific datasets, not a theorem.
[Abstract vs. §6] The abstract (as provided) claims experiments on 'coding (HumanEval), math (GSM8K), and reasoning (MMLU-Pro)', but Section 6 describes experiments only on HumanEval. No GSM8K or MMLU-Pro results appear in the main text or appendices. This discrepancy must be resolved—either the abstract is overclaiming, or the experiments are missing from the manuscript.

minor comments (5)

[§4.1, Eq. (10)] The notation n=floor(t/τ) and u=t−nτ is used without defining the range of u; specify that 0≤u<τ and that t is an integer number of attempts. Also clarify that Eq. (10) relies on the infinite-pool assumption where each reset draws an independent question.
[§5.3, Eq. (22)] The inference method assumes P0(p)=Beta(α,β) exactly, not merely the small-p power-law form used elsewhere. This is an additional modeling assumption and should be stated more prominently; the empirical validation on one model (llama-3.1-8b-instant) is limited.
[Fig. 3] The fit is truncated at the 15th round because of noise, but the choice of cutoff is not justified quantitatively. Please describe the criterion (e.g., Rn threshold) and show sensitivity of the inferred α to the cutoff.
[§2, References] The reference 'Tang et al. (2024)' is incomplete; full author list and venue are needed.
[§6] The paper states 'All the code used to generate the results of this paper will be posted on GitHub' but no link is provided. For reproducibility, include a URL or an anonymized repository link.

Circularity Check

0 steps flagged

No significant circularity: the pass@k-to-coverage mapping is a renewal-theoretic identity and the ReD optimality theorem is proved in the appendix; self-citations are standard restart results and not load-bearing.

full rationale

The paper's derivation chain is self-contained. Eq. (6) is the standard renewal equation expressing coverage@cost as the renewal function of F(t)=pass@t; this is a mathematical identity under the stated i.i.d. per-attempt model, not a fit or a definition of the target in terms of the input. Eqs. (10)-(11) restate known sharp-restart formulas for the CDF and mean of Tτ, and App. B rederives the key optimality E[Tτ]≤E[Tτ+1] from these formulas using the Chebyshev sum inequality, so the τ=1 claim does not reduce to a self-citation. The α-inference of Sec. 5 is an estimator derived from the survival curve Pn(p); its validation compares with a direct high-k pass@k fit on the same results matrix, which is a consistency check rather than a fitted-input-called-prediction. Self-citations to Eliazar & Reuveni, Pal & Reuveni, and Levi/Kazdan are bibliographic support for standard restart identities and the empirical power-law motivation; none carries the proof of the central theorem. The unqualified 'any budget' improvement claim in Sec. 1 and Sec. 4.2 is not actually established by the asymptotic-slope proof and is false for finite question pools (a simple N=2 counterexample suffices), but this is a correctness/scope gap, not circularity: the claim does not reduce to its inputs. The paper also explicitly scopes its model to a perfect verifier and independent attempts in Sec. 7, further clarifying that the renewal derivation is conditional rather than definitionally circular.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No new physical or architectural entities are posited; ReD is a policy, coverage@cost is a metric. The 'beta distribution' and 'power-law exponent' are modeling constructs, not entities with independent falsifiable handles. No free parameters are required for the core theoretical claims; the only fitted value is the estimated exponent α in Sec 5, which is an output of the method, not an input.

axioms (4)

domain assumption Each question has a fixed per-attempt success probability p_i, drawn i.i.d. from a population distribution P(p); attempts are independent.
Eq (1) in Sec 3.1 defines pass@k as an average over this distribution; all renewal derivations rely on i.i.d. completion times T^(i).
domain assumption Perfect verifier.
Stated in Sec 3.1 ('we will assume a perfect verifier') and repeated in Sec 7 limitations; if the verifier is imperfect, discarding 'solved' questions can remove unsolved ones.
domain assumption Infinite question pool / negligible probability of re-encountering a question, with finite-N approximations that neglect fluctuations.
Eq (8)-(12) use Tτ for a fresh question each reset; Sec 4.3 replaces t(n) by ⟨t(n)⟩, ignoring fluctuations in Rn and tn.
ad hoc to paper Small-p power-law form P(p) ≃ c p^{α-1}, and, for the exponent-inference section, P0(p) = Beta(α,β) exactly.
Eq (2) derives 1-pass@k ~ k^{-α}; Sec 5.3 assumes a Beta prior to derive the linear relation Eq (22), 'neglecting the initial relaxation phase.' This is an empirical ansatz, not derived from first principles.

pith-pipeline@v1.3.0-alltime-deepseek · 18305 in / 15885 out tokens · 158368 ms · 2026-08-03T06:55:33.899759+00:00 · methodology

0 comments

read the original abstract

The performance of large language models (LLMs) on verifiable tasks is usually measured by pass@k, the probability of answering a question correctly at least once in k trials. At a fixed budget, a more suitable metric is coverage@cost, the average number of unique questions answered as a function of the total number of attempts. We connect the two metrics and show that the empirically-observed power-law behavior in pass@k leads to a sublinear growth of the coverage@cost (diminishing returns). To solve this problem, we propose Reset-and-Discard (ReD), a query method of LLMs that increases coverage@cost for a given budget, regardless of the pass@k form. Moreover, given a pass@k, we can quantitatively predict the savings in the total number of attempts using ReD. If pass@k is not available for the model, ReD can infer its power-law exponent. Experiments on three LLMs across coding (HumanEval), math (GSM8K), and reasoning (MMLU-Pro) benchmarks demonstrate that ReD substantially reduces the required attempts, tokens, and USD cost to reach a desired coverage, while also offering an efficient way to measure inference power-laws. ReD's advantage is maintained for imperfect verifiers and outperforms the tested allocation baselines.

Figures

Figures reproduced from arXiv: 2601.21522 by Barak Hirshberg, Noam Levi, Sagi Meir, Shlomi Reuveni, Tommer D. Keidar.

**Figure 1.** Figure 1: coverage@cost normalized by the total number of questions, N = 164, against cumulative number of attempts for ReD and the standard (solve-to-completion) evaluation protocol for three models: (Left) llama-3.1-8b-instant, (Middle) llama-3.3-70b-versatile, and (Right) gpt-oss-20b. The dashed lines and shaded regions represent the mean and standard deviation over random realizations, respectively. The dashed-d… view at source ↗

**Figure 2.** Figure 2: coverage@cost normalized by the total number of questions, N = 164, of llama-3.1-8b-instant (standard, solve-to-completion versus ReD) compared to llama-70b-versatile and gpt-oss-20b (standard, solve-to-completion only), measured by (Left) Attempts, (Middle) Tokens, and (Right) USD cost. The dashed lines and shaded regions represent the mean and standard deviation over random realizations, respectively. 0 … view at source ↗

**Figure 3.** Figure 3: The ratio −⟨Rn⟩/⟨Rn+1 − Rn⟩ as a function of the round number n for llama-3.1-8b-instant, where Rn is the number of unsolved problems at round n. abling a shift from “Pay-per-Token” to “Outcome-as-aService.” Currently, users of commercial APIs bear the economic risk of model failure, paying for generations regardless of their validity. By utilizing ReD as a backend allocation strategy, similar to hierarc… view at source ↗

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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A review arguing that restarting random processes ("stochastic resetting") is a unified framework for predicting, inferring, and designing non-equilibrium dynamics in chemistry.

Reference graph

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