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arxiv: 2605.23362 · v1 · pith:TC4WMJGLnew · submitted 2026-05-22 · 💻 cs.LG · cs.IT· math.IT· math.ST· stat.ML· stat.TH

Instance-Optimal Estimation with Multiple LLM Judges on a Budget

Junghyun Lee , Sanghwa Kim , Yassir Jedra , Alexandre Prouti\`ere , Se-Young Yun This is my paper

Pith reviewed 2026-05-25 05:23 UTC · model grok-4.3

classification 💻 cs.LG cs.ITmath.ITmath.STstat.MLstat.TH
keywords budgeted estimationmulti-judge evaluationinstance optimalityinverse-variance weightingadaptive allocationLLM evaluationheteroskedastic estimation
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The pith

An adaptive algorithm using optimistically biased variance estimates matches the oracle inverse-variance weighted estimator rate for multi-judge LLM score estimation under a fixed budget.

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

The paper formalizes budgeted heteroskedastic multi-judge estimation, where a fixed budget must be allocated across prompt-response pairs and judges that differ in cost and reliability to minimize error in a bounded score vector. It first derives the oracle allocation that minimizes error for the inverse-variance weighted estimator when variances are known. For the realistic case of unknown variances, it introduces EST-IVWE, which constructs optimistically biased variance estimates to produce a stable empirical allocation. The central result is that EST-IVWE attains the oracle error rate up to lower-order budget terms, and this performance is instance-optimal by a matching local minimax lower bound proved via an Assouad-type argument that preserves local variance structure. A sympathetic reader cares because LLM evaluations are expensive and heterogeneous; a provably near-optimal allocation directly improves accuracy for any given spend.

Core claim

The central claim is that EST-IVWE, an adaptive procedure that builds and uses optimistically biased variance estimates, matches the error rate of the oracle inverse-variance weighted estimator up to lower-order terms in the budget. A matching local minimax lower bound, obtained via an Assouad-type in-expectation argument based on local perturbations, establishes that the proposed algorithms are instance-optimal. This bound is sharper than what Fano-type packing arguments can deliver because the latter lose the local variance information that determines the optimal allocation.

What carries the argument

The inverse-variance weighted estimator (IVWE) whose error is minimized by an oracle allocation depending on unknown query-judge variances; EST-IVWE extends this to the unknown-variance case by constructing optimistically biased variance estimates that stabilize empirical allocation without rate loss.

If this is right

  • EST-IVWE attains the oracle IVWE error rate up to lower-order budget terms even when variances are unknown.
  • A local minimax lower bound shows the achieved rate is instance-optimal for each fixed variance configuration.
  • The Assouad-type argument based on local perturbations yields an allocation-dependent lower bound that Fano-type arguments cannot recover.
  • Numerical comparisons on synthetic data and HelpSteer2 confirm lower error than uniform allocation under the same budget.

Where Pith is reading between the lines

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

  • The same optimistic-bias stabilization technique may extend to other budgeted allocation problems where measurement costs and noise levels are heterogeneous and initially unknown.
  • Local-perturbation lower-bound constructions could be applied to other estimation settings where global packing arguments erase the structure that governs optimal resource use.
  • The instance-optimality result implies that uniform or non-adaptive allocations are provably suboptimal on instances with strong variance heterogeneity.

Load-bearing premise

The adaptive algorithm can construct and leverage optimistically biased variance estimates to stabilize the empirical allocation without degrading the final estimator's rate.

What would settle it

On synthetic instances where the true variances are known, if the squared error of EST-IVWE exceeds the oracle IVWE error by more than lower-order terms in the budget for large enough budgets, the rate-matching claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.23362 by Alexandre Prouti\`ere, Junghyun Lee, Sanghwa Kim, Se-Young Yun, Yassir Jedra.

Figure 1
Figure 1. Figure 1: Experimental results on both synthetic and real-world datasets. All results are averaged [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Additional Experimental results on synthetic datasets. All results are averaged over [PITH_FULL_IMAGE:figures/full_fig_p050_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Full experimental results for the datasets (Complexity, Correctness, Helpfulness, Verbosity). Each column corresponds to an error metric ( [PITH_FULL_IMAGE:figures/full_fig_p052_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Impact of Cost Structure on Performances of [PITH_FULL_IMAGE:figures/full_fig_p053_4.png] view at source ↗
read the original abstract

Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as *budgeted heteroskedastic multi-judge estimation*. Given $K$ prompt-response pairs, $J$ judges with known costs, and unknown query-judge variances, the goal is to estimate a bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its error rate. Since this allocation depends on the unknown variances, we then address the practical unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and leverages *optimistically biased* variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a matching *local* minimax lower bound, which establishes the instance-optimality of the proposed algorithms. A key technical insight is that Fano-type high-probability arguments are too coarse for this problem: their packing construction loses the local variance structure that governs the optimal allocation. We instead use an Assouad-type in-expectation argument, based on local perturbations, which preserves this structure and yields the sharp allocation-dependent lower bound. Finally, we numerically validate the superiority of our approach over na\"ive uniform allocation on synthetic and HelpSteer2 datasets.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper formalizes budgeted heteroskedastic multi-judge estimation for LLM evaluations with heterogeneous judge costs and instance difficulties. It analyzes the inverse-variance weighted estimator (IVWE), derives its oracle allocation minimizing ℓ_p error under a fixed budget, and proposes the adaptive EST-IVWE algorithm that uses optimistically biased variance estimates to handle unknown variances while matching the oracle rate up to lower-order terms. It establishes instance-optimality via a matching local minimax lower bound derived from an Assouad-type in-expectation argument with local perturbations (avoiding coarse Fano packings), and validates the approach empirically against uniform allocation on synthetic data and the HelpSteer2 dataset.

Significance. If the central claims hold, the work delivers a practically relevant, instance-optimal framework for cost-efficient LLM-as-a-judge scoring that adapts to per-instance and per-judge variance heterogeneity. The local minimax lower bound that preserves the variance structure governing optimal allocation, together with the explicit adaptive procedure for unknown variances, constitutes a technical contribution beyond standard inverse-variance weighting. The empirical results on HelpSteer2 further support applicability.

major comments (2)
  1. [EST-IVWE algorithm and its analysis] The claim that EST-IVWE matches the oracle IVWE rate up to lower-order terms (abstract) rests on the optimistic bias construction stabilizing allocation without degrading the leading 1/sqrt(B) constant. The bias must be strong enough to avoid unstable allocations on high-variance instances yet weak enough that the resulting estimator retains the exact oracle leading term; an explicit bias definition and concentration argument showing the bias term is o(1/sqrt(B)) are required to confirm this.
  2. [Local minimax lower bound section] The Assouad-type local-perturbation argument is presented as yielding the sharp allocation-dependent lower bound. The specific local perturbation construction and the in-expectation calculation that retains the per-instance variance structure (rather than averaging it away) should be verified to ensure the lower bound exactly matches the oracle upper bound's leading constant.
minor comments (2)
  1. Clarify the precise meaning of 'lower-order terms in the budget' (e.g., whether o(1/sqrt(B)) or O(log B / sqrt(B)) is intended) and state the dependence on K, J, and p explicitly.
  2. The synthetic data generation process and the precise definition of the ℓ_p error metric used in the experiments should be described in more detail to allow reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight opportunities to strengthen the explicitness of our technical arguments, and we address each point below. We are prepared to revise the manuscript to incorporate additional details where needed.

read point-by-point responses

  1. Referee: [EST-IVWE algorithm and its analysis] The claim that EST-IVWE matches the oracle IVWE rate up to lower-order terms (abstract) rests on the optimistic bias construction stabilizing allocation without degrading the leading 1/sqrt(B) constant. The bias must be strong enough to avoid unstable allocations on high-variance instances yet weak enough that the resulting estimator retains the exact oracle leading term; an explicit bias definition and concentration argument showing the bias term is o(1/sqrt(B)) are required to confirm this.

    Authors: We agree that the optimistic bias construction merits a more explicit treatment to confirm the leading constant is preserved. In the revision we will add a dedicated subsection (or lemma) in Section 4 that (i) states the precise bias term (a multiple of the estimated standard deviation scaled by a slowly growing function of the number of samples per instance), (ii) proves that the resulting allocation deviates from the oracle allocation by an o(1/sqrt(B)) term in total variation with high probability, and (iii) shows via a direct calculation that this deviation contributes only lower-order terms to the final ℓ_p error. This will make the matching claim fully rigorous. revision: yes

  2. Referee: [Local minimax lower bound section] The Assouad-type local-perturbation argument is presented as yielding the sharp allocation-dependent lower bound. The specific local perturbation construction and the in-expectation calculation that retains the per-instance variance structure (rather than averaging it away) should be verified to ensure the lower bound exactly matches the oracle upper bound's leading constant.

    Authors: The local perturbation is constructed by adding an independent Rademacher perturbation of size Θ(1/σ_{k j}) to each instance-judge mean, with the scale chosen small enough to remain inside the bounded score interval. The in-expectation lower bound is obtained by linearity of expectation over the independent sign flips; because each coordinate's contribution appears separately in the total risk and the variance of the estimator for that coordinate is exactly the reciprocal of the total weight allocated to it, the per-instance variance structure is retained and the resulting lower bound matches the leading 1/sqrt(B) term of the oracle upper bound. We will insert a short clarifying paragraph after the main proof in Section 5 that spells out this coordinate-wise calculation. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations rely on independent technical arguments

full rationale

The paper analyzes the inverse-variance weighted estimator to derive an oracle allocation, then proposes EST-IVWE using optimistically biased variance estimates to match the oracle rate up to lower-order terms, with a matching local minimax lower bound obtained via a new Assouad-type in-expectation argument based on local perturbations. No load-bearing step reduces by construction to its inputs, fitted parameters renamed as predictions, or self-citation chains; the central technical insight (preserving local variance structure in the lower bound) is presented as novel and independent of the algorithm definition. This matches the expectation that most papers are non-circular when the proof techniques are self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions about bounded scores and heteroskedastic unknown variances; the main technical work is in the adaptive procedure and lower-bound construction. No free parameters or invented entities are introduced beyond the problem setup itself.

axioms (2)
  • domain assumption The target score vector is bounded.
    Explicitly stated as part of the estimation goal in the problem formulation.
  • domain assumption Judges have known per-query costs but unknown query-judge variances.
    Core modeling choice that defines the heteroskedastic multi-judge setting.

pith-pipeline@v0.9.0 · 5873 in / 1368 out tokens · 38573 ms · 2026-05-25T05:23:08.078375+00:00 · methodology

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