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arxiv: 2606.30931 · v1 · pith:W27FBLOWnew · submitted 2026-06-29 · 💻 cs.AI · cs.LG· cs.MA· math.OC· math.PR

RoPoLL: Robust Panel of LLM Judges

Anish Acharya , Kris W Pan , Brian Verkhovsky This is my paper

Pith reviewed 2026-07-01 01:30 UTC · model grok-4.3

classification 💻 cs.AI cs.LGcs.MAmath.OCmath.PR
keywords LLM evaluationrobust estimationgeometric medianHuber contaminationjury of judgesbias in LLM judgesByzantine robustness
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The pith

A single biased LLM judge produces unbounded error in any jury panel; the geometric median bounds it at breakdown point 1/2.

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

The paper shows that standard LLM jury consensus (PoLL) incurs unbounded bias under the Huber contamination model as soon as one judge fails in a typical LLM way such as sycophancy or mode collapse. The bias does not vanish with larger juries. RoPoLL keeps the same panel of judges but replaces the aggregation step with the geometric median, a robust estimator that achieves the optimal finite-sample breakdown point of 1/2 and delivers the parametric error rate up to a computational gap. Experiments across 13 judges, multiple benchmarks, and corruption regimes up to 50 percent confirm large gains over PoLL on biased attacks and even let small RoPoLL committees surpass much larger single models under contamination.

Core claim

Under the Huber contamination model, PoLL incurs unbounded bias under any positive contamination whenever a single judge fails in a biased, LLM-typical way. RoPoLL preserves the PoLL panel but replaces the aggregation function with the geometric median, which is tuning-free and attains the optimal finite-sample breakdown point of 1/2. Finite-sample error bounds and matching information-theoretic lower bounds agree on the rate sigma sqrt(d/N) while differing on the breakdown floor by a factor of sqrt(d).

What carries the argument

Geometric median: the point minimizing the sum of distances to the individual judge score vectors, used as the robust aggregator in place of majority vote or mean.

If this is right

  • RoPoLL dominates PoLL on every biased corruption type tested, by roughly 19 percent on cross-dimensional attacks at matched compute.
  • A 3-judge RoPoLL committee at 38B parameters outperforms Mistral-Large-3 at 675B parameters by 1.31 times on HelpSteer-2 under 30 percent bimodal-random corruption.
  • RoPoLL yields orders-of-magnitude improvement over PoLL against heavy-tailed Byzantine adversaries.
  • The finite-sample bound and minimax lower bound match on the parametric rate but leave a sqrt(d) gap on the breakdown floor.

Where Pith is reading between the lines

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

  • The same geometric-median replacement could be applied to other multi-model aggregation tasks such as ensemble decoding or preference tuning.
  • The statistical-computational gap points to a need for faster high-dimensional robust estimators that still achieve breakdown 1/2.
  • The contamination framing suggests testing whether training-data filtering benefits from similar robust aggregation before fine-tuning.

Load-bearing premise

Individual LLM judge errors behave like Huber contamination with the listed biased modes, and the geometric median is the right robust estimator for the consensus score.

What would settle it

A controlled simulation in which one judge is forced into sycophantic mode collapse while the others remain clean, then checking whether PoLL error grows without bound as contamination rate or dimension increases.

Figures

Figures reproduced from arXiv: 2606.30931 by Anish Acharya, Brian Verkhovsky, Kris W Pan.

Figure 1
Figure 1. Figure 1: POLL vs. ROPOLL under heavy-tailed cauchy-far corruption. RMSE vs. per-case corruption rate r (log y-axis) for the MEDIUM jury (N=3, ≈89B), with the best single open￾weight judge as a gray dashed reference; coordinate-wise MEDIAN is competitive with ROPOLL here and is omitted (full three-method comparison in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Naturally-occurring parser-failure rates motivate the contamination model. Horizontal bars per judge (sorted by parameter count, top = smallest) restricted to the 13-judge pool common to both benchmarks (Claude-Opus/Sonnet/Haiku-4.5 are HS 3-only and excluded here for panel alignment; their HS 3 statistics appear in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cross-dimensional corruption (Example 1). Three competent judges (blue dots) cluster around the truth y ⋆ = (2.5, 2.5) in the score box [0, K] 2 with K = 5. A corrupted judge outputs yˆcorr = (0, K): each coordinate individually lies in the plausible range [0, K] (red axis ticks), so any coordinate-wise estimator sees nothing anomalous on either axis. Jointly, however, the corrupted vector lies at ℓ2 dista… view at source ↗
Figure 4
Figure 4. Figure 4: Geometry of Theorem 1. Lemma 2 guarantees that at least (1−α−β)N judge outputs (blue dots) lie inside the cluster ball B(y ⋆ , ρ) of sub-Gaussian radius ρ (solid disk). Lemma 1 then forces the geometric median yˆGM (blue triangle) to lie inside the GM envelope B(y ⋆ , Cα+βρ) (dashed disk), regardless of where the remaining (α+β)N corrupted points (red ×) are placed—this is the distribution-free breakdown p… view at source ↗
Figure 5
Figure 5. Figure 5: Geometric core of Lemma 1. The contradiction hypothesis ∆ ≜ ∥x∗ − z∥2 > Cαr places x∗ outside the cluster ball B(z, r), so the ball subtends a cone of half-angle γ at x∗ with sin γ = r/∆ (right-angle at the tangent point shown). Every cluster point xj ∈ B(z, r) lies inside this cone, hence makes angle γj ≤ γ with the central ray x∗ → z, so cos γj ≥ p 1 − r 2/∆2 = α/(1 − α) when ∆ > Cαr with Cα = (1 − α)/ √… view at source ↗
Figure 6
Figure 6. Figure 6: Modulus of continuity for Theorem 2. Top: total variation between two equal-covariance Gaussians at separation ∆⋆ is 2Φ(∆⋆/2σ) − 1 (dimension-free; depends only on the line through the two centers). The overlap is shaded; when the overlap mass exceeds α/(1−α), the two Huber neighborhoods touch. Bottom: the contamination class Fα(y) = {(1 − α)N (y, σ2 Id) + αQ} is depicted as a cloud of distributions around… view at source ↗
Figure 7
Figure 7. Figure 7: Theory validation. (a) Empirical MSE of the arithmetic mean for N=10 equicorrelated clean Gaussian judges matches the closed-form prediction 1+(N−1)γ N dσ2 (Corollary 1) across the full range γ ∈ [0, 0.95]; the effective jury size Neff = N/(1 + (N−1)γ) saturates at 1/γ, motivating the three-judge committees of §6. (b) Under worst-case Huber contamination (α=0.3, σ=0.3, d=5, Dirac corruption at y ⋆ + 10 1),… view at source ↗
Figure 8
Figure 8. Figure 8: Parameter efficiency of ROPOLL juries vs. individual judges under bimodal-random corruption at r = 30%. RMSE vs. parameter count (log scale) for each dataset; gray circles are the 13 individual open-weight judges (four anchors labelled), dashed line is their log-linear scaling fit, and coloured stars mark the four ROPOLL juries (Medium/Mixed/Small/Tiny) at their aggregate parameter budget—all evaluated und… view at source ↗
Figure 9
Figure 9. Figure 9: Parameter efficiency of ROPOLL juries vs. individual judges under zeros corruption at r = 30%. RMSE vs. parameter count (log scale) for each dataset; gray circles are the 13 individual open-weight judges (four anchors labelled), dashed line is their log-linear scaling fit, and coloured stars mark the four ROPOLL juries at their aggregate parameter budget, all under identical 30% per-case corruption. zeros … view at source ↗
Figure 10
Figure 10. Figure 10: Parameter efficiency at the clean baseline (r = 0%). RMSE vs. parameter count (log scale) for each dataset; gray circles are the 13 individual open-weight judges, dashed line is their log-linear scaling fit, and coloured stars mark the four ROPOLL juries at their aggregate parameter budget. Clean counterpart of Figures 8 and 9. 0.95 1.00 1.05 1.10 1.15 1.20 1.25 zeros RMSE (ropoll) clean (0%) 1.4 1.5 1.6 … view at source ↗
Figure 11
Figure 11. Figure 11: Jury-size ablation: RMSE vs. jury size N. Mean RMSE across sampled N-judge subcommittees from each tier pool, under zeros/inverted/bimodal-random corruption. Left column: r = 0%; right column: r = 30%. Bands show ±1 standard deviation across combinations. 6.6 Jury-Size Ablation and Corruption-Type Dependence Two practical questions remain: how many judges does ROPOLL actually need, and is the geometric me… view at source ↗
Figure 12
Figure 12. Figure 12: POLL vs. MEDIAN vs. ROPOLL degradation curves. RMSE vs. per-case corruption rate r for the MEDIUM jury, one panel per (dataset × corruption type). Solid = ROPOLL, dashed = POLL, dotted = coordinate-wise MEDIAN. falls within the standard-deviation band, confirming the Corollary 1 prediction that the three-judge committees sit at the knee of the cost–accuracy frontier. Corruption-type ablation. Under zeros … view at source ↗
Figure 13
Figure 13. Figure 13: Per-attribute judge-score distributions (log y-axis). HelpSteer 2 and UltraFeedback show heavy mass concentration at the score extremes—parser fallback at 0 and sycophantic saturation at the maximum—which motivates the zeros and inverted corruption types used in §6. HelpSteer 3 (signed-preference scalar) is centered on 0 with light tails, consistent with cancellation of per-attribute biases under the sign… view at source ↗
Figure 14
Figure 14. Figure 14: Inter-judge Pearson correlation heatmaps (lower-triangle, annotated). Pairwise correlations averaged over evaluation attributes; cells labelled with their numeric value. Empirical mean off-diagonal correlations: γ¯HS2 = 0.49, γ¯UF = 0.71, γ¯HS3 = 0.49. These values support the γ ∈ [0.3, 0.5] assumption used in §6.1 to motivate three-judge committees via Corollary 1; the higher γ¯UF explains the smaller RO… view at source ↗
Figure 15
Figure 15. Figure 15: Mode Collapse corruption (Q = δ0). Corrupted judges output the zero vector, modeling parser failures or safety refusals. The mean is pulled linearly toward the origin; at α = 0.40 it lies roughly 40% of the way from y ⋆ to 0. The geometric median remains within the competent cluster because the majority of Euclidean distances still point toward y ⋆ . Inverted (Q = δK·1−y⋆ ). The worst-case anti-correlated… view at source ↗
Figure 16
Figure 16. Figure 16: Inverted corruption (Q = δK·1−y⋆ ). The worst-case Byzantine adversary: corrupted scores are perfectly anti-correlated with the truth. The corruption locus and y ⋆ lie on opposite sides of the score space. At α = 0.30 the mean is already displaced past the midpoint, while the geometric median remains close to y ⋆ . This is the sharpest demonstration of the breakdown-point advantage. 43 [PITH_FULL_IMAGE:f… view at source ↗
Figure 17
Figure 17. Figure 17: Biased Dimension corruption. Corrupted judges evaluate Attribute 1 correctly but catastrophically fail on Attribute 2 (scores collapse near zero). This partial competence is challenging for coordinate-wise methods because the corruption is invisible on one axis. The geometric median, operating on joint Euclidean distances, detects the anomaly in Attribute 2 and downweights the corrupted points across both… view at source ↗
Figure 18
Figure 18. Figure 18: Random hypercube corners (the canonical instance of the cross-dimensional class of Example 1, matching the empirical bimodal-random class of §6.3). Corrupted judges output an extreme vertex of {0, K} d chosen uniformly at random; per-coordinate the corruption marginal 1 2 (δ0 + δK) is plausible scoring, but the joint vector lies far from y ⋆ in ℓ2. The geometric median resists the cross-dimensional pull (… view at source ↗
Figure 19
Figure 19. Figure 19: Sycophantic corruption (Q = Uniform([K−1, K] d )). Corrupted judges produce scores clustered near the maximum, modeling the “everything is great” failure mode. The corrupted cloud sits in the upper-right corner; the mean drifts diagonally toward it while the geometric median stays anchored to the competent majority near y ⋆ . Summary. Across all three failure modes, the arithmetic mean acquires a bias pro… view at source ↗
Figure 20
Figure 20. Figure 20: Per-dimension MAE for each LLM judge on UltraFeedback ( [PITH_FULL_IMAGE:figures/full_fig_p046_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Per-dimension mean bias for each LLM judge on UltraFeedback ( [PITH_FULL_IMAGE:figures/full_fig_p047_21.png] view at source ↗
read the original abstract

The LLM Jury, a Panel of LLM Evaluators (PoLL) reporting consensus scores, has become a practical alternative to single-judge LLM evaluation, yet its statistical behavior remains poorly understood. We formalize the LLM Jury under the Huber contamination model and show that PoLL incurs unbounded bias under any positive contamination, regardless of jury size, whenever a single judge fails in a biased, LLM-typical way (mode collapse, sycophancy, safety refusal). Framing jury consensus as classical robust mean estimation, we propose RoPoLL (Robust Panel of LLM-as-Judge), which preserves the PoLL panel but replaces the aggregation function with a robust mean estimator, instantiated with the geometric median (GM): tuning-free, with the optimal finite-sample breakdown point 1/2. A finite-sample error bound and a matching information-theoretic minimax lower bound agree on the parametric rate sigma*sqrt(d/N) and differ on the breakdown floor by a factor of sqrt(d), a statistical-computational gap that polynomial-time RoPoLL pays relative to the intractable Tukey halfspace median. Across 13 open-weight judges (4B-675B), three reward-model benchmarks, and four corruption regimes at rates up to 50%, RoPoLL dominates PoLL on every biased corruption type: by about 19% on cross-dimensional attacks at matched compute, and by orders of magnitude on heavy-tailed Byzantine adversaries. A 3-judge RoPoLL committee at 38B beats Mistral-Large-3 (675B) by 1.31x on HelpSteer-2 under 30% bimodal-random corruption, an 18x parameter advantage at better accuracy; a Noisy-GT control confirms the premium is paid against biased contamination, not benign imprecision.

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

3 major / 1 minor

Summary. The manuscript formalizes the LLM Jury (PoLL) under the classical Huber contamination model, proves that standard averaging incurs unbounded bias under any positive fraction of biased judges (mode collapse, sycophancy, safety refusal), and proposes RoPoLL which retains the same panel but replaces the aggregator with the geometric median; it supplies a finite-sample error bound of order sigma*sqrt(d/N) together with a matching minimax lower bound (differing by a sqrt(d) factor on the breakdown floor), and reports that RoPoLL dominates PoLL by ~19% on cross-dimensional attacks and by orders of magnitude on heavy-tailed Byzantine corruptions across 13 open-weight judges, three reward-model benchmarks, and four corruption regimes up to 50%.

Significance. If the central claims hold, the work supplies a tuning-free, breakdown-point-1/2 aggregator for LLM panels together with explicit finite-sample and information-theoretic guarantees; the matching parametric rate, the 18x parameter-efficiency result (3-judge 38B RoPoLL vs 675B single model), and the Noisy-GT control that isolates bias from benign noise are concrete strengths that would be useful to the LLM-evaluation community.

major comments (3)
  1. [Abstract] Abstract: the claim that PoLL incurs unbounded bias 'under any positive contamination, regardless of jury size' is derived under the classical Huber mixture (arbitrary contamination independent of the clean data). The listed LLM-typical failures (sycophancy, safety refusal, mode collapse) are structured and input-dependent; this dependence can allow coordinated bias to pull the geometric median even when the contaminated fraction is below 1/2 and can invalidate the sigma*sqrt(d/N) finite-sample bound that assumes unstructured contamination. This assumption is load-bearing for both the motivation and the robustness guarantees.
  2. [Experiments] Experimental regimes (bimodal-random, heavy-tailed Byzantine, cross-dimensional): all four tested corruption models are non-adaptive or fully arbitrary. They therefore do not probe the input-dependent case highlighted above; the reported dominance (19% on cross-dimensional, orders of magnitude on Byzantine) cannot yet be taken as evidence that the same gains hold against the structured biases the paper itself lists as motivation.
  3. [Theoretical analysis] Theoretical bounds paragraph: the finite-sample upper bound and minimax lower bound are stated to agree on the rate sigma*sqrt(d/N) but to differ by a factor of sqrt(d) on the breakdown floor, producing a 'statistical-computational gap' that polynomial-time RoPoLL pays relative to the Tukey median. The manuscript should make explicit whether this gap affects the practical recommendation to use the geometric median or whether the gap is only asymptotic.
minor comments (1)
  1. [Abstract] Notation: the dimension d appears in the rate sigma*sqrt(d/N) but is never defined in the abstract; a one-sentence clarification of what the score vectors live in would help readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major point below, clarifying the scope of our theoretical model and experiments while acknowledging where additional discussion is warranted.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that PoLL incurs unbounded bias 'under any positive contamination, regardless of jury size' is derived under the classical Huber mixture (arbitrary contamination independent of the clean data). The listed LLM-typical failures (sycophancy, safety refusal, mode collapse) are structured and input-dependent; this dependence can allow coordinated bias to pull the geometric median even when the contaminated fraction is below 1/2 and can invalidate the sigma*sqrt(d/N) finite-sample bound that assumes unstructured contamination. This assumption is load-bearing for both the motivation and the robustness guarantees.

    Authors: The manuscript explicitly states that the formalization and unbounded-bias result for PoLL are derived under the classical Huber contamination model (arbitrary, unstructured contamination). The LLM-typical failure modes are presented as motivating examples of biased judges rather than as the precise contamination process used in the proofs. We agree that input-dependent structured attacks lie outside the current guarantees and could in principle affect both the geometric median and the finite-sample bound. We will revise the abstract to explicitly qualify the unbounded-bias claim as holding under the Huber model and add a short limitations paragraph noting that coordinated, input-dependent biases remain an open question for future analysis. revision: partial

  2. Referee: [Experiments] Experimental regimes (bimodal-random, heavy-tailed Byzantine, cross-dimensional): all four tested corruption models are non-adaptive or fully arbitrary. They therefore do not probe the input-dependent case highlighted above; the reported dominance (19% on cross-dimensional, orders of magnitude on Byzantine) cannot yet be taken as evidence that the same gains hold against the structured biases the paper itself lists as motivation.

    Authors: The four corruption regimes were chosen to match the classical Huber setting used in the theory (non-adaptive or fully arbitrary contamination). We acknowledge that these regimes do not directly test input-dependent structured attacks. The reported gains therefore demonstrate robustness under the modeled contamination but cannot be extrapolated to the structured case without further experiments. We will add an explicit limitations statement in the experimental section and a sentence in the conclusion identifying input-dependent attacks as important future work. revision: yes

  3. Referee: [Theoretical analysis] Theoretical bounds paragraph: the finite-sample upper bound and minimax lower bound are stated to agree on the rate sigma*sqrt(d/N) but to differ by a factor of sqrt(d) on the breakdown floor, producing a 'statistical-computational gap' that polynomial-time RoPoLL pays relative to the Tukey median. The manuscript should make explicit whether this gap affects the practical recommendation to use the geometric median or whether the gap is only asymptotic.

    Authors: The gap concerns only the breakdown floor (geometric median achieves 1/2 while the information-theoretic optimum can be slightly higher by a sqrt(d) factor in finite samples); the leading error term sigma*sqrt(d/N) is identical. Because the number of judges N is typically small (3–13) and the geometric median is polynomial-time computable, the gap does not alter the practical recommendation. We will add one clarifying sentence in the theoretical analysis section stating that the gap is primarily of asymptotic interest and does not change the recommendation to use the geometric median for LLM panels. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation applies standard Huber model and geometric median properties directly

full rationale

The paper's central theoretical claims (unbounded bias of the mean under positive Huber contamination, and GM's breakdown point of 1/2 with rate sigma*sqrt(d/N)) are obtained by direct invocation of classical results in robust statistics rather than by fitting parameters to the paper's LLM data, self-citation chains, or redefinition. No load-bearing step reduces to an input by construction; the Huber model and GM estimator are external, independently established objects whose properties are applied to the new LLM-jury setting. Experiments test the instantiated method but do not retroactively define the claimed bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the Huber contamination model and the assumption that LLM failures manifest as biased contamination rather than symmetric noise; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Huber contamination model applies to LLM judge outputs
    Invoked to formalize PoLL bias and to derive the unbounded-bias result.
  • standard math Geometric median achieves breakdown point 1/2
    Standard property of the geometric median used to motivate the choice of aggregator.

pith-pipeline@v0.9.1-grok · 5876 in / 1245 out tokens · 39241 ms · 2026-07-01T01:30:48.776094+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 2 internal anchors

  1. [1]

    (2025).Robust and Efficient Learning in High Dimensions from Noisy Data

    Acharya, A. (2025).Robust and Efficient Learning in High Dimensions from Noisy Data. PhD thesis, The University of Texas at Austin. Acharya, A., Hashemi, A., Jain, P., Sanghavi, S., Dhillon, I. S., and Topcu, U. (2022). Robust training in high dimensions via block coordinate geometric median descent. InInternational Conference on Artificial Intelligence a...

    work page 2025

  2. [2]

    ChatEval: Towards Better LLM-based Evaluators through Multi-Agent Debate

    Boucheron, S., Lugosi, G., and Massart, P. (2013).Concentration Inequalities: A Nonasymptotic Theory of Independence. Oxford University Press. Chan, C.-M., Chen, W., Su, Y ., Yu, J., Xue, W., Zhang, S., Fu, J., and Liu, Z. (2024). Chateval: To- wards better LLM-based evaluators through multi-agent debate.arXiv preprint arXiv:2308.07201. Chen, M., Gao, C.,...

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  3. [3]

    Dudley, R. M. (2002).Real Analysis and Probability, volume 74 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge. El Mhamdi, E. M., Guerraoui, R., and Rouault, S. (2018). The hidden vulnerability of distributed learning in Byzantium. InInternational Conference on Machine Learning, pages 3521–3530. 26 Esary, J. D., Proschan,...

    work page Pith review arXiv 2002

  4. [4]

    Minsker, S

    Springer. Minsker, S. (2015). Geometric median and robust estimation in Banach spaces.Bernoulli, 21(4):2308–

    work page 2015

  5. [5]

    R., and Feng, S

    Panickssery, A., Bowman, S. R., and Feng, S. (2024). LLM evaluators recognize and favor their own generations. InThe Thirty-eighth Annual Conference on Neural Information Processing Systems. Pitt, L. D. (1977). A gaussian correlation inequality for symmetric convex sets.The Annals of Probability, pages 470–474. Rockafellar, R. T. (1997).Convex analysis, volume

    work page 2024

  6. [6]

    Verbosity bias in preference labeling by large language models

    Princeton university press. Saito, K., Wachi, A., Wataoka, K., and Akimoto, Y . (2023). Verbosity bias in preference labeling by large language models.arXiv preprint arXiv:2310.10076. Small, C. G. (1990). A survey of multidimensional medians.International Statistical Review, 58(3):263–277. Stureborg, R., Alikaniotis, D., and Suhara, Y . (2024). Large lang...

    work page arXiv 2023

  7. [7]

    Large Language Models are not Fair Evaluators

    Cambridge university press. Wang, P., Li, L., Chen, L., Cai, Z., Zhu, D., Lin, B., Cao, Y ., Liu, Q., Liu, T., and Sui, Z. (2023). Large language models are not fair evaluators.arXiv preprint arXiv:2305.17926. Wang, Z., Dong, Y ., Delalleau, O., Zeng, J., Egert, G., Zhang, P., Kamalakara, A. S., and Kuchaiev, O. (2024). Helpsteer2: Open-source dataset for...

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  8. [8]

    Sums of continuous convex functions are continuous and convex, so F is continuous and convex

    (i) Existence.Each summand z7→ ∥z− ˆyi∥2 is the Euclidean norm of an affine function ofz, hence continuous and convex (see any standard reference on convex analysis). Sums of continuous convex functions are continuous and convex, so F is continuous and convex. For coercivity, fix any data point ˆy1; by the reverse triangle inequality F(z)≥ ∥z− ˆy1∥2 ≥ ∥z∥...

    work page 2000

  9. [9]

    The corrupted contribution has norm at most|Sc|<|S| , hence the total subgradient has norm at least|S|−|S c|>0 , contradicting the optimality 0∈∂F ′(z′)

    If ∥z′∥2 were unbounded as the adversary varies the corrupted points within their m-coordinate budget, then for the competent points i∈S the unit vectors (z′ − ˆyi)/∥z′ − ˆyi∥2 would all lie in a small cone (all pointing approximately from the bounded competent cluster towardz′), so their sum has norm at least|S|(1−o(1)) . The corrupted contribution has n...

    work page 1991

  10. [10]

    The number of iterations to reach toleranceϵis thereforeO(log(1/ϵ))

    Convergence.Vardi and Zhang (2000) prove that the modified Weiszfeld iteration converges to the unique geometric median at a linear rate whenever the data are not collinear: there exists ρ∈(0,1) depending on the data configuration with ∥z(t) − ˆyGM∥2 ≤ρ t∥z(0) − ˆyGM∥2. The number of iterations to reach toleranceϵis thereforeO(log(1/ϵ)). Cost.Each iterati...

    work page 2000

  11. [11]

    The argument is by contradiction: assume ∥x∗ −z∥ 2 > C αr and derive a violation of the optimality ofx ∗

    We give the proof of Minsker (2015), with the geometric setup made explicit. The argument is by contradiction: assume ∥x∗ −z∥ 2 > C αr and derive a violation of the optimality ofx ∗. For brevity write ∆≜∥x ∗ −z∥ 2 and let F(y)≜ Pk j=1 ∥y−x j∥2 denote the geometric-median objective. Since x∗ minimizes the convex function F on Rd, the one-sided directional ...

    work page 2015

  12. [12]

    corrupted (Zi = 1)

    vs. corrupted (Zi = 1). Conditional on Zi = 0, Assumption 4 states that ϵi ∈R d isσ-sub-Gaussian, i.e. for everyλ∈R d, E exp ⟨λ,ϵ i⟩ Zi = 0 ≤exp 1 2 σ2∥λ∥2 2 .(31) We now show, from (31) alone, Pr ∥ϵi∥2 > σ(C 1 √ d+t) Zi = 0 ≤exp(−c t 2),∀t >0,(32) for absolute constantsC 1, c >0. We prove (32) directly from (31) via a covering-net argument over the unit ...

    work page 2018

  13. [13]

    1 N NX i=1 Wi − 1 N NX i=1 EWi <−u # ≤exp(−2N u 2).(44) Combining (44) with the lower bound (43) on eachEW i: Pr

    Substituting s=c t 2 with c= 1/C 2 2 = 1/8: C2σ√s= C2σ √ ct2 =C 2σ t/C2 =σ t. Hence for allt≥0, Pr ∥ϵi∥2 > C1σ √ d+σt ≤exp(−c t 2),(38) which is exactly (32) with the same absolute constantsC 1 = 2√2 log 5≤4andc= 1/8. Remark on the explicit constants.The covering radius 1/2, net size 5d, and resulting prefactor C1 = 2√2 log 5are not optimized; sharper cha...

    work page 2013

  14. [14]

    The clean-rate term p d/N matches the upper bound exactly

    gives Ω(σ( p d/N+α/(1−α))) . The clean-rate term p d/N matches the upper bound exactly. On the breakdown floor the upper bound scales asCα+βσ √ d while the lower bound scales asσα/(1−α), leaving a gap of order √ d/α. The reason is structural: total variation between two equal-covariance Gaussians is dimension-free (Step 2.2 of the proof of Theorem 2), so ...

    work page 2018

  15. [15]

    For jointly Gaussian competent noise with positive score correlation, the cluster indicators are positively associated by Pitt’s Gaussian correlation inequality (Pitt, 1977; Esary et al., 1967; Joag-Dev and Proschan, 1983), so ¯γW ≥0 ; we are not aware of a clean general upper bound on ¯γW in terms of ¯γalone. In practice, ¯γW can be estimated directly fr...

    work page 1977

  16. [16]

    We invoke Le Cam’s two-point method (Tsybakov, 2009, Sec. 2.4): for any two parameter valuesy 0,y 1 ∈R d inducing observation distributionsF 0, F1 ∈ F α,σ, inf ˆy sup F∈{F 0,F1} EF [∥ˆy−y ⋆∥2]≥ ∥y0 −y 1∥2 4 · 1−TV(F ⊗N 0 , F ⊗N 1 ) .(59) The strategy is to construct (y0,y 1, F0, F1) maximising the right-hand side. Part 1 controls the parametric variance t...

    work page 2009

  17. [17]

    Hence (1−α)P 0 +αQ 0 = (1−α)P 1 +αQ 1 is a common element ofF α(y0)∩ F α(y1), establishing (62)

    = (1−α)(µ + −µ −) + (1−α)(µ − −µ +) = 0, using (63) (the ρ terms cancel). Hence (1−α)P 0 +αQ 0 = (1−α)P 1 +αQ 1 is a common element ofF α(y0)∩ F α(y1), establishing (62). Step 2.2 (Equal-covariance Gaussian TV is dimension-free).The total-variation distance between N(y 0, σ2Id) and N(y 1, σ2Id) depends only on ∆≜∥y 0 −y 1∥2: projecting onto the line y1 −y...

    work page 2018

  18. [18]

    This is not slack in the analysis but a real statistical–computational gap

    scales asCασ √ d while the lower bound scales asσα/(1− α); the gap is a √ d/α factor. This is not slack in the analysis but a real statistical–computational gap. The minimax-optimal estimator on the breakdown floor is the Tukey halfspace median (Tukey, 1975; Donoho and Gasko, 1992), whose exact computation is NP-hard for d≥3 (Johnson and Preparata, 1978; ...

    work page 1975