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One-Bit Distributed Mean Estimation with Unknown Variance

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Pith's one-line read A two-round adaptive protocol achieves the exact asymptotic MSE for one-bit mean estimation with unknown variance, and non-adaptive protocols provably fall behind for many log-concave distributions.

desk verdict The adaptive protocol and its analysis look solid, but the paper's headline claim—strict suboptimality of non-adaptive protocols—rests on an invalid Hellinger bound and the wrong branch of h, so the current version overclaims. read the letter →

arxiv 2501.18502 v4 pith:6PFGRZAG submitted 2025-01-30 cs.IT math.ITmath.STstat.TH

Ritesh Kumar , Shashank Vatedka This is my paper · ORCID
classification cs.ITmath.ITmath.STstat.TH
keywords one-bitcommunicationdistributedmeanestimationunknownvarianceadaptiveprotocolsminimaxMSElog-concavedistributionsgeneralizedGaussiansquaredHellingerdistance
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

The paper settles the leading constant of one-bit distributed mean estimation when the variance is also unknown. It shows that a simple two-round adaptive scheme—first a coarse estimate from a vanishing fraction of users, then a thresholded second round at the estimated mean—achieves asymptotic mean squared error $\sigma^2/(4 f_X(0)^2)$ per user. It proves a matching lower bound for all sequential one-bit adaptive protocols on symmetric strictly log-concave densities whose Fisher-information profile is maximized at the origin, so the scheme is asymptotically minimax. On the non-adaptive side, it gives a universal lower bound $0.1034\,\sigma^2/T(f_X)$; for generalized Gaussian shape parameter $\beta<1.85$ and for the hyperbolic secant this exceeds the adaptive value, so one-shot protocols are provably inferior. The paper's simulations confirm the gap.

What carries the argument

The non-adaptive lower bound is carried by a refined squared-Hellinger analysis. For two nearby means $\mu\pm\epsilon\sigma$, the shifted densities are decomposed into level sets of their pointwise difference, and Theorem B.1 bounds the per-user Hellinger distance between the induced one-bit transcripts by $\epsilon^2(T(f_X)+o(1))$, where $h(x)=2\varphi'(x)f_X(x)$, $h^*=\max_{x\ge0}h(x)$, and $T(f_X)=\int_0^{h^*}\varphi'(h^{-1}(t))h^{-1}(t)\,dt$. Le Cam's two-point method then converts this into the universal constant $0.1034$. The adaptive lower bound uses the Van Trees inequality (the Bayesian Cramér–Rao bound) together with Lemma D.1, which bounds the Fisher information of any single one-bit observation by $(2f_X(0)/\sigma)^2+\delta$; its reciprocal gives $\sigma^2/(4f_X(0)^2)$. The upper bound is quantile inversion at a broadcast threshold.

What would settle it

Evaluate $T(f_X)$ for the standard normal: if $0.1034/T(f_X)\le 1/(4f_X(0)^2)$, the lower bound does not certify a gap for the Gaussian, matching the paper's Figure 10. A decisive test would be to exhibit, for one distribution in the stated class, a non-adaptive threshold protocol whose large-$n$ MSE is below $0.1034\,\sigma^2/T(f_X)$, which would falsify Theorem 3.2.

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Extended reading notes

Core claim

The central claim, in the authors' terms, is that unknown variance changes the comparison between adaptive and non-adaptive one-bit mean estimation in a quantifiable way. Theorem 4.1 gives $\lim_{n\to\infty} n\,\mathrm{MSE}(\hat\mu_f)=\sigma^2/(4f_X(0)^2)$ for the two-round protocol, and Theorem 4.3 shows any sequential adaptive protocol satisfies $\lim n\,\mathbb{E}[(\hat\mu-\mu)^2]\ge\sigma^2/(4f_X(0)^2)$ under a shape condition on $\eta(x)=f_X(x)^2/(F_X(x)F_X(-x))$. Theorem 3.2, by contrast, gives $\lim \sup_\mu n\,\mathrm{MSE}(\hat\mu)\ge 0.1034\sigma^2/T(f_X)$ for every non-adaptive protocol over symmetric strictly log-concave $f_X=e^{-\varphi(x)}$. Comparing $C_{\mathrm{adapt}}=1/(4f_X(0)^2)$ with $C_{\mathrm{non}}=0.1034/T(f_X)$ shows that the non-adaptive floor is strictly higher for generalized Gaussian densities with $1<\beta<1.85$ and for the hyperbolic secant, establishing a positive gap.

Load-bearing premise

The strict-suboptimality comparison stands on the non-adaptive lower bound's regularity conditions—$f_X=e^{-\varphi(x)}$ with $\varphi$ symmetric, differentiable, strictly convex, and polynomially bounded—so for merely log-concave or heavier-tailed distributions (the Laplace is explicitly excluded) the paper proves no gap.

Editorial extensions

If this is right

  • For every distribution satisfying the conditions of Theorem 4.3, the two-round protocol is asymptotically minimax among sequential one-bit adaptive protocols, so no additional rounds can reduce the constant.
  • For generalized Gaussian densities with $1<\beta<1.85$ and for the hyperbolic secant, every non-adaptive one-bit protocol has asymptotic MSE at least $C_{\mathrm{non}}\sigma^2/n$ with $C_{\mathrm{non}}>C_{\mathrm{adapt}}$, giving a provable price for non-adaptivity.
  • The two-threshold non-adaptive scheme also yields a strongly consistent estimator of the scale parameter $\sigma$ with asymptotic MSE of order $1/n$, although the paper does not claim optimality for variance estimation.
  • The nonparametric multi-threshold estimator extends one-bit mean estimation beyond scale-location families, but its MSE is not $O(1/n)$ unless the support is bounded, so parametric structure is what delivers the fast rates.

Reading between the lines

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

  • A sharper distribution-dependent replacement for the universal constant $0.1034$ in Theorem 3.2 could extend the strict gap beyond $\beta<1.85$; the paper already conjectures its non-adaptive lower bound is loose.
  • The per-bit Fisher-information bound suggests a quantitative multi-bit analogue: most of the benefit of adaptivity is captured by the first bit, and extra bits should buy only a limited constant-factor improvement.
  • The layer-cake/Hellinger construction is not tied to scalar thresholds and could plausibly give non-adaptive lower bounds for high-dimensional mean estimation under coordinatewise one-bit constraints, one of the open directions named by the authors.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies one-bit distributed mean estimation for scale-location families with unknown variance. It proposes a two-threshold non-adaptive protocol, a multi-threshold non-parametric estimator, and a two-round adaptive protocol, and derives asymptotic normality and mean-squared-error limits for these schemes. It also gives lower bounds: a Van Trees-based lower bound for adaptive protocols and a Le Cam/Hellinger-based lower bound for non-adaptive protocols. The headline claim is that for generalized Gaussian distributions with 1 < beta < 1.85 and for the hyperbolic secant, the asymptotic minimax MSE of the best non-adaptive protocol is strictly larger than the σ²/(4 f_X(0)²) achieved by the simple adaptive protocol.

Significance. If correct, the paper would establish a clean separation between adaptive and non-adaptive one-bit protocols for a broad class of symmetric log-concave distributions. The adaptive-protocol half is a genuine contribution: the two-round protocol is simple, the claimed optimal constant σ²/(4 f_X(0)²) is plausible, and the lower-bound strategy via Van Trees and per-letter Fisher information bounds is credible. The non-adaptive lower bound is also an interesting approach. However, the main advertised strict-suboptimality gap is currently not supported: the numerical evaluation of T(f_X) in Table 2 uses the wrong branch of the non-monotone function h, and the same error propagates into Figures 3–6 and the claims in Sections 2.2 and 4.3. The paper therefore needs substantial correction before its central claim can be considered established. No code or data are provided, but the appendices contain lengthy proofs.

major comments (3)
  1. [§3.3, Theorem 3.2; §4.3, Table 2; Appendix B] The constant T(f_X) is evaluated on the wrong branch of h(x)=2φ'(x)f_X(x). For GGD beta=1.5, f_X(0)≈0.476, so the threshold encoder Y=1{X≤0} under μ=±ε (σ=1) gives Bernoulli probabilities 1/2 ± f_X(0)ε + o(ε). With the paper's H² convention, this yields H²(P_Y⁺,P_Y⁻) = 2 f_X(0)² ε² + o(ε²) ≈ 0.453 ε². Theorem B.1 therefore forces any valid T(f_X) to be at least ≈0.453, but Table 2 reports T=0.040. The resolution is that h is non-monotone: the proof's x_a(ε)=sup A_ε(a) selects the decreasing (right) branch of h, while the numerical T in Table 2 corresponds to the increasing (left) branch. Recomputing on the right branch gives T≈1.4 for beta=1.5, so C_non=0.1034/T≈0.074, far below C_adapt=1.1035. The claimed strict suboptimality gap for GGD beta<1.85 and for the hyperbolic secant is therefore not established; it is an artifact of the branch choice. All Table 2 entries need to be recomputed with the branch that is actually used in the proof.
  2. [§3.2, Theorem 3.1 and Appendix A] The asymptotic MSE formulas in Theorem 3.1 are algebraically incorrect. From Eq. (23), sqrt(n_i)(α_i − (θ_i−μ)/σ) converges to N(0, σ_i²/f_i²). Since n_i=K_i n, the corresponding sqrt(n) covariance is σ_i²/(K_i f_i²), not K_i σ_i²/f_i². The covariance matrix Σ in the delta-method step should therefore be diag(σ_1²/(K_1 f_1²), σ_2²/(K_2 f_2²)), and the factors K_1, K_2 belong in the denominator in the final formulas. This invalidates the displayed expressions for n·MSE(μ̂_c) and n·MSE(σ̂_c) and the simulation curves labeled 'Asymptotic (Non-adaptive)' in Section 5. It does not by itself invalidate Theorem 4.1, but it is a correctness error in a stated theorem used for benchmarking.
  3. [Appendix B, Theorem B.1] The proof of Theorem B.1 assumes that the superlevel set A_ε(a) is a symmetric interval and that h^{-1} is single-valued on [0,h^*]. This requires h to be unimodal, which is not implied by the stated assumptions (φ symmetric, differentiable, strictly convex, polynomially bounded). The custom density in Eq. (20), with its 0.0675 sin²(4x/2.023076) perturbation, is an example where h is not obviously single-peaked. If h has multiple local maxima, the equivalence x_a(ε)=h^{-1}(a/(2ε)+o(1)) can fail, and the change of variables that defines T(f_X) is not justified. The theorem statement should either add a unimodality condition on h or the proof should be extended to handle non-unimodal h.
minor comments (4)
  1. [Appendix A, Eq. (23)] The second line of Eq. (23) writes sqrt(n_2)(α_1 − ...) but should refer to α_2.
  2. [Appendix H, Lemma C.1] The displayed bound in Lemma C.1 contains '+exp(η log n_3/3)' inside the second factor. With the plus sign the right-hand side diverges as n_3→∞, so the lemma would not imply the claimed convergence. Lemmas H.1 and H.2 have a minus sign in the same place, suggesting a typo, but the statement should be corrected.
  3. [Appendix D, proof of Theorem 4.3] The sentence 'Using the fact that equation 58 holds for every δ>0' cites Eq. (58), which is the final lower bound; the intended reference appears to be the bound in Lemma D.1.
  4. [Notation, Theorem 3.1] The notation σ_i² in Theorem 3.1 collides with the true scale parameter σ; a different symbol, such as v_i, would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bounds are derived from external benchmarks and explicit asymptotic calculations, with no fitted parameter renamed as a prediction.

full rationale

The load-bearing claims are self-contained against external benchmarks. The adaptive achievable MSE in Theorem 4.1 follows from the delta method, the Berry–Esseen bound, and an in-paper concentration lemma (Lemma C.1); the adaptive lower bound in Theorem 4.3 uses the Van Trees inequality and borrows the Fisher-information lemma H.3 from Kipnis & Duchi, which is an external source, not the present authors. The non-adaptive lower bound in Theorem 3.2 uses Le Cam's two-point method and the squared Hellinger bound Theorem B.1, whose constants are derived from explicit functional integrals: alpha* is the maximum of the closed-form t(1 - sqrt(1 - exp(-2t))) and T(f_X) is a distribution-dependent integral, neither of which is fitted to simulation output. The only self-citation (Kumar & Vatedka 2023) appears in related-work context and does not carry any proof step. No equation is defined in terms of the quantity it is used to establish, and no fitted constant is later relabeled as a prediction. In particular, the numerical values in Table 2 are evaluations of the stated closed-form functionals, not calibrated parameters. A possible technical flaw in the branch of h^{-1} used to evaluate T(f_X) would be a correctness concern about the proof of Theorem 3.2, but it is not a circularity: an erroneous bound is not a definitional equivalence or a fit masquerading as a prediction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central results rest on standard information-theoretic inequalities and technical regularity conditions on the base density f_X. No new physical entities are introduced, and the only hand-picked constants appear in a purely illustrative custom density example.

free parameters (1)
  • Example density shape constants (1.48, 2.023076, 0.5, 0.0675, 4) = hand-designed constants
    Used only to construct the custom strictly log-concave example in Eq. 20 so that it has unit variance. They affect Table 1 but are not fitted to data and do not influence the general theorems.
assumptions (4)
  • domain assumption f_X is symmetric, differentiable, strictly log-concave, and phi is upper bounded by a polynomial (Theorems 3.2, B.1).
    This regularity is required for the refined squared Hellinger bound and for T(f_X) to be finite and meaningful.
  • domain assumption eta(x) = f_X(x)^2 / (F_X(x) F_X(-x)) is non-increasing in |x| and uniquely maximized at 0 (Theorem 4.3).
    Ensures that the Fisher information from one-bit messages is maximized when the quantization threshold is placed at the true mean, a key step in Lemma D.1.
  • domain assumption The prior density g on mu has bounded expected squared score E[(g'/g)^2] (Theorem 4.3).
    Required for the Van Trees inequality to give a non-vacuous lower bound as n goes to infinity.
  • standard math Standard asymptotic tools: CLT, delta method, Berry-Esseen, Le Cam's method, Van Trees inequality, and the Bhattacharyya coefficient factorization.
    These are used throughout the proofs and are not in dispute.

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Cite this review

Pith. "Pith review of One-Bit Distributed Mean Estimation with Unknown Variance." pith.science (2026). https://pith.science/paper/6PFGRZAG

@misc{pith2026250118502,
  author       = {Pith},
  title        = {Pith review of: One-Bit Distributed Mean Estimation with Unknown Variance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PFGRZAG}},
  note         = {Machine review of arXiv:2501.18502}
}
read the original abstract

In this work, we study the problem of distributed mean estimation with 1-bit communication constraints when the variance is unknown. We focus on the setting where each user has access to one iid sample drawn from a distribution belonging to a location-scale family, and is limited to sending just a single bit of information to a central server whose goal is to estimate the mean. We propose simple non-adaptive and adaptive protocols and show that both achieve asymptotic normality. We derive bounds on the asymptotic (in the number of users) Mean Squared Error (MSE) achieved by these protocols. For a class of symmetric log-concave distributions, we derive matching lower bounds for the MSE of adaptive protocols, establishing the optimality of our scheme. Furthermore, we develop a lower bound on the MSE for non-adaptive protocols that applies to any symmetric strictly log-concave distribution, using a refined squared Hellinger distance analysis. Through this, we show that for many common distributions, including a subclass of the generalized Gaussian family, the asymptotic minimax MSE achieved by the best non-adaptive protocol is strictly larger than that achieved by our simple adaptive protocol. We also demonstrate that increasing the number of bits per user can only marginally reduce the asymptotic MSE of adaptive protocols. Our simulation results confirm a positive gap between the adaptive and non-adaptive settings, aligning with the theoretical bounds.

Figures

Figures reproduced from arXiv: 2501.18502 by the authors.

Figure 1
Figure 1. Illustration of our non-adaptive protocol. We partition the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Our two-round adaptive protocol: The first [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Illustration of bounds on 1 σ2 limnÑ8 MSEpµˆq for the generalized Gaussian family parameterized by β ą 1. Here, Cnon denotes the lower bound on this quantity for non-adaptive protocols, whereas Cadapt denotes the corresponding upper bound for our simple adaptive protocol. As β increases, Cnon decreases while Cadapt increases. The curves intersect at β « 1.8488. This allows us to conclude that non-adaptive protocols … view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: Worst-case and average (over µ) MSE across four source distributions under one-bit protocols. Benchmarks are computed using fXp0q for each family; adaptive and non-adaptive curves are shown according to the legend. The curve labeled Asymptotic (Non-adaptive) represents…
Figure 6
Figure 6. Figure 6: Non-adaptive MSE for symmetric, strictly log-concave distributions with different [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Adaptive MSE across four symmetric, strictly log-concave distributions. Worst-case closely follow the [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: Adaptive MSE across four symmetric, strictly log-concave distributions with the special split [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: MSE versus the mean µ for the generalized Gaussian source with shape parameter β “ 1.50 and unit variance, using n “ 40000 users. Thresholds are fixed at θ1 “ ´0.8333 and θ2 “ 0.8333, and µ is varied over r´2.5, 2.5s. 0 8000 16000 24000 32000 40000 Total Number of User…

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    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.