Variance estimation in compound decision theory under boundedness

Authors: subhodh  kotekal
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-02 17:22:42

Abstract: The normal means model is often studied under the assumption of a known variance. However, ignorance of the variance is a frequent issue in applications and basic theoretical questions still remain open in this setting. This article establishes that the sharp minimax rate of variance estimation in square error is $(\frac{\log\log n}{\log n})^2$ under arguably the most mild assumption imposed for identifiability: bounded means. The rate-optimal estimator proposed in this article achieves the optimal rate by estimating $O\left(\frac{\log n}{\log\log n}\right)$ cumulants and leveraging a variational representation of the noise variance in terms of the cumulants of the data distribution. The minimax lower bound involves a moment matching construction.

Main paper: 
 introduction: 
 Consider the prototypical normal means model in compound decision theory,EQUATIONfor 1 ≤ i ≤ n, where the means µ = (µ 1 , ..., µ n ) ∈ R n and the noise level σ > 0 are parameters. As set out in the groundbreaking work of Robbins [47, 48, 67] , the empirical Bayes setting is closely related,EQUATIONwhere the prior G is treated as an unknown parameter in contrast to a fully Bayesian approach. The empirical Bayes and compound decision theory literatures, motivated by the advent of scientific technologies (e.g. microarrays) enabling large-scale, parallel experiments, have witnessed tremendous methodological successes [18, 15, 67, 27, 19, 20, 44, 26, 24, 62, 65, 12, 21, 34, 55] and deep theoretical developments [66, 54, 23, 68, 31, 30, 46, 49, 33, 2, 8, 9, 45, 52] .In the Gaussian contexts of (1) and (2), existing theoretical work has largely focused on mean estimation or hypothesis testing when the variance σ 2 is known. The goal is to produce an estimator or a test and evaluate its performance with respect to the benchmark achieved by an oracle (termed oracle Bayes in [18, 31] ) having access to information not available to the statistician (e.g. the prior G in (2) or the empirical distribution 1 n n i=1 δ µi in (1)). For estimation under square loss, the oracle estimator is µ * i = T * (X i ) whereEQUATION38th Conference on Neural Information Processing Systems (NeurIPS 2024).is the posterior mean, under the prior µ ∼ G in (2) or µ ∼ 1 n n i=1 δ µi in (1) , and is known as Tweedie's formula [16] . Here, f denotes the density of the marginal distribution of X and the second term in (3) is known as the Bayes correction. Tweedie's formula requires knowledge of the prior in order to compute f and is thus available only to the oracle.Broadly, there are two common approaches to mimicking the oracle [17] . Typically, the statistician either estimates the prior (known as g-modelling) or directly estimates the marginal distribution f (known as f -modelling), and plugs in the resulting estimate into Tweedie's formula (3) . Both approaches have been taken in the literature [66, 3, 2, 46, 28, 51] , with the nonparametric maximum likelihood estimator (NPMLE) [33, 31, 34, 65, 51] being an especially popular method for g-modelling; regret bounds have been established under various assumptions on the class of priors.Not much of the literature addresses the case of an unknown variance. The dearth of results is certainly not from want of motivation; estimation of the variance is necessary for mimicking the oracle estimator (3) , construction of confidence intervals, hypothesis testing, and estimation of the signal-to-noise ratio among other statistical tasks. Rather, variance estimation has been recognized as a challenging problem in a wide range of high-dimensional and nonparametric models [61, 50, 35, 13, 11, 60, 29, 36, 4, 42] . A few articles have investigated a heteroskedastic version of (1) or (2) with unknown variances [40, 41, 25] , but all of these articles essentially assume access to multiple i.i.d. observations per mean, thereby enabling direct estimation of each variance. The situation in (1) and ( 2) is quite different in that only one observation per mean is available; it is an honest sequence setting. Despite its clear importance, the fundamental limit of variance estimation in (1) and ( 2) has not been established in the literature. This article investigates variance estimation from a minimax perspective. We will work in the compound decision setting (1) , and so our results will also directly hold in the empirical Bayes setting (2) . For the development of the minimax theory, the following parameter space will be considered,EQUATIONwhere L > 0. We will focus on the case where L > 0 is some large universal constant, and we will notationally suppress it by writing Θ to refer to (4) . Furthermore, the choice of 1 in the bound ||µ|| ∞ ≤ 1 is not essential; our results essentially go through when it is replaced by some other universal constant. The boundedness conditions are imposed to avoid triviality. Without boundedness, the minimax risk is easily seen to be infinite, inf σ sup µ∈R n ,σ>0 E( σ2 -σ 2 2 ) = ∞, since the variance is not identifiable. In the empirical Bayes context (2) , the constraint ||µ|| ∞ ≤ 1 corresponds to the condition that G is supported on [-1, 1]; the class of priors with bounded support is a popular choice for study in the empirical Bayes literature [46, 37, 3, 31, 38] . This article's goal is to obtain a sharp characterization of the minimax rate of variance estimation in (1) under square loss over the parameter space (4) . Appendix E contains the notation used in this article.The marginal distribution of the data in the empirical Bayes model (2) can be written as X 1 , ..., X n iid ∼ G * N (0, σ 2 ) where * denotes convolution. The problem of estimating σ 2 is a special case of variance estimation in a semiparametric convolution model [42, 4, 43] , which encompasses generic noise distributions beyond Gaussian; the Fourier transform of the standardized (i.e. unit-scale) noise is assumed known. In their article [4] , Butucea and Matias impose regularity conditions on G by way of assumptions on its Fourier transform. Butucea and Matias point out it is essential G is less smooth than the standardized noise distribution. In the Gaussian case, they assume the Fourier transform Ĝ does not vanish for large frequencies and its modulus does not decay faster than the Fourier transform of the standard Gaussian distribution ω → e -ω 2 /2 . If G may be somewhat smooth in the sense | Ĝ(ω)| ≥ ce -α|ω| r for |ω| sufficiently large where 0 < r < 2, α > 0, and c > 0 is an arbitrary constant, then Butucea and Matias construct a Fourier-based estimator which achieves |σ 2 -σ 2 | 2 ≲ (log n) r-2 with high probability. In the rougher case | Ĝ(ω)| ≥ c|ω| -β for |ω| sufficiently large where β > 1, the faster rate |σ 2 -σ 2 | 2 ≲ (log log n/ log n) 2 is achieved with high probability. Matching lower bounds are also obtained.Though the results of [4] are sharp in their setting, the smoothness assumptions are unappealing and do not imply anything when G is only assumed to have bounded support. In particular, G being supported on [-1, 1] does not imply G must fall into one of the two cases considered by [4] . For example, the Uniform[-1, 1] distribution and the mixture 1 2 δ -1 + 1 2 δ 1 respectively have Fourier transforms ω → sin(ω)/ω and ω → cos(ω), both of which have zeros that are arbitrarily large. Without the condition that G has bounded support, the assumptions of [4] which stipulate | Ĝ(w)| is bounded away from zero for large |ω| are made to ensure identifiability of the variance. However, identifiability is automatically guaranteed when G is assumed to be supported on [-1, 1]. Though the assumptions of [4] can be safely abandoned, it is not clear how, if it all, the approach of Butucea and Matias can be modified without smoothness conditions. Setting this serious issue aside, the keen reader might intuit from an uncertainty principle that if G has a density g which is not identically zero, then the Fourier transform of g cannot decay too fast since it is compactly supported. In particular, it is well known it cannot decay faster than Ce -c|ω| , and so it might be guessed by taking r = 1 in the results of [4] that log -1 n may be optimal.Moving away from smoothness conditions and assuming only that G has bounded support, Matias [42] notices the moment generating function of G * N (0, σ 2 ) is finite everywhere and is given by [63] .M (λ) = exp(λµ)G(dµ) exp(λ 2 σ 2 /2A different perspective regards the marginal distribution X 1 , ..., X n iid ∼ G * N (0, σ 2 ) as a Gaussian mixture model with mixing distribution G. Employing a moment-based approach, Wu and Yang [63] construct an estimator which achieves|σ 2 -σ 2 | 2 ≲ k 2 n -1k with high probability under the assumption G is a k-atomic measure supported on [-1, 1] with k ≲ log n/ log log n. When k is a fixed constant, a matching minimax lower bound is obtained. Their result highlights a rapid deterioration in the convergence rate as k increases. In particular, in the generic case where G need not be atomic (which can be intuited as k → ∞), it might be expected the sharp rate will be logarithmic. Appendix A further discusses the results of [63] .Variance estimation has also been studied under sparsity or regularity assumptions on the means [11, 61, 5, 36] . In (1) under the assumption ||µ|| 0 ≤ k < n 2 , an estimator achieving|σ 2 -σ 2 | ≲ σ 2 k n log -1 (1 + k/ √ n)was constructed and shown to be minimax rate-optimal in [36] (see also [11, 7] ). In the context of (2), faster rates of convergence were obtained by [5] when regularity on the means was assumed in addition to the sparsity. Fourier-based estimators were employed by all of [36, 11, 5] . Optimal estimation with Hölder-type regularity and without sparsity assumptions was obtained by way of kernel smoothing in [61] . These approaches critically exploit the regularity and/or sparsity, and all appear to fail when only boundedness ||µ|| ∞ ≤ 1 is assumed.Beyond the sequence setting of (1) and (2), much recent attention has been directed at variance estimation in the linear regression model Y i = ⟨X i , β⟩ + σZ i where the noise Z i ∼ N (0, 1) and the design X i ∼ N (0, Σ) are independent. If Σ ∈ R p×p is known with bounded operator norm and ||β|| ≲ 1, then Dicker [14] showed the variance can be estimated at rate|σ 2 -σ 2 | 2 ≲ n -1 + pn -2with high probability; notably, consistent estimation is possible even with p ≥ n without any sparsity assumptions on β. Assuming β is k-sparse, [53] (see also [1, 22] ) established the upper boundn -1 +((k log p)/n) 2 .For particular regimes, optimality was established in [60] , and the impossibility of consistent estimation for the dense regime was conjectured in the setting n = o(p) and unknown Σ. Though some approaches were suggested in [13, 29] , variance estimation with unknown Σ was not settled until the impressive work of Kong and Valiant [35] , who showed consistent estimation with n = o(p) samples is surprisingly possible even when k = p. Assuming ||β|| ≲ 1 and Σ has bounded condition number, an estimator can be constructed achieving, with high probability, error |σ 2 -σ 2 | ≤ ϵ with sample complexity n = O(poly(log(1/ϵ))p 1-log -1 (1/ϵ) ). Without a condition number assumption on Σ, consistent estimation was shown to be possible with sample complexity n = O(poly(1/ϵ)p 1- √ ϵ ). Moreover, Kong and Valiant [35] show these complexities are essentially tight. Their estimator is based on a clever polynomial approximation scheme. Though their article also covers certain covariate distributions which are not Gaussian, the sequence settings of (1) and ( 2) are not covered.Our main result is a sharp characterization of the minimax rate of variance estimation over (4) in the compound decision setting (1) ,EQUATIONA few remarks are in order. Remark 1. Though the slow, logarithmic nature of ( 5) is not surprising given the Gaussian mixture model results of [63] discussed earlier, it is notable a rate faster than log -1 n is achievable. The set of means µ in the parameter space (4) have no exploitable structure beyond ||µ|| ∞ ≤ 1, and so intuition suggests to examine the tails of the observations to estimate σ 2 . This thinking may further suggest the least favorable prior might place substantial mass near ||µ|| ∞ ≈ 1, thus suggesting using an extreme order statistic.Though natural, this intuition does not fully use the statistician's knowledge the noise is Gaussian. Conceptually, if the signal distribution looks very different from a Gaussian, then the signal might be easily disentangled from the noise, enabling easy variance estimation. 1 From a lower bound perspective, the least favorable prior for the means should resemble the noise. Indeed, our lower bound argument in Section 3 involves constructing a compactly supported prior which shares a growing number of moments with a particular Gaussian distribution. For technical reasons elaborated on in Section 3, it turns out one can only match O(log n/ log log n) moments, which subsequently yields a lower bound of order (log log n/ log n) 2 for variance estimation.The quantitative constraint on the number of moments which can be matched in the lower bound construction inspires hope it may be possible to estimate at a rate faster than log -1 n, and it encourages developing some kind of method-of-moments estimator. Though such a strategy has seen success in a linear regression setting [35] , complications seem to appear in the sequence model (1) In regression, it is clear the second marginal moment is E(Y 2 1 ) = ⟨β, Σβ⟩ + σ 2 , and so variance estimation is (up to parametric rate) equivalent to estimation of ⟨β, Σβ⟩. By changing focus to this target, σ 2 is now viewed as a nuisance. The key advantage in regression is that the covariates are a source of fresh randomness that is independent of the noise. Since X i is correlated withEQUATIONβ⟩, and so on. In other words, unbiased estimators of the terms ⟨β, Σ k β⟩ n k=2 are easily constructed due to the availability of the random covariates {X i } n i=1 . The clever idea of Kong and Valiant [35] is to reach for polynomial approximation tools and approximate ⟨β, Σβ⟩ by the readily estimable ⟨β, Σ k β⟩ n k=2 . The sequence model (1) is equivalent to linear regression with a deterministic orthogonal design, which is quite different from the random design considered above. The target ||µ|| 2 is the target analogous to ⟨β, Σβ⟩. The only accessible randomness in (1) are the responses themselves{X i } n i=1 . The marginal moments 1 n n i=1 E(X k i ) = 1 n n i=1 E((µ i + σZ i ) k) all involve the nuisance σ, thus precluding an application of Kong and Valiant's idea. The interesting phenomenon that design properties can have substantial effects on minimax rates has been noted in the literature in other settings [11, 6, 59 ], and we suspect this is the cause for the slower rate (5) for variance estimation in the sequence model (1) . Remark 3. Given the central position of the Gaussian sequence models (1) and (2) in compound decision theory and empirical Bayes theory (as well as their status at the core of mathematical statistics more broadly), the cumulant-based variance estimator we propose in this article is designed to exploit the Gaussian character of the noise. It turns out incorporating noise information is, in some sense, essential; in Section 4 it is discussed that noise agnosticism implies the impossibility of consistent variance estimation. Interestingly, the results of Kong and Valiant [35] in the linear regression setting allow for an unknown noise distribution. 2 There appears to be subtle interplay between properties of the design and the noise; a careful study remains an open problem.Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: The abstract claims a minimax rate for variance estimation under the assumption of bounded means, which is the setting addressed in the paper.• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
a cumulant-based estimator: 
 As noted in Remark 1, the variance estimator we will propose is cumulant-based. Cumulants, though perhaps not as familiar, are related to the moments of a random variable. But, in contrast to moments, they have particularly nice attributes for the purpose of variance estimation in (1) and (2) . Some key properties of cumulants are briefly reviewed before the estimation methodology is developed.Suppose Y is a random variable such that its moment generating function M (λ) := E(e λY ) exists for all λ ∈ R. The cumulant generating function is defined by K(λ) := log M (λ), and it admits the power seriesK(λ) = ∞ r=1 κrλ rr! where κ r := K (r) (0) is defined to be the rth cumulant. Though κ 1 and κ 2 coincide respectively with the mean and variance of Y , the higher cumulants do not have a simple relationship with the moments. However, the first r cumulants are determined completely by the first r moments; likewise, the first r moments are determined completely by the first r cumulants. In fact, an explicit correspondence is available via Bell polynomials. Definition 1. The incomplete Bell polynomial B r,l for l ≤ r is given byB r,l (x 1 , ..., x r-l+1 ) = r! j 1 !j 2 ! • • • j r-l+1 ! x 1 1! j1 x 2 2! j2 • • • x r-l+1 (r -l + 1)! j r-l+1where the sum is taken over all sequences j 1 , j 2 , j 3 , ..., j r-l+1 of nonnegative integers such that j 1 + ... + j r-l+1 = l and j 1 + 2j 2 + 3j 3 + ... + (r -l + 1)j r-l+1 = r.EQUATIONEQUATIONwhere m r = E(Y r ) denotes the rth moment of Y . As is evident by its definition, the cumulant generating function plays nicely with convolutions. The cumulant generating function of the random variable Y 1 + Y 2 for independent Y 1 and Y 2 is given by the sum of the individual cumulant generating functions. Consequently, the rth cumulant of the sum is the sum of the rth cumulants.In the model (1), the second moment exhibits1 n n i=1 X 2 i = ||µ|| 2 /n + σ 2 + O P (n -1/2) due to the boundedness of ||µ|| ∞ and σ. It is clear estimation of σ 2 is equivalent to estimation of ||µ|| 2 /n up to a (negligible) parametric slowdown in the rate. Associated with any estimator Q for the quadratic functional||µ|| 2 n is a corresponding variance estimator σ2 = 1 n n i=1 X 2 i -Q. Hence, it sufficesto focus attention on quadratic functional estimation. The estimation methodology is more plainly motivated in the empirical Bayes context (2), so the following discussion will develop ideas in that setting.In the context of ( 2), the goal is to estimate the second moment of G. Let κ r and m r denote the rth cumulant and moment respectively of the marginal distribution G * N (0, σ 2 ) of the data. Likewise, let γ r and ν r denote the rth cumulant and moment respectively of G. Since the noise is mean zero and σ is bounded, the mean of G can be estimated at parametric rate, and so it is equivalent (up to a negligible parametric slowdown) to estimate the variance of G, i.e. γ 2 .It is not immediately clear how to identify γ 2 from the cumulants κ r of the marginal distribution of the data G * N (0, σ 2 ). Consider the moment generating function ofN (0, σ 2 ) is λ → exp(λ 2 σ 2 /2),and so all the cumulants (except the second) are equal to zero. Therefore, κ 1 = γ 1 , κ 2 = γ 2 + σ 2 , and κ r = γ r for r ≥ 3. The upshot is we are able to directly estimate all the cumulants of G, except the second, from{X i } n i=1. Unfortunately, it is impossible to generically reconstruct the second cumulant from the other cumulants. Indeed, the collection of all centered Gaussians differ only in their second cumulant (the rest are zero).It turns out it is possible to identify γ 2 by exploiting the boundedness of G's support. For r ≥ 2, define the functionEQUATIONwhere B r,l is an incomplete Bell polynomial (see Definition 1). Note by ( 7) that M r gives the rth moment associated to the cumulant sequence (γ 1 , γ, γ 3 , . . .) if it is a valid cumulant sequence. The following proposition asserts a variational representation of γ 2 .Proposition 1. If G is supported on [-1, 1], then γ 2 = sup {γ ∈ [0, 1] : |M r (γ)| ≤ 1 for all r ≥ 2}.Proof. For ease of notation, letγ * = sup {γ ∈ [0, 1] : |M r (γ)| ≤ 1 for all r ≥ 2}. It is clear γ 2 ≤ γ * since M r (γ 2 ) = ν r ∈ [-1, 1] for all r ≥ 2.To show the lower bound, fix δ > 0. For any τ 2 ≥ δ, observe that the cumulant sequence of the distribution G * N (0, τ 2 ) is (γ 1 , γ 2 + τ 2 , γ 3 , γ 4 , ...). Since G * N (0, τ 2 ) does not have all moments contained in [-1, 1], it follows γ 2 + τ 2 is not a feasible point. Since this holds for all τ 2 ≥ δ, it immediately follows γ * ≤ γ 2 + δ. Since δ > 0 was arbitrary, we have shown γ * ≤ γ 2 .Proposition 1 enables recovery of γ 2 from the other cumulants. However, it requires certifying all putative moments {M r (γ)}∞ r=2 live in [-1, 1], which is a difficult task given only a finite amount of data. One idea is to approximate γ 2 by certifying only that an rth putative moment lies in [-1, 1] for some large choice of r. Specifically, define for r ≥ 2,EQUATION) Proposition 2. If r is even, then |γ 2 (r) -γ 2 | ≤ C r for some universal constant C > 0.Proof. By (9) and Proposition 1, it followsγ 2 ≤ γ2 (r). Therefore, M r (γ 2 (r)) is the rth moment of G * N (0, γ2 (r) -γ 2 ). Since r is even, taking µ ∼ G and Z ∼ N (0, 1), it follows 1 ≥ |M r (γ 2 (r))| = E((µ + γ2 (r) -γ 2 Z) r ) = r l=0 r l E(µ r-l )(γ 2 (r) -γ 2 ) l/2 E(Z l ).If r -l is odd, then l must be odd since r is even, and so we must haveE(Z l ) = 0. It thus follows 1 ≥ l≤r, l even r l E(|µ| r-l )(γ 2 (r) -γ 2 ) l/2 E(Z l ) ≥ (γ 2 (r) -γ 2 ) r/2 E(Z r ). Hence, |γ 2 (r) - γ 2 | ≤ E(Z r ) -2/r = π -1/r 2 r/2 Γ r+1 2 -2/r . It follows from Stirling's approximation that for some small universal constant c > 0 whose value may change from instance to instance, we haveΓ r+1 2 2/r ≥ c π(r -1) r-1 2e r-1 2 2/r≥ cr. This immediately yields the claimed bound. The proof is complete.The methodological strategy is in place; an estimator will be constructed to estimate γ2 (r) for a well-chosen value of r to balance the estimator's variance with the bias from Proposition 2.We return to the compound setting of (1); there is now no "ground truth" prior G. However, a recurring theme of compound decision theory [48, 47] is that it mimics the empirical Bayes theory as if the prior were the empirical distribution of the means. Note the quadratic functional of interest ||µ|| 2 /n is precisely equal to the second moment of the empirical distribution of the means.To describe our estimation strategy, some preliminary development is necessary. For even r, set m r to be the rth moment of the distribution ( 1 n n i=1 δ µi ) * N (0, σ 2 ). For odd r, set m r = 0. Observe m 1 , m 2 , ... are the moments of the symmetrized distribution1 n n i=1 ( 1 2 δ µi + 1 2 δ -µi ) * N (0, σ 2). Thus, we are able to conceptually place ourselves in the context of Section 2.2 by making the choice of "prior" 3 G = 1 n n i=1 ( 1 2 δ µi + 1 2 δ -µi ).Let us adopt the notation of Section 2.2, namely let κ r denote the rth cumulant of G * N (0, σ 2 ), and let γ r and ν r denote the rth cumulant and moment respectively of G. As noted in Section 2.2, we have κ r = γ r for r ̸ = 2. Furthermore, note the quadratic functional of interest ||µ|| 2 /n is exactly γ 2 .The strategy is to estimate γ2 (r) given by ( 9) for a carefully chosen value of r. In pursuit of this strategy, estimators for the cumulants (except the second) will now be constructed. Define the moment estimatorsEQUATION)Cumulant estimators are obtained via plugging in to (6) ,EQUATIONfor r ̸ = 2. To estimate γ2 (r), an estimator of the function M r given by ( 8) is needed. Define the (random) function Mr : [0, ∞) → R given byEQUATION)For r ≥ 2 and ε > 0, define the estimatorEQUATION)where r, ε are tuning parameters to be chosen. The high-level justification behind γ2 (r) lies in the approximation,EQUATIONSuppose it could be shown Mr (γ) concentrates around M r (γ) uniformly over γ ∈ [0, 1]. If so, then the first approximation in ( 14) follows. The Bell polynomial structure is quite convenient as it facilitates a straightforward proof of the desired uniform concentration (see Appendix B.2); cumbersome empirical process theory is avoided. Since both γ2 (r) and γ2 (r) are defined in terms of supremums in ( 13) and ( 9), both Mr (γ 2 (r)) and M r (γ 2 (r)) will be near one, intuitively yielding the second approximation in (14) . Thus |M r (γ 2 (r)) -M r (γ 2 (r))| is small, and since M r is an 12) and ( 8) respectively. On the event E, we have |γ 2 (r)-γ 2 | ≤ ((r/2)!•2ε) 2/r + C r where γ2 (r) is given by ( 13) and C > 0 is a universal constant.Proof. It follows from the definition (9) of γ2 (r) and Proposition 1 that γ 2 ≤ γ2 (r). On the event E, it follows from the definition of γ2 (r) and |M r (γ 2 (r))| ≤ 1 that γ2 (r) ≤ γ2 (r). Therefore, on the event E, it follows by Lemmas 1 and 2 that 1 = M r (γ 2 (r)) ≤ M r (γ 2 (r)) ≤ Mr (γ 2 (r)) + ε ≤ 1 + 2ε. This implies 0 ≤ M r (γ 2 (r)) -M r (γ 2 (r)) ≤ 2ε, and so an application of Lemma 3 yields 0 ≤ γ2 (r) -γ2 (r) ≤ ((r/2)! • 2ε) 2/r . Combining this bound with the approximation error bound of Proposition 2 delivers the desired result.To obtain an estimation error bound, it remains to investigate the stochastic error, i.e. the typical magnitude of sup γ∈[0,1] | Mr (γ) -M r (γ)|, so that the first approximation in ( 14) is justified. The cumulant estimators {γ l } 1≤l≤r, l̸ =2 used to define Mr (γ) are constructed by plugging in sample moment estimators for the even moments. Heuristically, only a slow concentration rate can be expected because the number of moments r being estimated is large. Proposition 4. Suppose (µ, σ) ∈ Θ and c * , β > 0 are universal constants. If δ ≥ 1 log β n , then there exist universal constants C β, * , C, C ′ ≥ 1 depending only on β and c * such that the following holds.If 2 ≤ r ≤ 1 C β, * log n log log n and σ 2 ≥ c * r , then P µ,σ sup γ∈[0,1] Mr (γ) -M r (γ) ≤ Ce C ′ r log r √ nδ ≥ 1 -δ.With the stochastic error addressed by Proposition 4, the ingredients are in place to furnish a risk bound for the variance estimator associated to (13) . There is the slight technical point that Proposition 4 applies only when σ 2 ≳ 1 r . This condition may be an artifact of the proof, and the conclusion of Proposition 4 might continue to hold for small σ. However, Proposition 4 turns out to suffice for the purpose of variance estimation as we can incorporate a truncation step. The variance estimator is defined as follows. For an even integer r and a real ε > 0, defineEQUATIONwhere γ2 (r) is given by ( 13). , thensup (µ,σ)∈Θ E µ,σ |σ 2 -σ 2 | 2 ≲ log log n log n 2 ,where σ2 is given by (15) .Since ||µ|| ∞ ≤ 1 implies that max 1≤i≤n X i is typically no larger than 1+ 2σ 2 log n, the truncation choice 5 4 log n r essentially results in using the trivial estimator σ2 = 0 when σ 2 ≲ 1 r . On the other hand when σ 2 ≳ 1 r , the stochastic error result of Proposition 4 is meaningful and it can be shown1 n n i=1 X 2 i -γ2 (r) estimates σ 2 well.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes]Justification: The paper discusses how the proposed variance estimator relies on the Gaussian character of the noise.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
lower bound: 
 The slow convergence rate of the cumulant-based estimator of Section 2 turns out to be the sharp rate. Theorem 2. There exist universal constants C, c > 0 such that inf σ sup (µ,σ)∈Θ(2) P µ,σ {|σ 2 -σ 2 | 2 ≥ C( log log n log n ) 2 } ≥ c.The lower bound is proved through a moment matching technique employed in Le Cam's two point method. The intuition for the construction follows from appreciating the constraint ||µ|| ∞ ≤ 1.As discussed, the boundedness constraint is imposed to ensure σ 2 is identifiable. Though quite elementary, it is conceptually useful for our construction to examine why σ 2 is unidentifiable if no constraints are imposed. Specifically, it can be understood from a lower bound perspective by examining the reduction to the following Bayesian testing problem, H 0 : µ = 0 and σ 2 = 1 + τ 2 against H 1 : µ i ∼ N (0, τ 2 ) and σ 2 = 1. Observe that under both hypotheses the data are X 1 , ..., X n are i.i.d. draws from N (0, 1 + τ 2 ). The hypotheses are indistinguishable yet the variances differ by τ 2 under H 0 and H 1 . Therefore, any variance estimator incurs square loss of at least τ 4 ; taking τ → ∞ establishes infinite estimation risk is inescapable.Under the boundedness constraint ||µ|| ∞ ≤ 1, the choice of prior µ 1 , ..., µ n iid ∼ N (0, τ 2 ) is no longer feasible. One idea is to find a distribution G supported on [-1, 1] such that the marginal data distribution G * N (0, 1) is indistinguishable from N (0, τ 2 ) * N (0, 1). This can be achieved by constructing G to share a large number of moments with N (0, τ 2 ). To elaborate, if the first 2k-1 moments match and τ < 1, then (see Theorem 3.3.3 of [64] ) we have χ 2 (G * N (0, 1) || N (0, τ 2 ) * N (0, 1))≤ 16 √ 2k-1 τ 4k1-τ 2 . A well-known sufficient condition (e.g. see [58] ) for the desired indistinguishability is that the χ 2divergence is at most O(1/n). Hence, we seek a G through moment matching.How many moments can be matched under the constraint that G be supported on [-1, 1]? Since the odd moments of a centered Gaussian are all zero and taking G symmetric would thus match the odd moments, only the even moments require consideration. For k even, the kth moment of N (0, τ 2 ) is at least, by Stirling's approximation, cτ k e k log k/2 for some small constant c > 0. On the other hand, the kth moment of G must lie in [-1, 1]. Therefore, a necessary condition for the kth moments to match is τ k e k log k/2 ≲ 1. This crude reasoning already shows G can only match at most O(1/τ 2 ) moments.The actual construction of G relies on the technology of Gaussian quadrature. With Proposition 9 and Lemma 8, it can be shown that the k-atomic Gaussian quadrature G = k i=1 w i δ τ zi , where {z i } k i=1 are the roots of the kth Hermite polynomial, is supported on the interval [-τ 2 (4k -4), τ 2 (4k -4)] and shares the first 2k -1 moments with N (0, τ 2 ). A sufficient condition for G to be supported on [-1, 1] is k ≤ 1/(4τ 2 ). Therefore, a properly supported G which matches Ω 1/τ 2 moments of N (0, τ 2 ) can be constructed, validating that our crude reasoning from earlier is tight. The problem now boils down to choosing τ 2 . Since k ≍ τ -2 , it follows from the χ 2 -divergence bound implied by moment matching that choosing τ 2 ≍ log log n log n yields the desired 1 n bound on the χ 2 -divergence and is precisely the desired separation.Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof? 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Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. 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noise agnosticism: 
 The cumulant-based variance estimator proposed in this paper heavily relies on the Gaussian character of the noise, specifically that all cumulants (except the first-and second-order) of a Gaussian are equal to zero regardless of the Gaussian's mean and variance parameters. From a minimax perspective, it turns out it is a fundamental necessity to exploit information about the noise. Concretely, it can be shown that consistent variance estimation is impossible if nothing beyond subgaussianity (see Definition 2) is assumed. The impossibility can be seen by appealing to Le Cam's two-point method to reduce the problem to a two-point testing problem and using the following simple construction,H 0 : µ i iid ∼ Rademacher (1/2) , σ 2 = 1, and ξ i iid ∼ Rademacher (1/2) , H 1 : µ = 0, σ 2 = 2, and ξ i √ 2 iid ∼ Rademacher (1/2) * Rademacher (1/2). It is clear both noise distributions are 2-subgaussian. Furthermore, the data {X i } n i=1 are independent and identically distributed according to Rademacher (1/2) * Rademacher (1/2) under both hypotheses, and so it is impossible to distinguish H 0 and H 1 . Since the separation between the choices of σ 2 between the two hypotheses is 1, it follows consistent estimation is impossible. This is formally stated in the following proposition without proof. Proposition 5. For a > 0, let Ξ a := {P : P is a-subgaussian, has mean 0 and variance 1}. Let P µ,σ,P ξ denote the joint distribution of the observations from the sequence model,X i = µ i + σξ i for 1 ≤ i ≤ n where ξ i ∼ P ξ are i.i.d. Then inf σ sup (µ,σ)∈Θ(2),P ξ ∈Ξ2 E µ,σ,P ξ |σ 2 -σ 2 | 2 ≳ 1.In this sense it is necessary to exploit finer information about the noise, and so our cumulant-based estimator is designed to exploit the noise's Gaussian character in (1). Lemma 1. If r is even, then the function M r given by ( 8) is strictly increasing on the interval (γ 2 , ∞).Proof. Observe for γ > γ 2 that M r (γ) gives the rth moment of the distribution G * N (0, γ -γ 2 ). In other words, letting µ ∼ G and Z ∼ N (0, 1), we haveM r (γ) = E µ + √ γ -γ 2 Z r = r l=0 r l E(µ r-l )(γ -γ 2 ) l/2 E(Z l ).Consider that if r -l is odd, then l must also be odd since r is even, which implies E(Z l ) = 0. Therefore, we can writeM r (γ) = 0≤l≤r, l even r l E(|µ| r-l )(γ -γ 2 ) l/2 E(Z l ).Hence, M r is strictly increasing in γ on the interval (γ 2 , ∞) as claimed.Lemma 2. If r is even, then M r (γ 2 (r)) = 1 where M r is given by ( 8) and γ2 (r) is given by ( 9).Proof. Consider M r is a continuous function as it is a polynomial. Observe M r (γ 2 ) ≤ 1 by Proposition 1. Furthermore, consider since M r (γ) is the rth moment of G * N (0, γ -γ 2 ) for γ > γ 2 it follows lim γ→∞ M r (γ) = ∞. Furthermore, since Lemma 1 asserts M r is monotonically increasing on (γ 2 , ∞) and since γ2 (r) ≥ γ 2 , it follows by continuity that M r (γ 2 (r)) = 1.Lemma 3. If x ≥ y ≥ γ 2 and r is even, then x -y ≤ ((r/2)! (M r (x) -M r (y))) 2/rwhere M r is given by (8) .Proof. From the proof of Lemma 1 and taking µ ∼ G and Z ∼ N (0, 1), we have for γ ≥ γ 2 ,M r (γ) = 0≤l≤r, l even r l E(|µ| r-l )E(Z l )(γ -γ 2 ) l/2 .For 0 ≤ k ≤ r 2 and for γ ≥ γ 2 , it follows from even l thatM (k) r (γ) = 0≤l≤r, l even, l/2≥k r l E(|µ| r-l )E(Z l ) (l/2)! (l/2 -k)! (γ -γ 2 ) l/2-k ≥ E(|µ| r-2k ).Since M r (γ) is an r 2 -degree polynomial in γ, Taylor expansion along with the above bound yieldsM r (x) -M r (y) = r/2 k=1 M (k) r (y) (x -y) k k! ≥ (x -y) r/2 (r/2)! .Here, we have used that every term in the sum is nonnegative. In particular, Mr (y) ≥ 0 and (x -y) k ≥ 0 since x ≥ y. Rearranging gives the desired result.As the definition of Mr in (12) relies on moment estimators mr given by (10) and cumulant estimators γr given by ( 11), we will first collect results for these intermediate estimators. Further, recall from Section 2.3 that m r = 1 n n i=1 E(X r i ) if r is even and m r = 0 otherwise. Proof. The claim is trivially true for odd r, so consider even r. Write X i = µ i + σZ i where Z i iid ∼ N (0, 1). Observe by independence and|µ i | ≤ 1, we have Var( mr ) = Var 1 n n i=1 X r i = 1 n 2 n i=1 Var(X r i ) ≤ 1 n 2 n i=1 E(X 2r i ). By Jensen's inequality, we haveX 2r i = 2 2r µi 2 + σZi 2 2r ≤ 2 2r-1 µ 2r i + (σZ i ) 2r. Taking expectation yields E(X 2r i ) ≤ 2 2r-1 (µ 2r i + σ 2r (2r -1)!!) ≤ 2 2r-1 1 + σ 2r (2r -1) 2r-1 ≤ 4 r + σ 2r (4r) 2r , which yields the desired result.Corollary 1. If ||µ|| ∞ ≤ 1 and u > 0, then P µ,σ {max 1≤l≤r | ml -m l | > u} ≤ r4 r +r(1∨σ 2 ) r (4r) 2r u 2 n .Proof. The result follows by union bound and Chebyshev's inequality with Lemma 4. Proposition 6. Suppose (µ, σ) ∈ Θ(L) where Θ(L) is given by ( 4) and where L > 0 is a universal constant. There exists a large universal constant C > 0 such that the following holds. If u > 0 such that C r u 1∧σ r ∈ (0, 1), then on the event {max 1≤l≤r | ml -m l | ≤ u} we havemax 1≤l≤r, l̸ =2 |γ l -γ l | ≤ (1 + L r r!) • (r!) 2 • (2r) r • e C r 1∧σ r ru -1 .Proof. The argument borrows heavily from the proof of Lemma A.4 in [10] . For 1 ≤ k ≤ r with k ̸ = 2, we have by Definition 1,|γ k -γ k | ≤ k l=1 (l -1)!|B k,l ( m1 , ..., mk-l+1 ) -B k,l (m 1 , ..., m k-l+1 )| ≤ k l=1 (l -1)! k! j 1 !j 2 ! • • • j k-l+1 ! • m1 1! j1 m2 2! j2 • • • mk-l+1 (k -l + 1)! j k-l+1 - m 1 1! j1 m 2 2! j2 • • • m k-l+1 (k -l + 1)! j k-l+1where the sum is taken over all sequences j 1 , j 2 , ..., j k-l+1 of nonnegative integers such that j 1 + ... + j k-l+1 = l and j 1 + 2j 2 + 3j 3 + ... + (k -l + 1)j k-l+1 = k. By definition of ml given by (10) and the definition of m l given in Section 2.3, we have ml = m l = 0 for odd l. Consequently, we need only examine terms in the sum such that j l = 0 for all odd l, so let us now fix such a term.For even l, we have by Jensen's inequalitym l = 1 n n i=1 1 2 E(|µ i + σZ| l ) + 1 2 E(| -µ i + σZ| l ) ≥ E(|σZ| l ) where Z ∼ N (0, 1). Since l is even, E(|σZ| l ) = σ l E(Z l ) = σ l 2 l/2 Γ( l+1 2 ) √ π ≥ (c ′ σ 2 l) l/2for some small universal constant c ′ > 0 by Stirling's approximation. Consider (c ′ σ 2 l) l/2 ≥ min 1≤q≤r (c ′ σ 2 q) q/2 ≥ C -r (1 ∧ σ r ) since we take C > 0 to be a sufficiently large universal constant.Observe on the event {max 1≤l≤r | ml -m l | ≤ u}, we havem1 1! j1 m2 2! j2 • • • mk-l+1 (k -l + 1)! j k-l+1 ≤ m 1 + u 1! j1 m 2 + u 2! j2 • • • m k-l+1 + u (k -l + 1)! j k-l+1 ≤ m 1 1! j1 m 2 2! j2 • • • m k-l+1 (k -l + 1)! j k-l+1 exp k-l+1 i=1 j i u m i .To obtain the second line we have used that 0 ≤ mi i! ji ≤ mi+u i! ji for all 1 ≤ i ≤ k -l + 1 since j l = 0 for all odd l. We have also used the inequality 1 + x ≤ e x to obtain the third line. Since m i ≥ C -r (1 ∧ σ r ) for even i as established earlier, it follows by the identity j 1 + ...+ j k-l+1 = l that exp k-l+1 i=1 j i u mi ≤ e C r1∧σ r lu . Since j 1 + 2j 2 + 3j 3 + ... + (k -l + 1)j k-l+1 = k and j i = 0 for all odd i, it follows by Jensen's inequality (lettingEQUATIONLet us now prove an analogous lower bound. Again, since m i ≥ C -r (1 ∧ σ r ) for even i and j 1 + ... + j k-l+1 = l, we haveEQUATIONHere, we have used C r 1∧σ r u ∈ (0, 1), j l = 0 for all odd l, andm i ≥ C -r (1 ∧ σ r ) for even i to conclude 1 - mq-mq mq jq ≥ 1 -u mq jq≥ 0 for all q, thus obtaining the third line. We have also used C r 1∧σ r u ∈ (0, 1) to obtain the third-to-last line. Therefore, it follows from ( 16) and ( 17) that|γ k -γ k | ≤ k l=1 (l -1)! k! j 1 !j 2 ! • • • j k-l+1 ! • m k e C r 1∧σ r lu -1 ≤ m k |e C r 1∧σ r ku -1| • (k!) 2 • k k ≤ 2 k (1 + L k k!)|e C r 1∧σ r ku -1| • (k!) 2 • k k .The last inequality follows from ||µ|| ∞ ≤ 1 and Jensen's inequality. We have also bounded the number of admissible sequences j 1 , ..., j k-l+1 in the sum by k k by elementary counting since the constraint j i ≤ k must always be satisfied. Maximizing over 1 ≤ k ≤ r with k ̸ = 2 yields the desired result.Finally, we are able to state a concentration result about Mr . In the statement of Proposition 7, note the concentration occurs on the same event that the cumulant estimators concentrate around the true cumulants, which in turn occurs, from Proposition 6, on the same event the moment estimators concentrate.Proposition 7. Suppose (µ, σ) ∈ Θ(L) where Θ(L) is given by ( 4) and where L > 0 is a universal constant. On the event max 1≤l≤r,l̸ =2 |γ l -γ l | ≤ u , we have sup γ∈[0,1] Mr (γ) -M r (γ) ≤ (2r) r r 3 (r!)(r r ∨ u r )u,where Mr and M r are given by ( 12) and ( 8) respectively.Proof. For γ ∈ [0, 1], consider by Definition 1 thatMr (γ) -M r (γ) ≤ r l=1 r! j 1 !j 2 ! • • • j r-l+1 ! γ 2! j2 1≤a≤r-l+1, a̸ =2 γa a! ja - 1≤a≤r-l+1, a̸ =2 γ a a! ja ≤ r l=1 r! j 1 !j 2 ! • • • j r-l+1 ! |f j (γ) -f j (γ * )|where the inner sum is taken over all sequences j = {j 1 , ..., j r-l+1 } of nonnegative integers such that j 1 + ... + j r-l+1 = l and j 1 + 2j 2 + 3j 3 + ... + (r -l + 1)j r-l+1 = r. Here, we have defined the functionf j : R r-l+1 → R by f j (x) = 1≤a≤r-l+1, a̸ =2 xa a!ja and we have used the notation γ = (γ 1 , γ, γ3 , ..., γr-l+1 ) and γ * = (γ 1 , γ, γ 3 , ..., γ r-l+1 ). By Taylor expansion and Holder's inequality, we have|f j (γ) -f j (γ * )| ≤ ||∇f j (ξ)|| 1 • ||γ -γ * || ∞where ξ is some point on the line segment between γ and γ * . It is immediate that on the eventmax 1≤l≤r, l̸ =2 |γ l -γ l | ≤ u we have ||γ -γ * || ∞ ≤ u. Further consider |ξ a | ≤ |γ a | + u ≤ 2(|γ a | ∨ u) ≤ 2(a a ∨ u) by Lemma 9. A straightforward calculation shows ||∇f j (ξ)|| 1 ≤ 1≤a≤r-l+1, a̸ =2 1 {ja≥1} j a |ξ a | ja-1 (a!) ja 1≤b≤r-l+1, b̸ ∈{2,a} ξ b b! j b ≤ 1≤a≤r-l+1, a̸ =2 1 {ja≥1} j a |2(a a ∨ u)| ja-1 (a!) ja 1≤b≤r-l+1, b̸ ∈{2,a} 2(b b ∨ u) b! j b ≤ 2 r r 2 (r r ∨ u r ) ,where we have used 2(a a ∨ u) ≥ 1, j a ≤ r, and r-l+1 a=1 aj a = r. To summarize, we have shownMr (γ) -M r (γ) ≤ r l=1 r! j 1 !j 2 ! • • • j r-l+1 ! • (2 r r 2 (r r ∨ u r ))u ≤ (2r) r r 3 (r!)(r r ∨ u r )u for all γ ∈ [0, 1].As in the proof of Proposition 6, we have bounded the number of admissible sequences j 1 , ..., j r-l+1 by r r . The proof is complete.Proof of Proposition 4. Since (µ, σ) ∈ Θ and L > 0 is a universal constant, it follows by Corollary 1 that the eventE = max 1≤l≤r | ml -m l | ≤ δ -1/2 C 1 e C ′ 1 r log r √ n has P µ,σ -probability of at least 1 -δ, where C 1 , C ′ 1 > 0 are some universal constants. Furthermore, it is clear for any 2 ≤ r ≤ 1 C β, * log n log log n , we have δ -1/2 C1e C ′ 1 r log r √ n ≲ (log β/2 n)n C ′ 1 C β, * -12 , and so C β, * can be taken sufficiently large to ensure δ -1/2 C1e C ′ 1 r log r√ n decays at some rate that is polynomial in n. We will now apply Proposition 6 on the eventE with u = δ -1/2 C1e C ′ 1 r log r √ n. To use Proposition 6, it must be checked Cr 1∧σ r u ∈ (0, 1) where C > 0 is the universal constant from Proposition 6. Consider that σ 2 ≥ c * r , and soCr 1 ∧ σ r u ≤ Cr ∨ C √ c * r r r/2 u ≤ δ -1/2 C 1 √ n exp C ′ 1 + 1 2 r log r + log C √ c * + log C ≲ (log β/2 n)n C ′ 1 +1/2+log( C2 / √ c * ) C β, * -1 2 .Therefore, taking C β, * sufficiently large (depending on c * and β) implies Cr 1∧σ r u decays at some rate that is polynomial in n. It then follows from Proposition 6, the fact that r is at most logarithmic in n, and the inequality |e x -1| ≤ ex for x ∈ (0, 1),max 1≤l≤r, l̸ =2 |γ l -γ l | ≤ (1 + L r r!) • (r!) 2 • (2r) r • e Cr 1∧σ r ru -1 ≤ C 2 e C ′ 2 r log r u where C 2 , C ′ 2 > 0 are some universal constants (potentially depending on c * ). Note C 2 e C ′ 2 r log r u ≲ (log β/2 n)n C ′ 1 +C ′ 2 C β, * -1 2, and so again we can take C β, * sufficiently large to ensure C 2 e C ′ 2 r log r u decays at some rate that is polynomial in n. It then follows from Proposition 7 thatsup γ∈[0,1] Mr (γ) -M r (γ) ≤ C 3 e C ′ 3 r log r • C 2 e C ′ 2 r log r u = δ -1/2 C 1 C 2 C 3 e (C ′ 1 +C ′ 2 +C ′ 3 )r log r √ n ,where C 3 , C ′ 3 > 0 are some universal constants (potentially depending on c * ). The proof is complete.Proof of Theorem 1. Fix (µ, σ) ∈ Θ. ChooseC * := C β, * ∨ 2 (4(C ′ + 1)) ∨ 2 • inf K > 0 : 4(C ′ + 1) K -1 • 2K < -2 .Note C * is a positive universal constant since C β, * , C, and C ′ all are positive universal constants. The reason for this choice of C * will become clear later on. To bound the estimation risk, first considerEQUATION)EQUATION)To bound (18) , let Z i = X i -µ i and consider by Lemma 10,EQUATIONwhere we have used the result of [32] which gives E µ,σ (max1≤i≤n Z i ) ≥ σ √ π log 2 √ log n.We have also used ||µ|| ∞ ≤ 1 and 1 r ≲ log log n log n . To obtain the result of the theorem, it remains to show ( 19) has order at most log log n log n 2 . To do so, we split the analysis into two cases.Case 1: Suppose σ 2 < 1 r . Consider by Cauchy-Schwarz inequality,EQUATIONLetting Z i = X i -µ i , consider by Lemma 10 we haveP µ,σ max 1≤i≤n X i > 4 log n r ≤ P µ,σ max 1≤i≤n Z i -E µ,σ max 1≤i≤n Z i > 4 log n r -||µ|| ∞ -E µ,σ max 1≤i≤n Z i ≤ 2 exp   - 1 2σ 2 4 log n r -||µ|| ∞ -E µ,σ max 1≤i≤n Z i 2 +   . Consider E µ,σ (max 1≤i≤n Z i ) ≤ σ √ 2 log n ≤ 2 log n r and ||µ|| ∞ ≤ 1. Therefore, it follows that P µ,σ max 1≤i≤n X i > 4 log n r ≤ 2 exp -log n rσ 2 ≤ 2 n .With this in hand, it follows fromE µ,σ 1 n n i=1 X 2 i -γ2 (r) -σ 2 4 ≲ 1 that (21) ≲ n -1 ≲ log log n log n 2 as desired. Case 2: Suppose σ 2 ≥ 1 r . It is clear (19) ≤ E µ,σ 1 n n i=1 X 2 i -γ2 (r) -σ 2 2 . To bound this expectation, denote the event E = sup γ∈[0,1] Mr (γ) -M r (γ) ≤ ε . Since σ 2 ≥ 1 r , C * ≥ C β, * , and ε = Ce C ′ r log r √ nδwith C, C ′ as in Proposition 4 with the choice c * = 1, it follows by Proposition 4 that P µ,σ (E) ≥ 1 -δ. From the inequality (a + b) 2 ≤ 2a 2 + 2b 2 , it follows1 n n i=1 X 2 i -γ2 (r) -σ 2 2 ≤ 2 1 n n i=1 X 2 i -(γ 2 + σ 2 ) 2 + 2|γ 2 (r) -γ 2 | 2 . Since γ2 (r), γ 2 ∈ [0, 1] and since ε = Ce C ′ r log r √ nδ, an application of Proposition 3 yields2|γ 2 (r) -γ 2 | 2 ≤ 2 2Ce (C ′ +1)r log r √ nδ 4/r + C ′′ r 2 + 21 E cwhere C ′′ > 0 is a universal constant. Considere (C ′ +1)r log r √ nδ 4/r = exp 1 r (4(C ′ + 1)r log r -2 log n + 4 log log n) ≤ exp 1 r 4(C ′ + 1) C * log n log log n log 1 C * log n log log n -log n ≤ exp 4(C ′ + 1) C * -1 log n r . Note C * ≥ 2 • 4(C ′ + 1) ≥ 4(C ′ + 1), so it immediately follows 4(C ′ +1) C * -1 < 0. Therefore, it follows from r ≥ 1 2C * log n log log n that e (C ′ +1)r log r √ nδ 4/r ≤ exp 4(C ′ + 1) C * -1 • 2C * log log n ≤ 1 log 2 nwhere the second inequality follows from the fact that C * satisfies, by design, the inequality4(C ′ +1) C * -1 • 2C * ≤ -2. Therefore, we have shown 2 2Ce (C ′ +1)r log r √ nδ 4/r ≲ 1 log 2 n .To summarize, we have shownE µ,σ   1 n n i=1 X 2 i -γ2 (r) -σ 2 2   ≲ Var 1 n n i=1 X 2 i + E µ,σ |γ 2 (r) -γ 2 | 2 ≲ 1 n + 1 log 2 n + 1 r 2 + P µ,σ (E c ) ≲ 1 n + 1 log 2 n + 1 r 2 + δ ≍ log log n log n 2as desired. The proof is complete.Proof of Theorem 2. As seen in many minimax lower bound arguments of the literature, we proceed by reducing to a two-point testing problem. Without loss of generality we will assume n is larger than a sufficiently large universal constant. Let 0 < τ 2 ≤ 1 and we will choose it later. Set σ 2 0 = 1 + τ 2 and σ 2 1 = 1. For a distribution π supported on [-1, 1], denote the induced mixture P π,σ = P µ,σ π ⊗n (dµ). It follows by reverse triangle inequalityEQUATIONwhere the infimum in the penultimate line runs over all events A. By the Neyman-Pearson lemma, 1 -d TV (P 0,σ0 , P π,σ1 ) is the optimal testing risk for the hypothesis testing problemH 0 : X 1 , ..., X n iid ∼ N (0, 1 + τ 2 ), H 1 : X 1 , ..., X n iid ∼ π * N (0, 1).Specifically, we have d TV (P 0,σ0 , P π,σ1 ) ≤ 1 2(1 + χ 2 (π * N (0, 1) || N (0, 1 + τ 2 )))n -1 by Lemma 5. Hence, it suffices to bound the χ 2 -divergence to furnish a lower bound for (22) .We now construct π and we will pick τ 2 < 1 16 at the end of the proof to obtain the claimed lower bound. Let D denote the largest even number smaller than or equal to1 4τ 2 . Note D ≥ 2 since τ 2 < 1 8 . Let g D = Di=1 w i δ zi denote the D-point Gaussian quadrature of N (0, 1). Note that the atoms {z i } D i=1 are the zeros of the Dth degree Hermite polynomial (Lemma 6). Further note that this measure is 1-subgaussian and symmetric about zero D is even (Lemma 7). Take the prior distribution π = D i=1 w i δ τ zi . In order for π to be a valid choice, it must be verified π is supported on[-1, 1]. Lemma 8 gives us that |z i | ≤ √ 4D -4 since z i is a zero of the Dth degree Hermite polynomial. Observe |τ z i | ≤ τ 2 (4D -4) < 1 by our choice of D. Hence, π is supported on [-1, 1]and thus is a valid choice.We use a moment matching technique (Proposition 8). To do so, we first verify the first 2D -1 moments of π and N (0, τ 2 ) match. For any r ∈ {1, ..., 2D -1}, observe since g D is theD-point Gaussian quadrature of N (0, 1), it follows E Y ∼N (0,τ 2 ) (Y r ) = τ r E Z∼N (0,1) (Z r ) = τ r E Z∼g D (Z r ) = E Y ∼π (Y r ). Note both N (0, τ 2 ) and π are τ -subGaussian. Proposition 8, along with D ≥ 1 4τ 2 -2 ≥ 2 and τ 2 < 1 16 , thus implies χ 2 (π * N (0, 1) || N (0, τ 2 ) * N (0, 1)) ≤ 16τ 4D √ 2D -1 (1 -τ 2 ) ≤ 16 2 15 √ 3 exp - 1 4τ 2 log 1 τ 2 .Select τ 2 = log log n 16 log n and observe since n is larger than a sufficiently large universal constant. Therefore, d TV (P 0,σ0 , P π,σ1 ) ≤1 2 1 + 1 n n -1 ≤ 1 2 √e -1, which, when plugged into (22) , yields the lower boundinf σ sup (µ,σ)∈Θ(2) P µ,σ |σ 2 -σ 2 | ≥ log log n 32 log n ≥ 1 2 1 - 1 2 √ e -1 .The proof is complete.Definition 2. A probability distribution P is said to be a-subgaussian for a > 0 if for X ∼ P we haveE (exp (λ(X -E(X)))) ≤ e λ 2 a 2 2for all λ ∈ R. Lemma 5 (χ 2 tensorization [58] ).If P = n i=1 P i and Q = n i=1 Q i are product measures, then χ 2 (P || Q) = n i=1 (1 + χ 2 (P i || Q i )) -1.Proposition 8 (Theorem 3.3.3 [64] ). Suppose ν and ν ′ are two symmetric probability distributions that are ε-subgaussian for ε < 1. If the first D moments of ν and ν ′ are equal, then χ 2 (ν * N (0, 1) || ν ′ * N (0, 1)) ≤ 16√ D ε 2D+21-ε 2 . Proposition 9 (Gaussian quadrature [64] ). Suppose µ is a probability measure supported on E ⊂ R. If k ≥ 1, there exists a k-atomic distribution µ k = k i=1 w i δ xi supported on E such that for any polynomial p of degree at most 2k -1 we have p(x) dµ(x) = k i=1 w i p(x i ). Lemma 6 (Remark 2.7.2 [64] ). If k ≥ 1, then the k-point Gaussian quadrature of N (0, 1) has its atoms at the roots of the degree k Hermite polynomial He k .Lemma 7 (Lemma 2.7.3 [64] ). Suppose k ≥ 1 and g k is the k-point Gaussian quadrature of N (0, 1). For j ≥ 2k, we have E X∼g k (X j ) ≤ E Z∼N (0,1) (Z j ) when j is even and E X∼g k (X j ) = 0 otherwise. In particular, g k is 1-subgaussian. Proof. From (6.2.18) on page 120 of [56] the zeros of the kth degree physicist's Hermite polynomialH k (x) = (-1) n e x 2 d k dx k e -x 2 lie in the interval - √ 2(k-1) √ k+2 , √ 2(k-1) k+2 . Since k -1 ≤ k + 2, it imme-diately follows that we have the inclusion - Lemma 10 (Lemma 2.10.6 [57] ). If Z ∼ N (0, σ 2 I n ), thenEQUATIONP max 1≤i≤n Z i -E max 1≤i≤n Z i ≥ u ≤ 2 exp - u 2 2σ 2for u ≥ 0. We will frequently use the same notation for two probability densities p and q. For sequences{a k } ∞ k=1 and {b k } ∞ k=1 , the notation a k = o(b k ) denotes lim k→∞ a k b k = 0 and the notation a k = ω(b k ) is used to denote b k = o(a k ).For a point x ∈ R, the symbol δ x denotes the probability measure which places full probability mass at the point x. The symbol * denotes convolution and the same symbol will be used in the context of the convolution of probability measures as well as functions. Throughout, iterated logarithms will be used (e.g. expressions like log log n). Without explicitly stating so, we will take such an expression to be equal to some universal constant if otherwise it would be less than one. For example, log log n should be understood to be equal to a universal constant when n < e e . 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA]Justification: The paper does not pose risks for misuse.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [NA]Justification: The paper does not use existing assets.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.At a high-level, this is the core spirit of the results in the deconvolution literature (e.g.[4,42]). However, we do not adopt a Fourier-based lens due to the issues discussed earlier.Their results also allow for a quite general class of covariate distributions beyond centered multivariate Gaussians.The reader should understand this choice of "prior" as essentially the same as the usual compound decisiontheoretic choice of the empirical distribution of the means. However, it happens to be more convenient to take the symmetrized version.4 Note that in the Bell polynomial B r,l (γ1, γ, γ3, ..., γ r-l+1 ), it follows by Definition 1 that the power of γ is given by j2 which must satisfy 2j2 ≤ r.The choice of 4 is not crucial and could be replaced by some other constant with no change to the conclusion of Theorem 1. 
 
 Appendix:  


Decision: The paper provides a sharp minimax rate for the variance estimation for the normal means model.

All reviewers were supportive of accepting the paper.

For a camera-ready version, please take into the account the comments from all reviewers regarding clarifications, for instance: inclusion of experiments, more discussion about the Gaussianity assumption and bounded means as compared to literature, and consider changing the title.

Reviews: review: paper_summary: paper_summary: The paper studies the variance estimation of the normal means model and establishes the minimax squared error in terms of $n$, the number of observations. The results assume a bounded parameter space, where the absolute means and variance are at most 1 and $L^2$ (a large hidden constant), respectively. The estimator achieves the optimal rate through cumulant estimation.
strength_weakness: strength_weakness: - The results show only the dependence on $ n$ but not $ L$, the upper bound on $ sigma$. Given that the optimal rate in $n$ is only inverse logarithmic, it seems important to understand if there are any gaps in $L$ between the lower and upper bounds (though it's treated as a universal constant).
- The boundedness assumptions are weaker, but Gaussianity is critical for establishing the paper's results. From this perspective, the methods may not be as widely applicable as some prior ones.
- The paper lacks numerical experiments. It might be helpful to run the estimator on synthetic data to understand the hidden constants and the estimator's actual performance.
questions: questions: - Please feel free to clarify/respond regarding the above comments.
- Would it be possible to go slightly beyond Gaussianity and make the results more broadly applicable?
limitations: limitations: The comparisons are detailed, and the authors explain the limitations of their results. The paper is mostly theoretical, so potential negative societal impact may not apply.
paper_summary: paper_summary: This paper studies the problem of variance estimation in compound decision setting, with the assumption that means are bounded. Main contributions:

- The authors prove the minimax rate of variance estimation in the setting of the paper, with the proof of the lower bound and a proposed estimator achieving the convergence rate.
- The authors discuss the importance of the Gaussian assumption of the result.

Post rebuttal: I acknowledge that I have read the authors responses and other reviewers' comments.
strength_weakness: strength_weakness: Weaknesses:
- There is no discussions on the computation complexity of the proposed estimator, and no experiments.
questions: questions: Questions:
- In real applications (where $\mu's$ are not chosen to maximize the errors), how does the proposed method compare to other variance estimation methods, say what are the advantages and disadvantages?
limitations: limitations: NA.
paper_summary: paper_summary: This paper gives a sharp minimax rate of the variance estimation under mild assumption on a Gaussian-based model.
strength_weakness: strength_weakness: The sign in equation (5) and also in line 94 in not clear for me, does it means proportional to? This paper is pretty theortical-based one, and it would be better if there are some experiments.
questions: questions: N/A
limitations: limitations: N/A
paper_summary: paper_summary: This submission studies the variance estimation problem for Gaussians under the compound decision/empirical Bayes settings, assuming bounded means and variance. The main results are up-to-constants matching upper and lower bounds on the estimation rate, for mean squared error. For the upper bound, the proposed algorithm is based on estimating cumulants and using Bell polynomials to find the cumulant $\gamma_2$, which is almost directly related to the desired variance parameter. For the lower bound, the authors use a moment matching and $\chi^2$-divergence approach to show indistinguishability results.
strength_weakness: strength_weakness: Here are some constructive criticisms that I hope the authors will find useful to improve the paper.

- The title: I think it is a little too general and perhaps slightly misleading, given that there has been a lot of other works studying the Gaussian variance estimation problem in the compound decisions setting, under other assumptions. The authors should consider including something to the effect of "under bounded means/variance assumption".

- In Section 3, consider re-ordering the explanation. Lines 340-341 says "This can be achieved by constructing $G$ to share a large number of moments with $\mathcal{N}(0,\tau^2)$." without much explanation. For readers who haven't seen this kind of approach, it would be helpful to explain immediately to them why this moment matching is relevant. The prose on $\chi^2$-divergence etc. later on should be moved up in my opinion, instead of diving straight into a discussion on how many moments one can match.

- My main criticism is actually that I don't find the problem setting super well-motivated, although I don't necessarily hold this against this submission in particular. I understand that this problem is very well-studied in the literature under different assumptions. However, for one, I wish the motivation were more explicit in this submission, since I'm not familiar with the model until reading this paper. On top of that, perhaps more importantly, when I first saw problem definition on the first page, my immediate thought was "ah, this is *really* going to depend on the Gaussian noise isn't it", and had pretty much the same identifiability doubts as what turned out to be discussed in Section 4. Given that information-theoretically one pretty much needs to have quite a bit of information about the noise distribution, I'm not too convinced about the applicability of the results. Having said that, I again understand that the problem has been well-studied in the literature, so clearly there are others who care about the problem, so I won't hold this against this paper in particular (as evidenced by my "Accept" score).
questions: questions: - The parameter space $\Theta(L)$, I presume the estimator (through the many hidden constants in propositions and theorems) depend on the knowledge of what $L$ is? My next question is, how does the estimation rate depend on $L$? Similarly, how does the estimation rate depend on the bound on the $\ell_\infty$ norm of $\mu$?

- What is the concentration behavior/tails of the estimator? Theorem 1 is in mean squared error only.

- Line 283 mentions that $\hat{M}_r$ concentrates around the expectation $M_r$ uniformly over the input $\gamma$. Can you comment roughly on the technique to show that, or at least why it deserves to be true? I think it's useful to comment on that even in the main paper, since uniform concentration is usually an annoying thing to show, if not sometimes a technical gist of a problem.
limitations: limitations: Yes.

Comments: title: Response to rebuttal
comment: Thank you; I forgot that this submission phrases things in the squared error for the variance.
Thank you very much for reviewing our responses. As you correctly note, it is a linear rescaling with a factor of $B$, which scales the variance by $B^2$. However, when speaking of the "rate", it is referring to squared error, so squaring yields a factor of $B^4$.
I thank the authors for their rebuttal. It is good to see the experiment is done, but a larger/complex problem with real-world data is more preferable. Thanks for taking my suggestions on the symbols and moving the definition to the main text. Overall, I will keep my score.
Thanks for your responses. I'll maintain my current score, and I hope the authors will incorporate the promised revisions, including the explanation of the scaling with $L$ and $B$ at least in the appendix (fleshed out, even if it's a straightforward rescaling argument).

By the way, for the dependence on $B$, do you mean $B^2$ instead of $B^4$ or am I misreading/misunderstanding? It's just a linear rescaling of the variable with a factor of $B$, so it should change the variance by $B^2$?