Variance estimation in compound decision theory under boundedness
Authors: Subhodh Kotekal
Venue: NeurIPS 2024 poster (2024)
Keywords: compound decision theory
empirical Bayes
normal means
variance estimation
minimax
Primary area: learning_theory

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.

Variance estimation in compound decision theory under boundedness
Subhodh Kotekal Department of Statistics University of Chicago Chicago, II. 60637 @koteekal@@chicaaooocom
Abstract
The normal means model often studied under the assumption of known variance. However, ignorance of the variance is 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 (log log n/ log n) 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 (log n/ log log n) cumulants and leveraging a variational rep- resentation of the noise variance in terms of the cumulants of the data distribution. The minimax lower bound involves a moment matching construction.
Introduction
Consider the prototypical normal means model in compound decision theory,
X; ind N (ui,o²)
(I)
for < S. where the means (M1, on Ma) E R" and the noise level a > 0 are parameters. As set out in the groundbreaking work of Robbins [47, 48, 67], the empirical Bayes setting is closely related,
Pi G,
(2)
X, p. ind N(,,
where the prior G is treated as an unknown parameter contrast to 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, 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 o² is known. The goal produce estimator or 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 the statistician (e.g. the prior G' in (2) or the empirical distribution 1 En Sp, in (1)). For estimation under square loss, the oracle estimator 1 7 (X,) where
( = B(u X = x) = x f'(x)
(3)
f(x)
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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is the posterior mean, under the prior ~ G in (2) or Tweedie's '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 's formula requires knowledge of the prior in order to compute f and thus available only to the oracle.
-E= 6n. in (1), and is known as
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 /-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 maxi- mum likelihood estimator (NPMLE) [33, 31, 34, 65, 51] being an especially popular method for g 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 minimax perspective. We will work in the compound decision setting (I), 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,
(4) where L > O. 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 in the bound ||ll I is not essential; our results essentially go through when it is replaced by some other universal constant. The boundedness conditions imposed to avoid triviality. Without boundedness,
(L))= == (p,,o) E R" X (0, oo): ||p|| S 1 and o S L},
the minimax risk is casily seen to be infinite, infø sup "ERP"."" .a>0 E E[|0² - o2
= oc, since the
variance is not identifiable. In the empirical Bayes context (2), the constraint ||]| S corresponds to the condition that G is supported on [-1, 1]: the class of priors with bounded support is popular
choice for study in the empirical Bayes literature [46, 37, 3, 31, 38). This article's '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.
1.1
Related work
The marginal distribution of the data in the empirical Bayes model (2) can be written as X1,.... Xn, iid G+ CO, o²) where denotes convolution. The problem of estimating 0² is 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 & does not vanish for large frequencies and its modulus does not decay faster than the Fourier transform of the standard Gaussian distribution
e- = If G may be somewhat smooth in the sense [G(w)] 2 -awl" for |w| sufficiently large where 0 < < 2, a >> O, and e > 0 is an arbitrary constant, then Butucea and Matias construct a Fouricr- based estimator which achieves | - 0²12 S (log - with high probability. In the rougher case (G(w)] 2 c|w]- 8 for W sufficiently large where 3 > 1, the faster rate 102 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 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

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example, the Uniform[-1,1 1] distribution and the mixture $6_1 + 201 respectively have Fourier transforms w + sin(w)/w and w + cos(w), both of which have zeros that are arbitrarily large. Without the condition that G has bounded support, the assumptions of [4] which stipulate [G(w)) is bounded away from zero for large |سا are made to ensure identifiability of the variance. However, identifiability is automatically guaranteed when G is assumed to be supported on [-1.1]. 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 Delol and so it might be guessed by taking r = 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 (O, o²) is finite everywhere and is given by M (A) (f exp(ppp (An) exp(X²o² /2)) Therefore, o² is identifialle since lim X log (A) o²
due to the bounded support of G. With this observation in hand, Matias forms the empirical moment generating function M(AA) = - exx, chooses À x vlog n. and defines the estimator 6²_ 3z log M (A). Matias establishes the upper bound B( r² S log n
but no matching lower bound obtained. Though the use of the empirical moment generating function is clever, the simpler estimator e = maxi sign X// 12/22llogg n also achieves the same rate Jō² - 0²12 2 log with high probability as pointed out by [63].
different perspective regards the marginal distribution XI...., X, did G o²) as a Gaussian mixture model with mixing distribution G. Employing a moment-based approach, Wu and Yang [63] construct an estimator which achieves 1ō - o²|2 $ k2n t with high probability under the assumption G is a k-atomic measure supported on [-1, with & S logn/ log log n. When is fixed constant, matching minimax lower bound is obtained. Their result highlights rapid deterioration in the convergence rate as increases. In particular, in the generic case where G need not be atomic (which can be intuited as k - oo), 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 (I) under the assumption |lи[]o S < 5. an estimator achieving lā2 o²| x o2 n log 1 (1 + +++////i k//n) was constructed and shown to be minimax rate-optimal in [36] (see also [11, 7D. 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 ||a||oc S 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; (X,, B) + a Z where the noise Z, NO, 1) and the design X; ~ N(0, Σ) are independent. I E RPxP is known with bounded operator norm and ||3||| S then Dicker [14] showed the variance can be estimated at rate 182 0212 < n nn + pn -2 with high probability; notably, consistent estimation is possible even with p 2 n without any sparsity assumptions on B. Assuming /3 is k-sparse, [53] (see also [1. 22) established the upper bound n + +(kk logg)) /n) For particular regimes, optimality was established [60], and the impossibility of consistent estimation for the dense regime was conjectured in the setting n o(p) and unknown E. Though some approaches were suggested in [13, 29], variance estimation with unknown 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 || S and Σ has bounded condition number, an estimator can be constructed achieving, with high probability, error l02 0²1 < with sample complexity n O(poly (log(1 / - (1/e). Without condition number assumption on Σ, consistent estimation was shown to be possible with sample complexity n (ppoly(1/e (p -vE). 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.
3

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1.2 Main contribution
Our main result is a sharp characterization of the minimax rate of variance estimation over (4) in the compound decision setting (1),
inf sup Eng (102-o* (LOEE
0z12) -029)
log log log
(5)
A few remarks are in order.
Remark 1. Though the slow, logarithmic nature of (5) is not surprising given the Gaussian mixture model results [63] discussed earlier, it notable a rate faster than log I n is achievable. The set of means J in the parameter space (4) have no exploitable structure beyond ||a||0c S 1, and intuition suggests to examine the tails of the observations to estimate o². This thinking may further suggest the least favorable prior might place substantial mass near || = 1, thus suggesting using an extreme order statistic.
Though natural, this intuition does not fully use the statistician's '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." 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 compactly supported prior which shares growing number moments with a particular Gaussian distribution. For technical reasons elaborated on in Section 3, it turns out one can only match Olog n/ log log n) moments, which subsequently yields a lower bound of order (log log n/ log n)? 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 and it encourages developing some kind of method-of-momentt estimator. Though such strategy has seen success in a linear regression setting [35], complications seem to appear in the sequence model (1) as discussed in Remark 2. Cumulants, which are closely related to moments, have much more convenient properties, and so we develop cumulant based variance estimator which essentially requires estimating OOlog n/ log log n) cumulants, parallelling the lower bound.
Remark 2. Variance estimation in the sequence setting (1) turns out to be harder than in regression. Consider the linear regression model Y, = (X,, 3) + Zi where 3 E R" is unknown, and the noise Z ~ (O, 1) is independent of the design X; N (0, Σ). For comparison with the sequence model (1), consider the setting p = n and |||||
v || S 1. Without any condition number assumption on E, Kong and Valiant [35] construct an estimator achieving, with high probability, the rate (translating their sample complexity result to estimation risk) |02 2 S (log log n// n//log n) Though this result is not directly comparable to (5) since Kong and Valiant impose the fn-nrrm constraint ||:3|| S 1 whereas the space (4) imposes an -oom constraint more natural for compound decision theory, it helpful for intuition to appreciate how the regression setting differs from the sequence setting to yield a "faster" rate.
In regression, it is clear the second marginal moment is E(Y?) = (B, Egi + o2, and so variance estimation is (up to parametric rate) equivalent to estimation of (3, EB). By changing focus to this target, o² is now viewed are a source of fresh randomness that is independent of the noise. Since X; is correlated with Y, but independent of Z,, it is clear (X1, (8, B)) and so on. In other words, unbiased estimators of the terms {(3, Σ k== are easily constructed due to the availability of the random covariates {x,)=_ The clever idea of Kong and Valiant [35] is to reach for polynomial approximation tools and approximate (3, Σ83 by the readily
nuisance. The key advantage in regression is that the covariates
= (B, E² E(Y; (X1, Xx),
estimable (3, Σx })] k=22
The sequence model (1) is equivalent to linear regression with deterministic orthogonal design, which is quite different from the random design considered above. The target || is the target analogous (,, Σ, The only accessible randomness in (1) are the responses themselves {X,) t=1 n Σ E 1 Σ E(u + 2,)*) all involve the nuisance a, thus precluding an application of Kong and Valiant's idea. The interesting phenomenon that design
The marginal moments 1
At a level, this is the core spirit of the results in the deconvolution literature (e.g. [4. 42)). However. we do not adopt Sourier-based lens due to the issues discussed earlier.

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properties can have substantial effects on minimax rates has been noted in the literature in other settings [11, 6, 59], and we suspect this 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 their status the core of mathematical statistics more broadly), the cumulant-basse 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; careful study remains an open problem.
2 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 random variable. But, in contrast moments, they have particularly nice attributes for the purpose of variance estimation in (I) and (2). Some key properties cumulants are briefly reviewed before the estimation methodology is developed.
2.1 A brief review of cumulants
Suppose Y is random variable such that its moment generating function M := E(e exists for all À E R. The cumulant generating function is defined by K (A) :- log M (A). and admits the power series ((A) = Er= K.A" where Ky == (() is defined be the rth cumulant. Though K1 and coincide respectively with the mean and variance of Y, the higher cumulants do not have simple relationship with the moments. However, the first r cumulants are determined completely by the first moments; likewise, the first moments are determined completely by the first cumulants. In fact, an explicit correspondence is available via Bell polynomials.
y
Definition 1. The incomplete Bell polynomial B 1 is given by
r! jilje!... Br.(11,... 17-1-1)= = -
11
51
82
(r-
(
Tr-(41
de 111
11
21
where the sum is taken over all sequences j1, J.. fr- 1+1 of nonnegative integers such that f1 + ...+1. Jr-141 = l and jı + 2j2 + 3jz +... (r - 1 1)j
The correspondence between moments and cumulants is given by
Ky
(-1)1- 1)! (, m2 ..... my-141),
(6)
m, (KI R2, Ka, k41)
where m, = E(Y") 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 yi + Y2 for independent Yi and Yg is given by the sum of the individual cumulant generating functions. Consequently, the rth cumulant of the sum is the of the rth cumulants.
2.2 Identifying the noise variance from the marginal cumulants
In the model (1), the second moment exhibits _ - X? boundedness of [|д]] and a. It is clear estimation of o² is equivalent to estimation of || /n up to a (negligible) parametric slowdown in the rate. Associated with any estimator Q for the quadratic functional |
+ Op(n-1/2) due to the
corresponding variance estimator 8² Ei X? Q. Hence, suffices
"Their results also allow for quite general class of covariate distributions beyond centered multivariate Gaussians.
5

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to 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 Kr and m, denote the rth cumulant and moment respectively of the marginal distribution G * N (0, o²) of the data. Likewise, let "rr and vr denote the rth cumulant and moment respectively of G. Since the noise is mean zero and a 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. 72-
is not immediately clear how to identify 7/2 from the cumulants H,, of the marginal distribution of the data G (N, d²). Consider the moment generating function of (0, o²) is À + exp(X2o²) /2), and so all the cumulants (except the second) are equal to zero. Therefore, K1 = K,,K2 7/2 + o², and Ki "r for 2 3. The upshot is we able directly estimate all the cumulants of G, except the second, from (X, )x))²=1
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 72 by exploiting the boundedness of G's support. For 2 2, define the function Mp [0, oo) + RR with
M)(() Σ Brl(1 Y, fr....7--- 1=1
(8)
where B, is an incomplete Bell polynomial (see Definition 1). Note by (7) that M, gives the rth moment associated to the cumulant sequence (71 7. Ya....) it a valid cumulant sequence. The following proposition asserts variational representation of y2-
Proposition 1. If G is supported on [-1, 1). then 72- = sup (7 E [0 [M+(7)] S 1 for all r 1 2}.
Proof. For ease of notation, y sup (7 E [0, MM,((>]] S
for all 2 2]. It clear 7/2 S
7" since M,(12) V4 [-1, for all 2 2. To show the lower bound, fix 8 > 0. For any 72 8, observe that the cumulant sequence of the distribution G * (0, 7²) is (71, 72 +T², 73: 74..... Since
G' + (O, r²) does not have all moments contained in [-1, it follows 12 + is not feasible
point. Since this holds for all 72 2 8, it immediately follows ? I 7/2 + 5. Since 5 > 0 was arbitrary,
we have shown Y" S 7/2
Proposition I enables recovery of 722 from the other cumulants. However, it requires certifying all putative moments (M+ live in [1,, which is a difficult task given only finite amount of data. One idea is to approximate 7/ by certifying only that an rth putative moment lies in [-1, 1] for some large choice of r. Specifically, define for r > 2.
(r(r) == sup fr E [0, 1] IM,(7)) 13.
(9)
Proposition 2. Ifr is even, then l52(2) - 72l S c for some universal constant C' > 0.
Proof. By (9) and Proposition 1, follows 72 S 52(r). Therefore, Mr is the rth moment of G * [00, (r) 72). Since is even, taking / G and Z ~ [0, 1). it follows 1 2 (f2(1))) = E((u= + (z)(r) - -))
Σi= (1) E(p ( (f2(() /2) If -1 odd, then I must be odd since is even, and so we must have E(Z") O. It thus follows 1 2 Eigr, Leenn "= - 72) E(Z') 2 (5a(r) - -2) (>"""//))) Hence, 152(r) ा
72 < (E(Z")-2/ --1r 1/r (2r/21 ( (x41) -/rr It follows from Stirling's 's approximation that for some small universal constant > whose value may change from instance to instance, we have 2/r
((#)) 2/r 2 ( (
2 cr. This immediately yields the claimed bound. The
proof is complete.
The methodological strategy is in place; an estimator will be constructed to estimate 52(r) for well-chosen value of to balance the estimator's 's variance with the bias from Proposition 2.

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2.3 Methodology
We return to the compound setting of (I); 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 the prior were the empirical distribution of the means. Note the quadratic functional of interest || /n is preciscly equal to the second moment of the empirical distribution of the means.
To describe our estimation strategy, some preliminary development necessary. For even r. set my to be the rth moment of the distribution [ >E= 5p.) * . (0, r²). For odd r, set my = 0. Observe Wi, 2/..... ... are the moments of the symmetrized distribution ( >ー= (55,, + ===))) NO, o²). Thus, we are able to conceptually place ourselves the context of Section 2.2 by making the choice of "prior" G = Ei= ( + 22__,)). ). Let us adopt the notation of Section 2.2, namely let Kr, denote the rth cumulant of G [ o²), and let "r and v, denote the rth cumulant and moment respectively of G. As noted in Section 2.2, we have Rr = " for + Furthermore, note the quadratic functional of interest || is exactly 7/2
The strategy is to estimate (2()) given by (9) for a carefully chosen value r. pursuit this strategy, estimators for the cumulants (except the second) will now be constructed. Define the moment estimators
1 E X
if is even,
m
(10)
0
otherwise.
Cumulant estimators are obtained via plugging in to (6).
3r = E--1--1u 1)1 (m,, m2 m,-1+1
(1I)
1=1
for r * To estimate 52 (r). an estimator of the function M, given by (8) is needed. Define the (random) function M, [0, xc) - given by
M, Σ
fr-11))
(12)
For r 2 2 and E > 0, define the estimator
(r(r) == sup {7 E [0, [M((7) s $1+e}
(13)
where r, tuning parameters to be chosen. The approximation.
behind (2(r) lies in the
(f(f(())) (M((2(r) x Mx(f2(r))
(14)
Suppose it could be shown M,(7) concentrates around M2((7) uniformly over 7 [0, If so, then the first approximation in (14) follows. The Bell polynomial structure is quite convenient as it facilitates straightforward proof of the desired uniform concentration (see Appendix B.2); cumbersome empirical process theory is avoided. Since both (r( and (r(r) are defined in terms of supremums in (13) and (9), both M, (r)) and M, (rz( will be near one, intuitively yielding the second approximation in (14). Thus (M, M, (f(())) is small, and since M, is an -edegree polynomial", intuition suggests (T) (2)r can be controlled via an f-degree Taylor expansion. Combining with the approximation error bound provided by Proposition 2 delivers bound on li2 (r) 72/. as the following result states.
Proposition 3. Fix E > 0 and even Let ε = { Suu M, - (M,))))((77) se} where M, and M, are given by (12) and (8) respectively. On the event E, we have (r2( S ( + where (r( is given by (13) and C > 0 is universal constant.
'The reader should understand this choice of "prior" as essentially the same as the usual compound decision- theoretic choice of the empirical distribution of the means. However, it happens to be convenient take the symmetrized version.
"Note that in the Bell polynomial Bra(m Jay... -1-1,)), it follows by Definition I that the power of is given j2 which must satisfy 2/2
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Proof. It follows from the definition (9) of (r()) and Proposition that "2 S 72(r). On the event E, it follows from the definition of (r(r) and (32())) 1 that (r(r) S (2(r). Therefore, on the event E, follows by Lemmas I and 2 that 1 = (f()(r)) - (f(((()) S (12((()) + S
(r)) - 2s, and an application of Lemma 3 yields 5 (r) (r) & ((r/2)! 22))/rr Combining this bound with the approximation error bound of Proposition 2 delivers the desired result.
1+ 2z. This implies 0 S M, (f2(((r))
To obtain estimation enror bound, it remains to investigate the stochastic error, i.e. the typical magnitude of ( [M)(1) M,(()), so that the first approximation in (14) is justified. The cumulant estimators (5x ( sisgr, 2yt used to define M,(7) are constructed by plugging in sample moment estimators for the even moments. Heuristically, only slow concentration rate can be expected because the number of moments r being estimated is large.
Proposition 4. Suppose (H, O) E ④ and c², >> 0 are universal constants. If 2 log" then there exist universal constants Ca C,C² depending only on 3 and et such that the following holds. log
and o² then Pu.o supp7eu..1 MM(1)) MM(()) S C'
2 1 - 8.
S
Cp.e log log n2
Vnó
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 o² ž +. This condition may be an artifact the proof, and the conclusion of Proposition 4 might continue to hold for small a. 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
integer and real 0, define
x? Fa(r)) 1 1=1
0² =
{ mmaNIsism X:34V 18g
(15)
where (r) is given by (13).
Theorem 1. Let 5 log? Let Cs C. and C' be the universal constants from Proposition 4 corresponding c² and 3 2. There exists universal constant C* 2 Cp,+ such that the
log n following holds. I is the largest even integer less than equal to t log lug then
and Ce rlog
Vnb
(log log n log
Ep. (102- - -o?1) S (M,O)E0
sup
where 6² is given by (15).
Since || implies that maxigiga X; is typically no larger than 1+ / 20 log n., the truncation
choice 4 log essentially results using the trivial estimator à² when x² S + On the other
hand when o² Z 1. the stochastic error result of Proposition 4 is meaningful and it can be shown DL X? (r) estimates 6² well.
3 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, O such that inf, suP(n.))e(()) Pp 0222
C(
Log
1ogg))?] 2 c.
log n
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 || S 1. As discussed, the boundedness constraint is imposed to ensure o² is identifiable. Though quite
'The choice of 4 is not crucial and could be replaced by some other constant with no change to the conclusion of Theorem 1.
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elementary, it is conceptually useful for our construction to examine why O² is unidentifiable if no constraints are imposed. Specifically,, it can be understood from lower bound perspective by examining the reduction to the following Bayesian testing problem, Ho
0 and o² = 1 + 72 against H1 : /4: ~ N(0, 72) and o² = 1. Observe that under both hypotheses the data are X1..., X,, are i.i.d. draws from (0,, 1 + The hypotheses are indistinguishable yet the variances differ by 7² under Ho and Ht, Therefore, any variance estimator incurs square loss of at least r²; taking Do establishes infinite estimation risk is inescapable.
sid Under the boundedness constraint ||4]| [0 < 1, the choice of prior P1. Mn ~ N( 7²) is no longer feasible. One idea is to find distribution G supported on [-1, 1] such that the marginal data distribu- tion G N(, (0, 1) is indistinguishable from (V, (*), * (0, 1). This can be achieved by constructing G to share large number of moments with (0, r.). To elaborate, if the first 2k 1 moments match and
T < 1, then (see Theorem 3.3.3 of [64) we have x N (0, 1) || I (0, (O, & vIK-T T-
16 44
well-known sufficient condition (e.g. see [58]) for the desired indistinguishabiiity is that the x²- divergence is at most (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 & even, the kth moment of NN r²) is at least, by Stirling's 's approximation, c log k/2 for some small constant the kth moment of G must lie in [-,, Therefore, a necessary condition for the kth moments to match is n e log < 1. This crude reasoning already shows G can only match at most O(1/79) moments.
O. On the other hand,
The actual construction of relies on the technology of Gaussian quadrature. With Proposition 9 and Lemma 8, it can be shown that the k-atomic Gaussian quadrature G = Ei W;ōrz,, where (= are the roots of the kth Hermite polynomial, is supported on the interval and shares the first 2k - 1 moments with (0, A sufficient condition for G to be supported on [-1, 1] is k S 1/(47²). Therefore, a properly supported G which matches Ω (1/r²) moments of (0, can be constructed, validating that our crude reasoning from earlier is tight. The problem now boils down to choosing 72. Since k x 7-2 2 it follows from the x" divviveree bound implied by moment matching that choosing -²x log log yields the desired 1 logn
[-
- 4), /
bound on the x² divvergeee and is precisely the desired separation.
4 Noise agnosticism
The cumulant-bass 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 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 nothing beyond subgaussianity (see Definition 2) is assumed. The impossibility can be seen by appealing to Le Cam's 's two-point method to reduce the problem to two-point testing problem and using the following simple construction, iid
Ho : p4 Rademacher (1/2), 0² = 1, and & sid
Rademacher (1/2).
H1 : p 0. = and &vV kid Rademacher (1/2) * Rademacher (1/2). It is clear both noise distributions are 2-subgaussian. Furthermore, the data {X; } are indepen- dent and identically distributed according to Rademacher (1/2) Rademacher (1/2) under both hypotheses, and so it is impossible to distinguish Ho and Hi. Since the separation between the choices of 2² between the two hypotheses is 1, it follows consistent estimation impossible. This is formally stated in the following proposition without proof.
Proposition 5. For > let E. == (P: P is a-subgaussian, has mean O and variance 1}. Let PuooPP denote the joint distribution of the observations from the sequence model, X; 14 + oof 1 5 i where a Pg are i..dd Then inf, sup( Ptez, Eu.o.P (IØ2 - 2² 1
In this sense it is necessary to exploit finer information about the noise, and so our cumulant- baaed estimator is designed to exploit the noise's 's Gaussian character in (1).

---

Acknowledgments and Disclosure of Funding
This work was supported part by NSF Grant ECCS-2216912. The author thanks Chao Gao for helpful discussions.
References
[1] Bayati, M., Erdogdu, M. A., and Montanari, A. (2013). Estimating LASSO Risk and Noise Level. In Advances Neural Information Processing Systems.
[2] Brown, L. D. and Greenshtein, E. (2009). Nonparametric empirical Bayes and compound decision approaches estimation of a high-dimensional vector of normal means. Ann. Statist. 37(4):16851174.
[3] Brown, L. D., Greenshtein, E., and Ritov, Y. (2013). The Poisson compound decision problem revisited. J. Amer. Statist. Assoc. 108(502):741-79..
[4] Butucea, C. and Matias, C. (2005). Minimax estimation of the noise level and of the deconvolu- tion density semiparametric convolution model. Bernoulli 11(2):309-340.
[5] Cai, T. T. and Jin, J. (2010). Optimal rates of convergence for estimating the null density and proportion of nonnull effects in large-scale multiple testing. Amn. Statist. 38(1):100-144
[6] Carpentier, A., Klopp, O., Löffler, M., and Nickl, R. (2018). Adaptive confidence sets for matrix completion. Bernoulli 24(4A):2429-244) 2460.
[7] Carpentier, and Verzelen, N. (2019). Adaptive estimation of the sparsity in the Gaussian vector model. Ann. Statist. 47(1):93-126.
[8] Castillo, I. and Mismer, R. (2018). Empirical Bayes analysis of spike and slab posterior distributions. Electron. J. Stat. 12(2):3953-400)
[9] Chen, J. (2017). Consistency of the MLE under mixture models. Statist. Sci. 32(1):47-63.
[10] Chen, X., De, A., and Servedio, R. A. (2020). Testing noisy linear functions for sparsity. In STOC 20-Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pp. 610-623.
[11] Comminges, L., Collier, O., Ndaoud, M., and Tsybakov, A. B. (2021). Adaptive robust estimation in sparse vector model. Ann. Statist. 49(3):1347-1377.
[12] Deb, N., Saha, S., Guntuboyina, A., and Sen, B. (2022). Two-component mixture model in the presence of covariates. J. Amer. Statist. Assoc. 1 1880544)
[13] Dicker, L. H. (2014). Variance estimation in high-dimensional linear models. Biometrika 101(2)2299-284.
[14] Dicker, L. H. and Zhao, S. D. (2016). High-dimensional classification via nonparametric empirical Bayes and maximum likelihood inference. Biometrika 103(1):
[15] Efron, B. (2008). Microarrays, empirical Bayes and the two-groups model. Statist. Sei. 23(1):1- 22.
[16] Efron, B. (2011). Tweedie's 's formula and selection bias. J. Amer. Statist. Assoc. 106(496):1602- 1614.
[17] Efron, (2014). Two modeling strategies for empirical Bayes estimation. Starist. Sei. 29(2):285 301.
[18] Efron, B. (2019). Bayes, oracle Bayes and empirical Bayes. Statist. Sci. 34(2):177-20
[19] Efron, B. and Morris, C. (1972). Empirical Bayes on vector observations: an extension of Stein's method. Biometrika 59:335-347.
10

---

[20] Efron, B. and Morris, C. (1972). Limiting the risk of Bayes and empirical Bayes estimators. II. The empirical Bayes case. J. Amer. Statist. Assoc. 67:130-139.
[21] Efron, B., Tibshirani, R., Storey, J. D., and Tusher, V. (2001). Empirical Bayes analysis of microarray experiment. J. Amer. Statisl. Assoc. 96(456):1151-111
[22] Fan, J., Guo, S., and Hao, N. (2012). Variance estimation using refitted cross-validation in ultrahigh dimensional regression. J. R. Stat. Soc. Ser. B. Stat. Methodol, 74(1):37-65.
[23] Ghosal, S. and van der Vaart, A. W. (2001). Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann. Statist. 29(5):1233-124)
[24] Ignatiadis, N., Saha, S., Sun, D. L., and Muralidharan, O. (2023). Empirical Bayes mean estimation with nonparametric errors via order statistic regression on replicated data. J. Amer. Statist. Assoc. 118(542)99879999
[25] Ignatiadis, N. and Sen, B. (2023). Empirical partially Bayes multiple testing and compound x² decisions. ArXiv:2303.022 028887 [math, stat].
[26] Ignatiadis, N. and Wager, (2022). Confidence intervals for nonparametrie empirical Bayes analysis. J. Amer. Statist. Assoc. 1176539):1499-1166
[27] James, W. and Stein, C. (1960). Estimation with quadratic loss. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I, pp. 361-379. Univ. Califomia Press, Berkeley-Los Angeles, Calif.
[28] Jana, S., Polyanskiy, Y., and Wu, Y. (2022). Optimal empirical Bayes estimation for the Poisson model via minimum-distance methods. ArXiv:2209.0132. [math, stat].
[29] Janson, L., Foygel Barber, R., and Candès, E. (2017). EigenPrism: inference for high dimen- sional signal-to-noist ratios. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79(4):1037-1065.
[30] Jiang, W. (2020). On general maximum likelihood empirical Bayes estimation of heteroscedastic IID normal means. Electron. J. Stat. 14(1):2272-22
[31] Jiang, W. and Zhang, C.-H. (2009). General maximum likelihood empirical Bayes estimation of normal means. Ann. Statist. 3744):6447-1684
[32] Kamath, G. Bounds on the Expectation of the Maximum of Samples from Gaussian.
[33] Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Statist. 27:887-906.
[34] Koenker, R. and Mizera, I. (2014). Convex optimization, shape constraints, compound decisions, and empirical Bayes rules. J. Amer. Statist. Assoc. 109(506):674-65..
[35] Kong, W. and Valiant, G. (2018). Estimating Learnability in the Sublinear Data Regime. In Advances in Neural Information Processing Systems.
[36] Kotekal, S. and Gao, C. (2024). Optimal estimation of the null distribution in large-scale inference. ArXiv:2401.0665 [math, stat].
[37] Li, J.. Gupta, S., and Liese, F. (2005). Convergence rates of empirical Bayes estimation in exponential family. J. Statist. Plann. Inference 111
[38]
Liang, T. (2000). On empirical Bayes test for normal mean. Ann. Statist. 28(2):648-655 655. [39] Lindsay, G. (1989). Moment matrices: applications in mixtures. Ann. Statist. 17(2):722-740.
[40] Lu, M. and Stephens, M. (2016). Variance adaptive shrinkage (vash): flexible empirical Bayes estimation of variances. Bioinformatics 32((2):3428-35 3434-34344
[41] Lu, M. and Stephens, (2019). Empirical Bayes estimation of normal means, accounting for uncertainty in estimated standard errors. ArXiv:1901.1061 [stat].

---

[42]
Matias, C. (2002). Semiparametric deconvolution with unknown noise variance. European Series in Applied and Industrial Mathematics. Probability and Statistics 6:271-292.
[43] Meister, A. (2006). Density estimation with normal measurement error with unknown variance. Statist. Sinica 16(1):195-211.
[44] Morris, C. (1983). Parametric empirical Bayes inference: theory and applications. J. Amer. Statist. Assoc. 78(381):47-65.
[45] Polyanskiy, Y. and Wu, Y. (2020). Self-regularizing Property of Nonparametrie Maximum Likelihood Estimator in Mixture Models. ArXiv:2008.088 08844 [math, stat].
[46] Polyanskiy, Y. and Wu, Y. (2021). Sharp regret bounds for empirical Bayes and compound decision problems. ArX1v:2109.03943 [cs, math, stat].
[47] Robbins, H. (1951). Asymptotically subminimax solutions of compound statistical decision problems. In Proceedings of the Second Berkeley Symposium on Mathematical Statisties and Probubility, 1950, pp. 131-148.
[48] Robbins, H. (1956). An empirical Bayes approach to statistics. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, vol. I, pp. 157- 163.
[49] Saha, S. and Guntuboyina, A. (2020). On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising. Ann. Statist. (8(2):738-762.
[50] Shen, Y.. Gao, C., Witten D., and Han, F. (2020). Optimal estimation of variance in nonpara- metric regression with random design. Ann. Statist. 48(6):3589-3618.
[51] Shen, Y. and Wu, Y. (2022). Empirical Bayes estimation: When does g g-modeling beat f- modeling in theory (and in practice)? ArXiv:2211.0 Stfi:22211.1200 12692 [math, stat].
[52] Singh, R. S. (1979). Empirical Bayes estimation Lebesgue-exponential families with rates near the best possible rate, Ann, Statist. 7(4):890-902.
[53] Sun, T. and Zhang, C.-H. (2012). Scaled sparse linear regression. Biometrika 9794):879-898.
[54] Sun, W. and Cai, T. T. (2007). Oracle and adaptive compound decision rules for false discovery rate control. J. Amer. Statist. Assoc. 102(479):011-912
[55] Sun, W., Reich, B. J., Cai, T. T., Guindani, M., and Schwartzman, A. (2015). False discovery control in large-scale spatial multiple testing. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77(1):59-83.
[56] Szegö, G. (1939). Orthogonal Polynomials, volume Vol. 23 of American Mathematical Society Colloquium Publications. American Mathematical Society, New York.
[57] Talagrand, M. (2021). Upper and Lower Bounds for Stochastic Processes: Decomposition Theorems, volume 60 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / Series of Modern Surveys in Mathematics. Springer International Publishing, Cham.
[58] Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. Springer, New York.
[59] Verzelen, N. (2012). Minimax risks for sparse regressions: ultra-high dimensional phenomenons. Electron. J. Stat. 6:38-90.
[60] Verzelen, N. and Gassiat, E. (2018). Adaptive estimation of high-dimensional signal-to-noise ratios. Bernoulli 24(4B):3683-3710
[61] Wang, L., Brown, L. D., Cai, T., and Levine, M. (2008). Effect of mean on variance function estimation in nonparametric regression. Ann. Statist. 36(2):646 664..
[62] Wang, W. and Stephens, M. (2021). Empirical Bayes matrix factorization. J. Mach. Learn. Res. 22:Paper No. 120, 40.
12

---

[63] Wu, Y. and Yang, P. (2020). Optimal estimation of Gaussian mixtures via denoised method of moments, Ann. Statist. 48(4):1981-2007,
[64] Wu, Y. and Yang, P. (2020). Polynomial methods in statistical inference: theory and practice. CIT 17(4):402-586
[65] Xing, Z., Carbonetto, P., and Stephens, M. (2021). Flexible signal denoising via flexible empirical Bayes shrinkage. Mach. Learn. Res. 22:Paper No. 93, 28.
[66] Zhang, C.-H. (1997). Empirical Bayes and compound estimation of normal means. Statist. Sinica 7:181-193.
[67] Zhang, C.-H. (2003). Compound decision theory and empirical Bayes methods. Ann. Statist. 31:379-390.
[68] Zhang, F. and Gao, C. (2020). Convergence Rates of Empirical Bayes Posterior Distributions: Variational Perspective. ArXiv:2009.0396) [math, stat].
13

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Appendices to "Variance estimation in compound decision theory under boundedness"
The appendices are organized as follows. Appendix A discusses the connection to Gaussian mixture model results of [63]. The remaining proofs for the results of Section 2.3 are presented in Appendix B and the proof of Theorem 2 is presented in Appendix C. Appendix D contains auxiliary definitions and results. Finally, Appendix E describes the notation used in the main text and the appendices.
A Relationship to Gaussian mixture model
Viewing the marginal distribution Yang [63] assume G is a k-atomic distribution with bounded support and, for S log log show that the variance estimator furmished by Lindsay's 's algorithm [39] achieves |62 -2 S k2n with high probability. minimax lower bound of order n è is obtained, establishing optimality for fixed k.
did G [(, o²) as a Gaussian mixture model, Wu and log
The main focus of Wu and Yang's article estimation of G with respect to the Wasserstein 1-distance. In the case where the variance o² is known, they develop denoised method of moments (DMM) estimator and prove achieves the minimax estimation rate. Notably, their results cover the case where G has a continuous density (see Theorem 5 in [63]); in this case, G is approximated by finite logn
mixture with CTg log atoms (for some small constant 0), and the approximation is estimated by the DMM estimator.
When x² unknown, Wu and Yang study Lindsay's s algorithm (Algorithm 3 in [63]). However, log their article [63] only addresses the case k S Clog logn and no analogue of Theorem is offered; no error bound for the variance estimator is offered either. For continuous G, natural idea is to mimic the known variance case by approximating the continuous G by k-atomic measure and running Lindsay's ' algorithm with this choice k. One may hope the fact that the data actually come from the continuous G and not the k-atomic approximation does not pose problem.
Unfortunately, appears the analysis in [63] of Lindsay's 's algorithm strongly relies on the assumption that the true data-genee generating distribution G' is an atomic measure. To illustrate, let us examine the proof on page 1997 in [63]. Denote र = G * N (0, o²) and π = G NO, o²) where G, d2 are the outputs of Lindsay's algorithm. Consider the case Following their proof, though Proposition in [63] cannot now be directly invoked since G is not k-atomic, one may hope a modification of Proposition 3's proof might be possible. However, Proposition 3 relies on Lemma 13 (found in the supplementary material of [63]), which would like to apply to the measures & N(0, σ²) and G. Though Lemma 13 does not require & to be an atomic distribution, it does require G to be k-atomic and the proof of Lemma 13 makes essential use of this assumption. It is not clear whether this can be circumvented in straightforward manner. It is an interesting open problem to obtain an analogue of Theorem 5 in the case of unknown variance, and also to establish whether or not Lindsay's 's algorithm can achieve the optimal estimation rate.
ô, and denote G' = G * N0, 0² o²).
B Upper bound
The high-level justification behind the development of the estimation methodology presented in Section 2.3 is the approximation (14). As discussed, the two key pieces are the handling of the stochastic error sup,ye(o.1) (M+(7) M,(r)) and the approximation error lf2(r) (2(r),, the latter of which is controlled through Taylor expansion of M,. Analytic properties of M, are presented in Appendix B.1 and the concentration of M, is presented in Appendix B.2
B.1 Analytic properties of M,
The following results pertain the function M, given by (8) and are made with the context of Section
2.2 in force.
14

---

Lemma 1. If is even, then the function M, given by (8) is strictly increasing on the interval (12,00).
Proof. Observe for y > 72 that M,(r) gives the rth moment of the distribution + NO 72).
In other words, letting μ G and Z ~ (0, 1). we have
M2(+) = F. ((1 VT -2 Σ -0
Consider that if is odd, then I must also be odd since is even, which implies E(Z') 0. Therefore, we can write
M,()) = Σ (1)
USISr
Hence, M, is strictly increasing in on the interval 00) as claimed.
Lemma 2. Ifr is even, then Mf(f2(r)) 1 where M, given by (8) and (r()r) is given by (9).
Proof. Consider Mp is a continuous function as it is a polynomial. Observe M, S 1 by Proposition 1. Furthermore, consider since M,(() is the rth moment of G' * N0, 72) for y > ?/2 it follows lim (7) = do. Furthermore, since Lemma asserts M, is monotonically increasing on (12, oo) and since (r) 2 2/2 it follows by continuity that M, (52(r)) =.
Lemma 3. If x 2 y 2 "/2 and even, then x - y S ((,(2) M,(r))) 2/r where M, given by (8).
Proof. From the proof of Lemma and taking μ G and Z ~ (O, 1), have for y 72.
Mr(7)= = Σ ()E11))-)
Osten. Leven
For S z/2 and for 7 2 72, follows from even that
(1/2)1 (1/2 - k)) (7 722 E((u)-222)
(7) = Σ
Oster, I oven. 1/22k
Since M1(r) is an dedegre polynomial in y, Taylor expansion along with the above bound yields
r/2 (x
y)*
(a
M,(x) - Mr(1) - Σ (g)
k!
(r/2)!
km
Here, we have used that every term the sum nonnegative. (x y)* 2 since 2 y. Rearranging gives the desired result.
particular, (*)((y) 2 and
B.2
Concentration of M,
As the definition of M, (12) relies on moment estimators vi, given by (10) and cumulant estimators f given by (11), we will first collect results for these intermediate estimators. Further, recall from Section 2.3 that m, 1 Eh_ E(())) if even and m, = 0 otherwise.
Lemma 4. If ||o I 1. then Var(m,) ("rr)"
15

---

Proof. The claim is trivially true for odd r, so consider even r. Write X, = + oZ. 1). Observe by independence and 1us1 I 1, we have Var(m,) =
where Zg iid
Var
E==1 X
= n E Var(XT) I +/- Ei= E(X?)). By Jensen's inequality, we have 2r 2 (n7 + (o)z))r) Taking expectation yields
1
X²" 22r [ + oZ.
E(X?) < 2 (n," + o (2r - 1)!!) s 2 (1 + o 2r (2r 1 1)2r-1) S 4" + +o 00 (4r)2r,
which yields the desired result.
Corollary 1. lf S 1 and u >> 0, then Pa. (maxisisr Im mil > n) S r +"*r((vv")
a
Proof. The result follows by union bound and Chebyshev's inequality with Lemma 4.
Proposition 6. Suppose (n, a) E (I.) where ((L) is given by (4) and where is a universal constant. There exists a large universal constant C > O such that the following holds. If u >> 0 such that C.u
1/ar E (0, 1), then on the event {max1 Im mul S u) we have
max 18a 1--ml=---- S (1 L'r!) (r!)² (2r)"
eInorru - 1/.
Proof. The argument borrows heavily from the proof of Lemma A.4 in [10]. For S & S r with k * 2, we have by Definition! 1,
1ix 7kl
(1 BR, m. m+11+1) (r.
k
k!
(1 - 1)! Σ
j1!j2! ... jo-
1=1
ju
fk-141
1k-141
mi
m22
11--111
M1
m2 j2
mk-
1!
2!
(k: (+=1
1!
2!
(4: - 1+111
where the sum taken over all sequences J1, ... . fk-1+1 of nonnegative integers such that ji + jk--+1 I and j1 + 2j2 + 3js
+ (k- 1 + 1)jk-141 = k. By definition of my given by (10) and the definition of my given in Section 2.3, we have m; = m3; 0 for odd 1. Consequently, we need only examine terms in the sum such that jr = 0 for all odd 1, so let now fix such term.
For even 1, we have by Jensen's 's inequality my 1 Σm= E (4s + +z + E p4 + oZ 2
E( Z()) where Z ~ (0, 1). Since / is even, E((oZ(') = of E(Z') = of TT for some small universal constant > by Stirling's 's approximation. Consider (d'o21)1/2 >
N (do21)//
min 2 C-1(1 A σ²) since we take C > be a sufficiently large universal constant.
Observe the event (maxigis, I me| u). have
m
di
114-141
26-1+1
1
2!
(k
1+11
Mi
+
u
m2 + u
52
(M + u
fe-141
1!
2!
(k 16--++1 1)!
54-1+1
k 111
m,
m2
my m1-1141
u
exp
2!
(k: 1 1)!
m,
To obtain the second line we have used that 0 <
< (m for all 1 < S h: - 1 + since ji = for all odd 7. We have also used the inequality + < e to obtain the third line. Since
( fy ji
m C-" C="(LAo") for even as established earlier, it follows by the identity jı +... + j6-141 = (
16

---

that exp for all odd i, it follows by Jensen's inequality (letting Y ~ G (N, x² that m ...mm- ++1 I E(Y
J4 m,
einorlu tu, Since j1 + + 3j3 + t... 1-1)++1j----- k and ji
--1441
< I1: B(Y
11:-1 k- B(y #) S I1: E(Y m.. This
inequality yields
j1
Ja-1+1
m2
M2
m
1!
2!
(A: - 1 1)!
7701
m2
da
(k m1-1+1
36-141
+ mi (ensorlu-1)
(16)
1!
2!
1)!
Let us now prove an analogous lower bound. Again, since m, 2 C- A a²) for even i and
J1 + ... + J L, have
mi
j1
m2
j2
mik 1+1
du t11
2!
kk 1 + 1)!
m1
J1
m2
ja
m4-1+1
j6-1+1
1!
2!
(k: 1 1)!
h
jK-i+1
m1
m1
m4-1+1 - m 1+1
m
7k-141
m7
J1
m2
d2
m=-1+1
jk-1+1
u
u
jk-l+1
1!
2!
(kc 1 1)!
mi
mi--+1
T00
-1t-1+1
ja-1+1
C
N
Añ u
1!
2!
(k - 1 1)!
771
172
82
mk--44
Sk-1+1
C
IV
1!
2!
(k: 1 + 1)!
1 1 o"
J1
m2
d2
mk-1+
S1-1+1
C
m&
1 1 or lu
1!
2!
(k 1+ 1)!
16-111
m1
m2
m
(eRorru- Iu
(17)
1!
2!
(k: - 1 1)!
Here, we have used
C
E (0, 1), jr = for all odd I, and my 2 C- A o²) for even i to 1 1 Ju
Ino
conclude (1 my m
z for all g, thus obtaining the third line. We have also lN E 10 to obtain the third-to-last line. Therefore, it follows from (16) and (17) that
used C
K! 1i - -kl < E (1 - 1)! 1 jilje! jo-141!
or e
Iu
1=1
$ m& le korku - 11- (k!) A.
< 2t (1 + L kt) T ku 11 (k!)² k.h.
The last inequality follows from || I and Jensen's 's inequality. We have also bounded the number of admissible sequences Jk. in the sum by A* by elementary counting since the constraint je < & must always be satisfied. Maximizing over < k < r with k + 2 yields the desired result.
Finally, we are able to state a concentration result about M, 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.
17

---

Proposition 7. Suppose (M, o) (L)) where (L)) is given by (4) and where L > 0 is a universal constans. On the event
maxister, f - mu S uj we have
T92
- (2r) Vu"ju,
sup xE(0.1]
where M, and M, are given by (12) and (8) respectively.
Proof. For E [0, 1]. consider by Definition that
M,((1) (1)
r! ji i1
y 5o
Ra
3u
Na
ja
fr 141
2!
a!
1sa$r-1ti 1+1, af2
Igagr-141. 1+1, ax/2
r!
(f((9) - fs(r''1
j1lja!
1m1
where the inner sum is taken over all sequences j {51, Jy of nonnegative integers such that ji + ... + -r-i41 = and ji 2j2 3ja + + (r - 1 1)1--1+1 = r. Here, we have defined
the function fj R RP-1-1
E by fz(z) I1sosr- [Eg] Jo and have used the notation a
(f2, y, (.......-441)) and x- (11.7 7a ... 7-1) By Taylor expansion and Holder's inequality, we have
((((9) / ) < |/ || 701lo where & is some point the line segment between & and y², It is immediate that the event
max Inn - xu $ " we have |1 - 7oo S u. Further consider |sal S 1Tyal + u S 2(haal V Ty22
w) < 2 (a V u) by Lemma 9. A straightforward calculation shows
|Vb()) S
Σ
U121130
&
(a!)aa
15a5r-141, a*2
1+1
(2(2)
12(a V u))
2 u)
Σ
11.21/40
(a!)ia
141, a#2
1
(2,a)
22' (r" V Vu)),
where we have used 2a Vu) 2 1, S rr and Er= aja = r. To summarize, we have shown
- -M(17) (1,(7)-M,(1))
$ ΣΣ ji ... jا (2" V V (2n) Vu")u
for all E [0, As in the proof of Proposition 6, we have bounded the number of admissible sequences J1-+1 by r". The proof is complete.
Proof of Proposition 4. Since (u, O and L 0 is universal constant, it follows by Corollary 1 that the event
E 8= max Im ml S 8- 1/2 C Cirlillg logr
1slsr
has Pu.o-probabbili of at least 6, where C2, Cf > 0 are some universal constants. Furthermore, Ci
it is clear for any 2 < r S Co.- log logn' we have 8 112 Efrlor
lng
Vn 2 (log
8/2 n)n S... and so
18

---

CB can be taken sufficiently large to ensure 8-1/2 1/2 Gyelinlog decays at some rate that is polynomial
vnn
n. We will now apply Proposition 6 on the event & with u 8 1/1 Cirrle To use Proposition Č
vn
6, it must be checked
Inor E (0, 1) where C' > O is the universal constant from Proposition 6.
Consider that o² = and so
E v
C
rt/2]
InO
4 S
Ve*
C1
(ci
+ 1 2
logr + logr+
C ve*
+ log Č 2))
8 1/2
exp
log
Vn
13/24220000
2 312 n)n
Cz..
Therefore, taking Cg.- sufficiently large (depending on e* and 3) implics TAgr u decays at some rate that is polynomial in It then follows from Proposition 6, the fact that is most logarithmic in and the inequality let S Ex for x (0,1). 1).
1 ml - (1 + L'rl) (r!)² (2r)" e 1 S C2e
CyeCrr
Car log Grrlgg
max <i 122
where C2, C22 > 0 are some universal constants (potentially depending on c*). Note
S
ii (log%/2 8/2 n)n 33 and so again can take Cg.. sufficiently large to ensure C2e C2rlog yu decays at some rate that polynomial in n. It then follows from Proposition that
(M)(1) - M, s Carroroog logr (Cze ---1/2 ( ve(0.1]
sup
Vn
where C3, C33 0 universal constants (potentially depending c²). The proof is complete.
B.3 Proof of Theorem 1
Proof of Theorem I. Fix (), E O. Choose
CB,x (4(( + 1)) 22v inf { K >0
(1(C'+1)
+1) -1) 2K 2K
Note C* is a positive universal constant since Cs,., 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 consider
log n
Ep,o (102 - 02(3) - s
X, < 41
(18)
Isi5a
2
+ E pnp
Σ x? (2(r)) - o²
maxigigo X,24 Ing
(19)
1=1
19

---

To bound (18), let Z; = X; 14 and consider by Lemma 10,
log
o 110
max Isish
X; S 4V
log
I 00 Pu.
max Z: Isis
max Zz 1345h
2
Fig.
max Zz
AV
11 1000
Isisn
r
ENi (maxisisn Zi) - 4// lug
1101100
S "2 2xxeexp exp
20²
Vlogn n
log +
S 20 exp
0-1
2
V log
log log log n
2
(20)
where we have used the result of [32] which gives Fygo (maxisigh 7,) 2 va log Vlog n. We have also used ||m||s0 lo
S
and 1 x
log logn
To obtain the result of the theorem, remains show (19)
log n
has order at most
log log log n
To do so, we split the analysis into two cases.
Case 1: Suppose d² J. Consider by Cauchy-Schward inequality,
n x² (2(() -22 - Pu.
log n
(19) < Ep,o
X; > AV Isisn
max
(21)
Letting Z; = X; - 1ti consider by Lemma 10 we have
log
Ph.
max X; 41
ISiS
r
log
Zi 21) Igisn }
< Pumo
max 1si5n
Zi
Enp
Zi Isisn
max
4V r
1|A1100
max
1
logn
< exp
41
1111l00
Eppo
max Z, Isisn
20²
Consider En,o (maxisisn Z,)= S oo lognn S V
2loen and ||a| S Therefore, follows
that Pao { mmxisisn X, > >4/4 >/////o logn
2 exp
logn ra
2
With this hand, it follows from n'
Eng (I En En X?- -f((r) - 0zl() S 1 that (21) S n1
log log a
as desired.
Case Suppose x² 2 J. It is clear (19) S En,o En X? - 02(²). To bound this expectation, denote the event E (sup MM(() - M,(7) 5o]. Since o² L. C >
Cineringr with C,C' as in Proposition with the choice c" = 1, it follows by 8. From the inequality (a + b)2 s 2a 2b2 follows
Cg.-, and ε = Vno Proposition that Pi.o (E) 1 Ei_, x? (2(r) o² < E x² (12 02) 2l2((r) 72l2. Since (r),,, 72
20

---

[0, and since & Cellrlogr Vnó
an application of Proposition yields
4/r
2Ce(C*+1)rlog log
C"
2112(1) 7212 2
21g
Vnō
p2
where C" is universal constant. Consider
4/r
(C' +1grlogr
_ (4(C" + 1or 2 2log n + 4 log (
= exp
Vnō
( (4(Ct +1
logn log
log n
S exp
log m
C+
log logn
C*
log loglog
[4(C' + 1)
log
- exp
-1
C*
Note C* 2 2. 4(C" + 1) 2 1(C' + +1), so it immediately follows
4(C'+1) C³ - < 0. Therefore,
follows from
1
log that
20"
log log a
(CC'++1)
4/r
(((C²+))
1
S exp
2C log logn S
Vnd
C-
log² n
where the second inequality follows from the fact that C satisfies, by design, the inequality
4(C'+11 -1) 2C* < 2. Therefore, we have shown?? +CCe0'+1)rlog 4/r Vno
log* n To summa-
C.
rize, we have shown
En,
fr(r) Σ -22(r) - 0² 1=1
S Var
Σ
x?
+ Eppo (1i2(r) - Y221 (²)
n
i=1
1
1
1
+ Puol(E
71
log"
n
72
1
1
+
+ 5
log n
72
log log 71
log
as desired. The proof is complete.
C Lower bound
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 < 7² s 1 and we will choose it later. Set of -1+7* 72 and of 1. For distribution π supported on [-1, 1], denote the induced mixture P=o== 5 Pp. n 0n (dp). It follows by reverse triangle inequality
f
inf sup (ML))0(2)
Puo 102-0211
1
inf
Po.0
102- -0ol
72
+ PT.
102 12--09 >
72
2
1
inf
Po.00 1022 102--15 1/5
72
32 1|>
?
&
2
2
inf (Po.ou (A) + Pr.o, (A9)
2
A
= (1 drv (Po.oo Pr.o))),
(22)
2
21

---

where the infimum in the penultimate line runs over all events A. By the Neyman-Pearsee lemma, 1 dTv Pr.o.) the optimal testing risk for the hypothesis testing problem
Hn X XI,,.. Xn idd
N(0, 1 +
H1 X XI,.., Xn od π NO,
Specifically, we have drv (Po P.) S 2/2 x ((.117))(01++1 by Lemma 5. Hence, it suffices to bound the x² divergence to furnish lower bound for (22).
We now construct and we will pick 7² < bound. Let D denote the largest even number smaller than or equal to 47 Note D Z 2 since 72 < atoms D { are the zeros of the Dth degree Hermite polynomial (Lemma 6). Further note that this measure is 1-subgaussian and symmetric about zero as D is even (Lemma 7). Take the prior distributtin π = Ei=1 widre,. In order for π to be a valid choice, it must be verified π is supported on [-1, Lemma gives us that |2,| I V4D 4 since 2; is a zero the Dth degree Hermite polynomial. Observe |T S T - 4) < 1 by our choice of D. Hence, π is supported on [-1, 1] and thus is valid choice.
1 at the end of the proof to obtain the claimed lower
8. Let 9D Σe wodz., denote the D-point Gaussian quadrature of N(0, 1). Note that the
We use a moment matching technique (Proposition 8). To do so, we first verify the first 2D 1 moments of x and N(O0, match. For any r E (1....,D 2D - -1}, observe since ID is the D-point Gaussian quadrature of (), it follows "v-No,,r=)Y")
= N', (Z")
T" Fiz-go (7*) = Pa (Y"). Note both 72) and æ are r-subGaussian. Proposition 8, along
with Z I - 2 2 and 7² < L thus implies
I6.
1674D
162
exp (-4r2 log (32) ))
x * N(O, N),, N(),
V2D
12D-1(1-
111 15/3
472
Select 72
log log 16 Ing n
and observe
162
-4 n (1 log (16 log log
1
(T ( NO,, II ~) NOO, 1)) S 15.3
exp
log log
n
since is larger than sufficiently large universal constant. Therefore, drv UTv(Po.00 PR.O)) / 1)"h 1 1 - which, when plugged into (22), yields the lower bound
inf sup Puo 102 102-021 02l (2L))0(12)
log log
( 1--ve-1).
32 log n
The proof is complete.
D Auxiliary definitions and results
Definition 2. A probability distribution P is said to be a-subgaussian for a > 0 for X ~ P we have E (exp (A(X- E(X)))) S e x²,² for all E IR
1= P, and Q Di Qi are product measures, then Lemma 5 (x2 tensorization [58]). IfP Ø
(P) (1= Q:)) 1.
Proposition 8 (Theorem 3.3.3 [64]). Suppose v and v' are two symmetric probability distributions that are e-subgaussian for e 1. If the first D moments of and v are equal, then x²(v (O, || N(O, 1)) < 16 20122
VD 1-25.
Proposition 9 (Gaussian quadrature [64]). Suppose μ is probability measure supported F. C R. If 2 1, there exists a k-atomic distribution Pk E wedz. supported on E such that for any polynomial of degree at most 2k 1 we have J p(x) du(z)) = Σ wxp(x:).
Lemma 6 (Remark 2.7.2 [64]). If 2 1, then the k-point Gaussian quadrature of N(O, 1) has its atoms at the roots of the degree l Hermite polynomial Heg.
22

---

Lemma 7 (Lemma 2.7.3 [64]), Suppose k 2 1 and is the k-point Gaussian quadrature of ((.0 1). For j 2 2k, we have Ex-gx (X') (z- when j even and Ex-ge (X') otherwise. In particular, 9s. is 1-subgaussian.
Lemma 8. If k 1, then the zeros of the kth Hermite polynomial lie in [-V 4k - 4, v 4k 4
Proof. From (6.2.18) on page 120 of [56] the zeros of the kth degree physicist's ' Hermite polynomial
Hx(x) (-1)ae" e"
lie in the interval
-2(k-1) 1j
/2(k-1
Since
+ 2, it imme-
vk+2
k+2
diately follows that we have the inclusion
12(k-))
<2(k-1)
c 2 2, / 2]. Since
vk42
vk+2
have the following correspondence between physicist's 's and probabilist's Hermite polynomials
(x))) 24/2 Hex /2r,, it follows the zeros of He, lie in [-v4k- 4, v 4). The proof is complete.
Lemma 9. The rth cumulant "Ir of a distribution G supported on [-1, 1] satisfies 1701 rt.
Proof. Let Py denote the kth moment of G and note |P&l S 1. From (6) it follows 7/-
Ei E 1,, 1/2..., ..., 1y-241). Since the coefficients of the Bell polynomial B are all positive and |ux| S 1, it follows hrl S Ei 1)!B,, 1,..., 1) = --- ()
where [7} is Stirling number of the second kind. Consider (7 - 1)4() 5-40 S Therefore, hr| S Ei= 1r11 < r" as desired.
Lemma 10 (Lemma 2.10.6 [57]). I (O0 In), then
u²
P
max I8iS
Zi-
E
max Zi
2 u
< 2exp
-0
for 2 O.
E
Notation
For a, b E I the notation a S b denotes the existence of a universal constant c >> such that a S cb. The notation a 2 is used denote b S a. Additionally a denotes a S and a Z b. The symbol := is frequently used when defining a quantity or object. Furthermore, we frequently use abb: = max(a, b) and AAb = min(a, b). We generically use the notation 4 denote the indicator function for an event A. For two probability measures P and Q on measurable space A), the total variation distance is defined as drv Q) := su IP(A) - Q(A). If P is absolutely
continuous with respect to Q, then the x2 diveergee is defined as (P(IQ) = Jx 15 -1) dQ.
We will frequently use the same notation for two probability densities and For sequences {ax k==1 and ) the notation a = o(bb) denotes lim4 d2 0 and the notation a (bb) used by
to denote by = o(az). For a point æ E IR, the symbol 8 denotes the probability measure which places full probability mass at the point 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, 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 e
23

---

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29


Reviews:
[{"review_id": "JUwLratwwi", "reviewer": "Reviewer_uxch", "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.", "questions": "- Please feel free to clarify/respond regarding the above comments.\n- Would it be possible to go slightly beyond Gaussianity and make the results more broadly applicable?", "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.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The major strength is determining the exact min-max rate regarding sample size, which requires both deriving a concrete estimator and establishing a lower bound (through moment matching).\n- The results hold under relatively weak conditions, i.e., the boundedness of the parameter space. In comparison, prior research makes assumptions like smoothness, which might be less practical.\n- The use of cumulants on top of the Gaussian character of the noise is innovative and might be of independent interest.", "weaknesses": "- 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).\n- 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.\n- 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your helpful feedback! Our responses below address the points raised in your review. \n\n[Weakness 1] Please see the global response to all reviewers regarding the dependence of the rate on $L$. \n\n[Weakness 2 and Question 2] We completely agree with your comment that Gaussianity is critical to the results. Indeed, the applicability is consequently limited. It is not at all clear to us how to generalize the cumulant methodology, and it would be an interesting, challenging direction for future work. The key property we use is that the variance of Gaussian noise only shows up in the second cumulant; this enables us to state a result like Proposition 1 to identify the noise variance from the marginal cumulants. For a different noise distribution, a different/analogous version of Proposition 1 needs to be established as the noise variance can now appear in other cumulants. The difficulty is reminiscent of the challenges of moment-based approaches discussed in Remark 2. \n\n[Weakness 3] We have conducted an experiment in the attached pdf, and the details are in the global response to all reviewers.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "joHQXiTCy9", "reviewer": "Reviewer_tWCj", "summary": "This paper studies the problem of variance estimation in compound decision setting, with the assumption that means are bounded. Main contributions:\n\n- 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.\n- The authors discuss the importance of the Gaussian assumption of the result.\n\nPost rebuttal: I acknowledge that I have read the authors responses and other reviewers' comments.", "questions": "Questions:\n- 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": "NA.", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Strengths:\n- This paper is well-written and well-organized.\n- The authors provide the minimax rate of variance estimation in the setting that means are bounded and noise follow the Gaussian distribution.\n- The authors discuss the noise agnosticism that the estimator is designed for Gaussian noises.", "weaknesses": "Weaknesses:\n- There is no discussions on the computation complexity of the proposed estimator, and no experiments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your comments! Below, we address some of the points raised in your review.\n\n[Weakness 1] We have conducted an experiment in the attached pdf, and the details are in the global response to all reviewers. \n\n[Question 1] Most other variance estimation procedures in the literature (beyond the simple maximum-based estimator and the naive second moment estimator discussed in the experiment) require assumptions about the structure of the means; for example, the means are distributed according to a $k$-atomic distribution, spike-and-slab, etc. However, these structural assumptions can be unrealistic in some applications. The advantage of our method is that the means need only be assumed bounded. \n\nThe disadvantage of our cumulant-based method is that it cannot, in its current form, adapt to any underlying structure of the $\\mu$'s. If the means are truly structured, previous estimators can achieve faster convergence rates than the proposed cumulant-based estimator. As you point out, the choice $r \\asymp \\frac{\\log n}{\\log \\log n}$ in our estimator is really made to protect against the worst case choice of $\\mu$'s. It is an interesting question whether $r$ could be chosen in a data-driven way to adapt to the underlying structure of the $\\mu$'s; this way, nothing but boundedness is assumed, yet existing structure could be exploited. At the current moment, it is not clear to us how to construct an adaptive version of our estimator; it is a nice direction for future work.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dVuYO9kJKA", "reviewer": "Reviewer_9ofA", "summary": "This paper gives a sharp minimax rate of the variance estimation under mild assumption on a Gaussian-based model.", "questions": "N/A", "limitations": "N/A", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The theoretical results are solid and complete, and the related reference are discussed and linked with their own work.", "weaknesses": "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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback! Our response to your review is below.\n\n[Weakness 1] We have conducted an experiment in the attached pdf, and the details are in the global response to all reviewers. Yes, the symbols $\\asymp$ and $\\lesssim$ mean ``up to universal constants''. The notation we use is defined in Appendix E, and in the revision we will note this in the main text to make the reader aware.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Jp8r0Rjo0I", "reviewer": "Reviewer_RbL1", "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.", "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 $\\mu$?\n\n- What is the concentration behavior/tails of the estimator? Theorem 1 is in mean squared error only.\n\n- 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": "Yes.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is very very well-written, with intuition properly explained and motivated. It is refreshing to see a properly-written paper being submitted to NeurIPS/ICML.\n\nTechnically, it is also nice that the authors settle the estimation rate up to constants for this problem. I also find the cumulant-based approach interesting --- I don't think I've seen many papers (at least in recent memory) that forego moments and deal with cumulants instead.", "weaknesses": "Here are some constructive criticisms that I hope the authors will find useful to improve the paper.\n\n- 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\".\n\n- 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.\n\n- 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).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your very helpful comments and thorough feedback! Our responses below address the points raised in your review. \n\n[Weakness 1] We agree the title can be revised for clarity. In the revision, we will revise the title to ``Variance estimation in compound decision theory under boundedness''.\n\n[Weakness 2] We agree the explanation would improve after the suggested re-ordering. We will do this in the revision.\n\n[Weakness 3] This comment is acknowledged and well-taken. Indeed, we are leaning heavily on statistical tradition and the existence of a substantial literature to motivate consideration of the problem studied in our article, and especially so to justify the Gaussian assumption. \n\n[Question 1] Yes, our estimator relies on knowledge of $L$. Please see the global response to all reviewers regarding the dependence of the rate on $L$. The question regarding the rate's dependence on the bound on the $\\ell_\\infty$ norm of $\\mu$ is very related. Suppose we are in the setting where we know $||\\mu||\\_\\infty \\leq B$ and $\\sigma \\leq L$. We can transform the data without loss of any information as $Y_i = X_i/B$, and so we have $Y_i \\overset{ind}{\\sim} N(\\theta_i, \\tau^2)$ where $\\theta_i = \\mu_i/B$ and $\\tau = \\sigma/B$. Now note $||\\theta||_\\infty \\leq 1$ and $\\tau \\leq L/B$, and so we are in the original setting of our paper. Estimation of $\\tau^2$ from $Y$ is equivalent to estimation of $\\sigma^2$ from $X$ (up to an appropriate scaling). For the same reasons outlined in the global response, estimation of $\\tau^2$ is essentially only interesting in the case $L/B \\asymp 1$. In this case, the minimax rate of estimation of $\\tau^2$ is $(\\log\\log n / \\log n)^2$, which thus implies the minimax rate of estimating $\\sigma^2$ is $B^4 (\\log\\log n/\\log n)^2$.\n\n[Question 2] The question regarding the concentration behavior of the estimator appears to us to be quite tricky. As can be surmised from Proposition 3, the fluctuations of the error $|\\hat{\\gamma}\\_2(r) - \\gamma\\_2|$ can be related to the fluctuations of $\\sup\\_{\\gamma \\in [0, 1]} |\\hat{M}\\_r(\\gamma) - M\\_r(\\gamma)|$. The latter random variable involves polynomials of growing degree (since we choose $r$ to grow), and so it appears difficult to us to obtain concentration results beyond that obtained by a simplistic analysis using very crude tools.\n\n[Question 3] It turns out the structure of the Bell polynomials makes showing uniform concentration very painless. The relevant analysis occurs in the proof of Proposition 7. In particular, recalling definitions from our paper, consider $|\\hat{M}\\_r(\\gamma) - M\\_r(\\gamma)| \\leq \\sum_{l=1}^{r} |B\\_{r, l}(\\hat{\\gamma}\\_1, \\gamma, \\hat{\\gamma}\\_3,...,\\hat{\\gamma}\\_{r-l+1}) - B_{r, l}(\\gamma\\_1, \\gamma, \\gamma\\_3,...,\\gamma\\_{r-l+1})| \\leq \\sum_{l=1}^{r} \\sum \\frac{r!}{j\\_1! j\\_2! \\cdot\\cdot\\cdot j\\_{r-l+1}!} \\left|\\frac{\\gamma}{2!}\\right|^{j\\_2} \\cdot W\\_{\\mathbf{j}}$ where $W\\_{\\mathbf{j}}$ is a random variable which does not depend on $\\gamma$. Here, the inner sum is taken over appropriate sequences $\\mathbf{j} = \\{j\\_1,j\\_2,...,j\\_{r-l+1}\\}$ as in the definition of Bell polynomials (Definition 1). Therefore, it is immediate to obtain the deterministic inequality $\\sup\\_{\\gamma \\in [0, 1]} |\\hat{M}\\_r(\\gamma) - M\\_r(\\gamma)| \\leq \\sum\\_{l=1}^{r} \\sum \\frac{r!}{j\\_1! j\\_2! \\cdot\\cdot\\cdot j\\_{r-l+1}!}\\cdot W\\_{\\mathbf{j}}$, and now the right hand side no longer involves a supremum. The Bell polynomial structure is very convenient. In the revision, we will make a comment about it in the main text.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank all of the reviewers for their helpful comments. Below, we address those points which were raised by multiple reviewers.

Two reviewers asked about the dependence of $L$ in the minimax rate. Thank you both for the comment; it is well-received. We would like to make the case that $L \asymp 1$ addressed in the article is essentially the only interesting setting with a gap in the literature. In the small $L$ case where $L \lesssim \frac{1}{\sqrt{\log n}}$, it can be shown quite easily that the minimax rate is $L^4$ and is achieved by the trivial estimator $\hat{\sigma}^2 = 0$. Of course, one can then ask to capture the rate dependence on $L$ in the intermediate setting $\frac{1}{\sqrt{\log n}} \lesssim L \lesssim 1$, but we feel this case is of very limited interest and relevance. On the other side, the large $L \gtrsim 1$ case can be essentially addressed by existing literature. To elaborate, note one can transform the data by $Y_i = X_i/L$ which then is distributed as $Y_i \sim N(\theta_i, \tau^2)$ where $\theta_i = \mu_i/L$ and $\tau^2/L^2$. It is clear then that $\tau^2/L^2 \leq 1$ and $||\theta||\_\infty \leq L^{-1}$. To estimate $\sigma^2$, we can equivalently estimate $\tau^2$ and then rescale. Since $L$ is large, intuitively the nuisance $\theta_i$ should not have much effect, and so we can directly use $\hat{\tau}^2 = \frac{1}{n} \sum_{i=1}^{n} Y_i^2$. It is straightforward to show $E(|\hat{\tau}^2 - \tau^2|^2) \lesssim n^{-1} + L^{-4}$. It turns out this simple estimator can actually be optimal for estimating $\tau^2$. For example, in the scaling $L = n^\alpha$ where $\alpha > 0$ is fixed, $\hat{\tau}^2$ achieves the rate $n^{-1} + n^{-4\alpha}$, which is well known to be the minimax rate for estimating $\tau^2$, e.g. it can be seen as a consequence of reference [61]. Consequently, the rescaled estimator $\hat{\sigma}^2 = L^2\hat{\tau}^2$ must also be minimax rate optimal for $\sigma^2$. Therefore, in our view, essentially the only interesting case where new methodology is needed is $L \asymp 1$. We had thus focused our article on this setting.


Three reviewers pointed out that an experiment would be helpful. We have conducted an experiment in the attached pdf. The implementation details are as follows. We considered $20$ different various sample sizes $n$ spread evenly on the log-scale between $e^{7}$ and $e^{10}$. For each sample size $n$, we ran $100$ simulations each for four different data-generating processes. In particular, we chose four different priors $\mu \sim G$ being the Rademacher distribution ($G = \frac{1}{2}\delta_{-1} + \frac{1}{2}\delta_1$), a rescaled Beta distribution ($\mu = 2q - 1$ where $q \sim \text{Beta}(5, 1)$), the uniform distribution ($G = \text{Uniform}[-1, 1]$), and a Gaussian prior ($G = N\left(0, \frac{\log\log n}{\log n}\right)$). Note though the Gaussian prior is not supported on $[-1, 1]$, it is informative to consider since it is quite close to (more precisely, matches many moments with) the prior constructed in the lower bound argument of Section 3. For each choice of prior $G$, we sampled $\mu_1,...,\mu_n \overset{iid}{\sim} G$, and generated data $X_i \,|\, \mu_i \sim N(\mu_i, \sigma^2)$ with $\sigma = 2.25$. We computed the maximum-based estimator $\hat{\sigma}^2\_{\max} = \left(\max\_{1 \leq i \leq n} \frac{X_i}{\sqrt{2\log n}}\right)^2$, the naive second-moment estimator $\hat{\sigma}\_{\text{mom}}^2 = \frac{1}{n} \sum\_{i=1}^{n} X\_i^2$, and our cumulant-based estimator $\hat{\sigma}^2$ with the choice of tuning parameters $r = \left\lfloor \frac{\log n}{\log\log n}\right\rfloor$ and $\varepsilon = n^{-0.45}$. We computed the square error for each estimator, then computed the average square error across the $100$ simulations; these are the dots plotted in the figure. The error bars in the plot are standard errors. 

From the experimental results, we can see that our cumulant-based estimator performs consistently well no matter the prior specification. In contrast, the maximum-based estimator (which achieves the upper bound $\frac{1}{\log n}$ as pointed out by reference [63]) performs consistently worse than the cumulant-based estimator across all priors. The behavior of the mean square error of the naive second-moment estimator is unsurprising, namely it related to $(E(\mu^2))^2$. Consequently, it does well in the Gaussian and Uniform prior settings where this quantity is low, but worse in the other settings where it is higher.

Meta-review: 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.
Decision: Accept (poster)