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F∈Gkµ(F) (3) = min F∈Fkµ(F), (4) where Fk=cl(Gk), simply adding a boundary to the locally open parts of the set. We refer to Fkas the likely set of distributions for the subset Ω k. To specify a lower bound for xtk, we consider two cases: where the likely set Fkis empty and where it is not empty. If Fkis not empty we d... | https://arxiv.org/abs/2502.17223v1 |
xtkwill be erroneous. But such a choice can be part of a valid lower bound over a family of distributions. Thus, the full specification of our order-conditioned bound, conditioned on the total order T, is as follows: B∗(xtk) =( ∞, ifFk=∅ min F∈Fkµ(F),otherwise.(6) Below we will prove a variety of results about this bou... | https://arxiv.org/abs/2502.17223v1 |
Our goal is to show that Equation 6 has two properties. The first is validity: that there exists no Fwith E[F]< B∗(xtk) such that Prob F(Ωk)> α. The second is conditional optimality: that the bound cannot be increased for anysample without breaking validity (or changing the ordering of the bounds). If Fk=∅, both of the... | https://arxiv.org/abs/2502.17223v1 |
be less than the means of Gk, and hence there will be a distribution in Gkfor which the bound is invalid. Hence, the bound cannot be made any larger. Notice that this result does not depend upon the value of the bound for any other sample, as long as the ordering is fixed. Thus, conditioning on the order makes it possi... | https://arxiv.org/abs/2502.17223v1 |
admissible) .LetB∗ Tbe a conditionally- optimal bound with respect to a sample space Ωand with respect to a total order T. Furthermore, assume that it is injective. Then it is admissible with respect to the full family of bounds Bover Ω. Proof. To prove this, we must show that there is no other valid bound that dominat... | https://arxiv.org/abs/2502.17223v1 |
such likelihoods (that are all conditioned on the same parameters) inherits the same properties. Proof. Multinomial probabilities are just polynomials with positive coefficients, so continuity and differentiability and non-negativity are trivial. Since any probability distribution in the open simplex assigns a non-zero... | https://arxiv.org/abs/2502.17223v1 |
of the sample space Ω. Then we have min cl{F∈F:Prob F(Ωj)>α}µ(F)< min cl{F∈F:Prob F(Ωk)>α}µ(F). That is, the minimum of the constrained means under Ωjmust be strictly less than the minimum of the constrained means under Ωk. Proof. LetFbe a distribution on the open probability simplex Gsuch that Prob F(S) =α. Now consid... | https://arxiv.org/abs/2502.17223v1 |
reverse the order of xAandxBin the total order to form a new total order in which xBprecedes xA, then the sequence of bounds would be governed by the mean of the yellow dot distribution (center of Figure 5) and then the mean of the red dot, which is lower. That is, by reversing the order of xAandxBin the total order, w... | https://arxiv.org/abs/2502.17223v1 |
results in an improved bound, we refer to this as a breakable tie . If swapping the elements in the order continues to result in a tie, we refer to this as an unbreakable tie . 3.3.2 Implications of breakable and unbreakable ties Suppose that a bound Awith a tie can be improved by swapping the order of the tied element... | https://arxiv.org/abs/2502.17223v1 |
that never leads to an admissible bound. In the next subsection, we consider the special case of conditionally-optimal bounds that depend upon an ordering in which the lowest possible sample is notthe first sample in the total order over the sample space. We refer to these at degenerate bounds, and they illustrate many... | https://arxiv.org/abs/2502.17223v1 |
0/∈Φ. Let FΦ=cl{F∈ F:Prob F(Φ)> α}. Note that every distribution in FΦhas a probability of Φ of at least α. We start by observing that F0is the only distribution in Fwhose mean is 0. Thus, Equation 14 can only hold true if F0∈ FΦ. Suppose F0∈ FΦ.For any α >0, the probability of at least one sample in Φ must be non-zero... | https://arxiv.org/abs/2502.17223v1 |
a conditionally optimal bound function based on that total order. Note that while there are always exactly N! orderings over a sample space, not all of the conditionally optimal bound function need be distinct. That is, the bound functions conditioned on two different orderings may be equivalent. LetK≤N! be the number ... | https://arxiv.org/abs/2502.17223v1 |
is as high as possible for any bound in which xBis the second to last element of the total order after xA. Hence, we also have that A+(xB) =A(xB). Using the same arguments, we conclude that B+(xA) =B(xA) and B+(xB) =B(xB). In other words, A+andB+have the same performance on the samples xAandxB, respectively, as the bou... | https://arxiv.org/abs/2502.17223v1 |
are the necessary conditions for Lemma 4.1, then there must be at least two admissible bounds. With two admissible bounds, there can be no optimal bound. For general multinomial distributions, we need to generalize the above result in several ways. We start with sample size. Let S={0,1}but with sample size n >2. Rather... | https://arxiv.org/abs/2502.17223v1 |
optimal with respect to a total order, but nevertheless non-admissible. This can only occur when the bound produces ties, that is, equivalent bounds for two or more samples in a sample space. –Among bounds with ties, there are two distinct sets: those with breakable ties and those with unbreakable ties. The latter boun... | https://arxiv.org/abs/2502.17223v1 |
On High-Dimensional Linear Regression On a class of high dimensional linear regression methods with debiasing and thresholding Ying-Ao Wang wya@bit.edu.cn School of Mathematics and Statistics Beijing Institute of Technology Beijing 100081, People’s Republic of China Yunyi Zhang zhangyunyi@cuhk.edu.cn School of Data Sci... | https://arxiv.org/abs/2502.17261v2 |
perform statistical inference across a wide range of linear regression methods, thereby expanding the available choices of linear regression methods for practical implementation. Beyond the simplification of theoretical study of existing linear regression methods, the propose of our work also establishes a basis for ge... | https://arxiv.org/abs/2502.17261v2 |
2015; Alexanderian et al., 2021; Zhang and Chen, 2022), among others. However, due to differing mathematical frameworks, these new regulariza- tion methods have not garnered widespread attention in the statistical community. Thanks to developments in machine learning, the combination of statistics and inverse problems ... | https://arxiv.org/abs/2502.17261v2 |
addition, we introduce thresholding techniques to enhance variable selection accuracy and preserve the sparsity of the estimator. This combined debiasing and thresholding procedure facilitates valid statistical inference. •Gaussian approximation theorem & bootstrap algorithm for statistical inference: The limiting beha... | https://arxiv.org/abs/2502.17261v2 |
to study a broad class of regularization methods within a unified framework. Math´ e (2004) demonstrated the satu- ration of methods for solving linear ill-posed problems in Hilbert spaces by introducing the concept of qualification for a wide class of regularization methods. Hofmann and Math´ e (2007) proposed a gener... | https://arxiv.org/abs/2502.17261v2 |
al. (2013); Naseri et al. (2021)). 3 Debiasing and thresholding in linear regression This manuscript focuses on the high-dimensional linear regression model Yn=Xnβββ+eeen, (3) where Yn= [Y1, . . . , Y n]T∈Rnrepresents the response vector, Xn= [xij]1≤i≤n,1≤j≤p∈ Rn×pdenotes the fixed (non-random) design matrix, which is ... | https://arxiv.org/abs/2502.17261v2 |
n I+rα(1 nXT nXn) gα(1 nXT nXn)XT nYn. Definition 2 (Slightly modified from Definition 2.3 in Zhang and Hofmann (2020)) .A linear regression method (4)for equation (3)generated by the generator function gα(λ)(0< λ≤Cλ)is said to have a monomial qualification of order dif the following inequality holds sup λ∈(0,Cλ]|rα(... | https://arxiv.org/abs/2502.17261v2 |
B 1. Calculate ˆθθθand˜θθθ3defined in (6), along with bσ2=1 nnX i=1 Yi−pX j=1xijˆθj 2 andeσ2=1 nnX i=1 Yi−pX j=1xij˜θj 2 . (7) 2. Unlike the previously defined optimal threshold, this threshold is not the average value that minimizes the errors ∥ˆθθθ−βββ∥and∥˜θθθ−βββ∥but is adjusted to different quantiles for e... | https://arxiv.org/abs/2502.17261v2 |
regression. From (49)and(24), the thresholded and debiased estimators of Spectral cut-off regression can be calculated as follows: ˆθθθα=VΛ−2 αΛUTYn×111i∈bNbn, ˜βββα=V(2I−Λ2Λ−2 α)Λ−2 αΛUTYn, where Λαrepresents Λafter truncation of singular values. Thus, the debiased and thresholded Spectral cut-off regression estimator... | https://arxiv.org/abs/2502.17261v2 |
singular vectors of matrix Xn, respectively. Consequently, we obtain βββ(t) =sX j=11−e−λjt λ1 2 j(Yn,uj)vj=:g(t,XT nXn)XT nYn, where g(t, λ) =1−e−λt λ. Replacing t= 1/α, we obtain the generator and bias functions of the Showalter regression gα(λ) =1−e−λ α λ, r α(λ) =e−λ α. Additionally, for all d >0, sup λ∈(0,Cλ]e−λ αλ... | https://arxiv.org/abs/2502.17261v2 |
gα(λ) = 1 λ 1−η+√ η2−4λ 2√ η2−4λe−η−√ η2−4λ 2α +η−√ η2−4λ 2√ η2−4λe−η+√ η2−4λ 2α , η2>4λ, 1 λ 1−e−η 2α η√ 4λ−η2sin√ 4λ−η2 2α + cos√ 4λ−η2 2α , η2<4λ, 1 λh 1−e−η 2α η 2α+ 1i , η2= 4λ, and rα(λ) = 1−λgα(λ) = η+√ η2−4λ 2√ η2−4λe−η−√ η2−4λ 2α−η−√ η2−4λ 2√ η2−4λe−η+√ η2−4λ 2α, η2>4λ, e−η... | https://arxiv.org/abs/2502.17261v2 |
decreasing on the interval (0, α). (D2-2) By using the conclusions of (Gorenflo et al., 2014, Corollary 3.7), rα(λ)satisfies inequality |rα(λ)|=Eϑ−λ α ≤Cϑα α+λ, where Cϑ≤1for0< ϑ < 1. Hence, for any fixed λ > 0,rα(λ) =Eϑ −λ α ≤Cϑα α+λ:=Rα(λ), and Rα(λ)is an increasing function with respect to α. Additionally, Rα(λ)i... | https://arxiv.org/abs/2502.17261v2 |
β >−1. utilizing the results from (Szeg, 1975, equations (7.33.1) and (4.7.3)). Hence we can find that |rk(λ)|= (1−∆tλ)k+1 2C(ω+1 2) k−1(√ 1−∆tλ) C(ω+1 2) k−1(1) ≤(1−∆tλ)k+1 2, k≥1, ω > −1. Thus, the bias functions of Landweber regression and Nesterov acceleration regression ex- hibit a similar structure. Consequently,... | https://arxiv.org/abs/2502.17261v2 |
hence ∥E˜βββα−βββ∥2≪ ∥Eˆβββα−βββ∥2. This implies that when there are sufficiently many samples, the order of magnitude of the variance term remains relatively unchanged, while the order of magnitude of the bias term significantly decreases. In summary, when dealing with large sample sizes, our analysis recommends using... | https://arxiv.org/abs/2502.17261v2 |
being truncated should be significantly larger than the βibeing truncated. Additionally, Assump- tion 4 (c) ensures the sparsity of βββ. We begin our investigation into the consistency of our new class of linear regression methods. It should be noted that without additional assumptions, the linear regression estimators... | https://arxiv.org/abs/2502.17261v2 |
n(ασ−1)/2×log−3/2(n), n−1/3×log−3/2(n) . (B)ασ<1/2,p=o nασ×log−3(n) and max i=1,···,p, l=1,2,···,n 1 τisX k=1vik nulk[1 +rα(λk n)]gα(λk n)p λk =O n−ασ×log−3/2(n) . According to error decomposition of the debiased estimator ˜βββα(25), the quantity nX l=1 1 τisX k=1vik nulk[1 +rα(λk n)]gα(λk n)p λk! el asymptotical... | https://arxiv.org/abs/2502.17261v2 |
= 0, (38) and if d >2(αβ+η−δ) δ, we have lim n→∞sup x≥0 P max i=1,2,···,p ˆθi−βi τ∗ i≤x −H∗(x) = 0. (39) 5.3 Best worst case error Consider the following admissible set of noisy data ¯Bσ(Xnβββ) :=ˇYn∈Rn:∥ˇYn−Xnβββ∥ ≤√nσ .‘ Letˇβββbe a solution from the general linear regression method (4), with Ynreplacing any ele... | https://arxiv.org/abs/2502.17261v2 |
GB RAM using Python 3.12.4. All experiments in this section are implemented for the following three subsection: 6.1 Sparse case In this subsection, we generate the design matrix Xn, the parameters vector βββand error vector eeenthrough the following strategies. •Design matrix Xn: Define Xn= [x1,···, xn]Twith xi= (xi1,·... | https://arxiv.org/abs/2502.17261v2 |
Therefore, this situation is not the focus of our study. 6.1.1 Case I: n=p= 1000 In this case, ς= 1 is set for the conventional discrepancy principle, with kmax= 5000. The iteration step size is ∆ t= 5×10−4for HBF and SOAR regression, ∆ t= 5×10−7 for Landweber, Showalter, and Nesterov regression, and ∆ t= 5×10−5for FAR... | https://arxiv.org/abs/2502.17261v2 |
reduction, adding thresholds is ineffective. Table 2 presents the average errors of the proposed estimators ˆθθθand˜θθθ(as defined in (6)), along with the average errors of bσ2andeσ2(as defined in (7)). It also shows the coverage probabilities of the confidence regions (8) and (9), based on 1000 numerical simulations9,... | https://arxiv.org/abs/2502.17261v2 |
of regularized regression methods. By comparison with thresholded estimators ˆθθθ, the debiasing process achieves consistent and substantial improvements across various metrics, demonstrating its broad applicability. For Ridge regression, the debiasing process markedly enhances the alignment of Cov- erage II with the n... | https://arxiv.org/abs/2502.17261v2 |
0.4050 ARκ6.5214 657 6.4346 6.2624 6.0640 0.1285 0.1695 SOAR 6.5133 151 6.4274 6.1553 6.0324 0.1585 0.1750 Nesterov 6.5207 1891 6.4337 6.2634 6.0638 0.1285 0.1695 FAR 6.4862 292 6.3987 6.0091 5.9602 0.1830 0.1670 LS 301.72 301.64 0.9985 32 On High-Dimensional Linear Regression SC (kr= 702) 6.5101 7.4283 6.5101 7.4283 0... | https://arxiv.org/abs/2502.17261v2 |
loss of generality, we focus on the non-sparse case where n=p= 1000. After defining the design matrix, parameter vector, and error vector, we proceed to evaluate the errors associated with various regression methods under two distinct iteration termination criteria. The first criterion is the adjusted optimal stopping ... | https://arxiv.org/abs/2502.17261v2 |
traditional regularization methods, such as Ridge regression, and modern iterative regularization methods. Therefore, it can be considered that the optimal estimates they achieve are effectively the same. 6.3 Inverse source problems (ISP) in partial differential equations (PDEs) As mentioned in the introduction, after ... | https://arxiv.org/abs/2502.17261v2 |
data q1is computed on a finer mesh Ξ, with a mesh size of h= 0.1293, consisting of 599 nodes and 1128 elements. Additionally, artificial noisy data are generated as follows: qσ 1,2(x) =q1,2(x) + randn(0 , σ) for all x∈Γ∩Ξ, where randn(0 , σ) denotes the random value from a normal distribution with mean 0 and variance σ... | https://arxiv.org/abs/2502.17261v2 |
Coverage I denotes the empirical coverage probability of ˆf, while Coverage II denotes that of ˜f. The nominal coverage probability is fixed at 1 −α∗= 95%. The overscore represents the sample mean computed across 1000 independent simulations. The number of bootstrap replicates is set to B= 500, and ∥ · ∥∞,Γ indicates t... | https://arxiv.org/abs/2502.17261v2 |
0.15 -0.5 -0.5 -0.0500.050.10.15 Figure 4: Confidence intervals for SOAR regression and its debiased estimator. From Figures 3 to 8, as well as 10 (correspondingly, Figures 5 to 8, and 10 in Appendix B), it can be observed that when the general estimators ˆfexhibit issues such as overfitting or underfitting in confiden... | https://arxiv.org/abs/2502.17261v2 |
as follows: Xn=Udiag{p λ1,···,p λs}VT:=UΛVT, (49) where U,V,{√λi}s i=1 represents the singular system of matrix Xn. 0< λs≤ ··· ≤ λ1are ordered eigenvalues of the square matrix XT nXn.U= [uij]n×sandV= [vij]p×sin equation (49) are respectively n×sandp×sorthonormal matrices, satisfying UTU=VTV=Is, Isdenotes the s×sident... | https://arxiv.org/abs/2502.17261v2 |
4, we have ˆθθθ−βββ 2 2 =X i∈Nbn ˆθi−βi2 +X i/∈Nbnβ2 i ≤2X i∈Nbn sX j=1vijζjrαλj n 2 + 2X i∈Nbn sX j=1vij ngα(λj n)λ1 2 jnX l=1uljel 2 +X i/∈Nbnβ2 i ≤2X i∈Nbn sX j=1vijζjrαλj n 2 + 2X i∈Nbn sX j=1vij ngα(λj n)λ1 2 jnX l=1uljel 2 +Cbcbn−vbX i/∈Nbn|βi| ≤2|Nbn| max i=1,2,···,p sX j=1vijζjrαλj n... | https://arxiv.org/abs/2502.17261v2 |
n)p λknX l=1ulkel!2 =σ2X j∈NbnnX l=1 sX k=11 nqikgα(λk n)p λkulk!2 =σ2X j∈NbnsX k=11 n2q2 ikg2 α(λk n)λk ≤4σ2|Nbn| λs, we have 1 nnX i=1 X j∈Nbnxij ˆθj−βj 2 =Op n2αβ−2dδ|Nbn|+n−2η|Nbn| . For the third term in (58), from Assumption 5 we have 1 nnX i=1 X j /∈Nbnxijβj 2 ≤CλX j /∈Nbnβ2 j≤CλbnX j /∈Nbn|βj|=O n−... | https://arxiv.org/abs/2502.17261v2 |
ασ≤1/2such that for j= 1,2,···, n, i= 1,2,···, k, σ2−bσ2 =Op n−ασ ,max i,j|γji|=o min n(ασ−1)/2×log−3/2(n), n−1/3×log−3/2(n) , 48 On High-Dimensional Linear Regression (B) There exists a constant 0< ασ<1/2such that for j= 1,2,···, n, i= 1,2,···, k, σ2−bσ2 =Op n−ασ , k=o nασ×log−3(n) ,max i,j|γji|=O n−ασ×log−3... | https://arxiv.org/abs/2502.17261v2 |
i= 1,···, p, and D2= diag{1 n[1 +rα(1 nλ1)]gα(1 nλ1)λ1 2 1,···,1 n[1 +rα(1 nλs)]gα(1 nλs)λ1 2s}. So from Lemma 14, there exists a constant C′which only depends on σ, Cλsuch that sup x∈R P max i=1,···,p nX l=1tile∗ l ≤x+Cn−δ1! −P max i=1,···,p nX l=1tile∗ l ≤x!! ≤C′Cn−δ1 1 +p log(f) +q |log (Cn−δ1)| . 51 Y. Wang, Y. Z... | https://arxiv.org/abs/2502.17261v2 |
get P max i=1,2,···,p ˜θi−βi τi≤x ≤P max i=1,2,···,p ˜θi−βi τi≤x\eNbn=Nbn +P eNbn=Nbn ≤P max i=1,···,p nX l=1tilel ≤x+Cn−δ1! +Cnαp+mνb−mη ≤P max i=1,···,p nX l=1tile∗ l ≤x! +Cnαp+mνb−mη + sup x≥0 P max i=1,···,p nX l=1tilel ≤x! −P max i=1,···,p nX l=1tile∗ l ≤x! + sup x∈R P max i=1,···,p nX l=1tile∗ l ≤x+Cn−... | https://arxiv.org/abs/2502.17261v2 |
∥zzz∥, zzz= [z1,···, zn]T. Then, ˇYn∈¯Bσ(Xnβββ) and equation (74) becomes ∥ˇβββα(ˇYn)−βββ∥2=∥βββα−βββ∥2+σ2 ∥zzz∥2⟨zzz,1 ng2 α(1 nXnXT n)XnXT nzzz⟩ +2√nσ ∥zzz∥⟨1 ngα(1 nXnXT n)zzz,1 ngα(1 nXnXT n)XnXT nXnβββ−Xnβββ⟩, If an appropriate value of zzzis selected (Wang et al., 2024), it follows that the final term on the righ... | https://arxiv.org/abs/2502.17261v2 |
0.02 0.04 0.5 0.06 0.08 0.1 0 0.12 0.14 0 -0.5 -0.5 -0.0200.020.040.060.080.10.120.14 Figure 6: Confidence intervals for HBF regression and its debiased estimator. -0.02 0.5 0 0.02 0.04 0.5 0.06 0.08 0.1 0 0.12 0 -0.5 -0.5 -0.02 0.5 0 0.02 0.04 0.5 0.06 0.08 0.1 0 0.12 0 -0.5 -0.5 -0.0200.020.040.060.080.10.12 Figure 7... | https://arxiv.org/abs/2502.17261v2 |
regularization methods for inverse problems. Acta Numerica , 27:1–111, 2018. Radu Bot ¸, Guozhi Dong, Peter Elbau, and Otmar Scherzer. Convergence rates of first-and higher-order dynamics for solving linear ill-posed problems. Foundations of Computa- tional Mathematics , 22(5):1567–1629, 2022. Peter B¨ uhlmann. Statist... | https://arxiv.org/abs/2502.17261v2 |
Mainardi, and Sergei Rogosin. Mittag-Leffler Functions, Related Topics and Applications . Springer-Verlag, Heidelberg, 2014. Markus Grasmair, Markus Haltmeier, and Otmar Scherzer. Sparse regularization with lq penalty term. Inverse Problems , 24(5):055020, 2008. Weihong Guo, Jing Qin, and Wotao Yin. A new detail-preser... | https://arxiv.org/abs/2502.17261v2 |
with high- dimensional responses. Journal of the American Statistical Association , 117(540):1738– 1750, 2022. Jianliang Li, Peijun Li, and Xu Wang. Inverse source problems for the stochastic wave equations: Far-field patterns. SIAM Journal on Applied Mathematics , 82(4):1113–1134, 2022. Qing Li. A comprehensive survey... | https://arxiv.org/abs/2502.17261v2 |
Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B , 58(1):267–288, 1996. Andrej Nikolaevich Tichonov, Aleksandr Sergeeviˇ c Leonov, and Anatolij Georgievich Jagola. Nonlinear Ill-Posed Problems . Chapman and Hall, London, 1998. Andrei Nikolaevich Tikhonov. ... | https://arxiv.org/abs/2502.17261v2 |
algorithm. Inverse Problems in Science and Engineering , 24(7):1186–1204, 2016. 67 Y. Wang, Y. Zhang and Y. Zhang Ye Zhang, Patrik Forss´ en, Torgny Fornstedt, M˚ arten Gulliksson, and Xiaoxia Dai. An adap- tive regularization algorithm for recovering the rate constant distribution from biosensor data. Inverse Problems... | https://arxiv.org/abs/2502.17261v2 |
INVARIANCE PRINCIPLE FOR THE GAUSSIAN MULTIPLICATIVE CHAOS VIA A HIGH DIMENSIONAL CLT WITH LOW RANK INCREMENTS MRIGANKA BASU ROY CHOWDHURY, SHIRSHENDU GANGULY Abstract. Gaussian multiplicative chaos (GMC) is a canonical random fractal measure obtained byexponentiatinglog-correlatedGaussianprocesses,firstconstructedinth... | https://arxiv.org/abs/2502.17412v1 |
one to study the other. The first is theory of Gaussian multiplicative chaos. 1.1.Gaussian multiplicative chaos (GMC). This was introduced by Kahane in [Kah85] as a mathematical model for energy dissipation in turbulence, making rigorous a program initiated by Mandelbrot in [Man05]. More precisely, Kahane gave a rigoro... | https://arxiv.org/abs/2502.17412v1 |
1.3). Subsequently, in [KK24], the question of an invariance principle in this context was investigated. Namely, whether the multiplicative chaos corresponding to S∞,a(t)is related to that corresponding to S∞,g(t).Note that they cannot be exactly the same since the k= 1term in the sum has a global multiplicative non-Ga... | https://arxiv.org/abs/2502.17412v1 |
particular instance involves testing whether a sample covariance matrix, properly centered and scaled, is close to a symmetric Gaussian matrix, which is related to testing the presence of latent geometry in random graph models [BDER16]. In [BG18], Bubeck and the second named author developed an entropy based argument t... | https://arxiv.org/abs/2502.17412v1 |
ofn a(1) i, a(2) io i⩾1 andn g(1) i, g(2) io i⩾1such that µγ,g≪µγ,a≪µγ,g, that is, the multiplicative chaos measures are mutually absolutely continuous. Natural higher dimensional analogues of the above theorem hold as well. We elaborate more on this in Section 3.4 later. As already mentioned, this solves the main open... | https://arxiv.org/abs/2502.17412v1 |
short outline of the key ideas next. 1.3.Idea of the proof. We first discuss the proof of the central limit theorem since that involves all the new ideas in the paper. Given this, the proof of Theorem 1.1 essentially follows from the arguments in [KK24]. The main difficulty in proving Theorem 1.2 is that the increments... | https://arxiv.org/abs/2502.17412v1 |
that the proof proceeds by discretizing the model into a hierarchical model, i.e., comparing the log-correlated process Sn,a(t)to a model on a tree (akin to how branching random walk is a, simpler to analyze, hierarchical proxy for the Gaussian free field), which reduces the problem to a finite dimensional one. We will... | https://arxiv.org/abs/2502.17412v1 |
d[Φ] t=1 nnX i=1vivT i1t<τidt= Γ2 tdt,Γt:=vuut1 nnX i=1vivT i1t<τi. It will be convenient to introduce the compact notation dΦt=VtdBtwhere B= (B1,···, Bn)and Vtis the matrix with the ithcolumn Vi tequal to1√nvi1t<τi.At this point we apply a change of variable justified by the following Lemma 2.1 to alternatively expres... | https://arxiv.org/abs/2502.17412v1 |
(2.4) NowAsisanaverageof nindependent, meanzeromatrices, whichbymatrixconcentrationestimates will have small operator norm. Now observe that if Asis small in operator norm, we expect√As+psU≈√psUallowing us to bound ∥Γs−Gs∥.This leads us to consider perturbations of square roots of positive semi-definite matrices. The u... | https://arxiv.org/abs/2502.17412v1 |
a, b⩾0. Asimilarboundholdsfor √P+Q+δI−√P+Q . Combining these bounds via the triangle inequality, we have p P+Q−√ P ⩽ p P+Q−p P+Q+δI + p P+Q+δI−√ P+δI + √ P+δI−√ P ⩽√ δ+p ∥Q∥+√ δ= 3p ∥Q∥, finishing the proof. □ 2.4.Bounding the operator norm of As.Returning to the problem at hand, Lemma 2.2 now allows us to bound the ga... | https://arxiv.org/abs/2502.17412v1 |
to be sub-polynomial in n.The remainder of the integral will be treated via direct first moment arguments used to bound eT 1(Γs−Gs) 2 2. While the computations appear shortly, it is worth remarking at this point that we can control the second part of the integral using moment arguments since the vectors vihave L2 norms... | https://arxiv.org/abs/2502.17412v1 |
i ∞⩽n−1 2nX i=1∥vi∥∞· |εi|⩽n−(r−1), (2.11) (since ∥vi∥∞≤√ d≤√nby hypothesis) we may replace aibya′ i(and τibyτ′ i), which we do in the sequel, with a (deterministic) additional coupling error of at most n−(r−1). However, having replaced aibya′ i, to match variances, we now replace gibyg′ i, where g′ iis Gaussian with V... | https://arxiv.org/abs/2502.17412v1 |
choice of s⋆above. Combining the bounds established above, we may convert (2.10) to read (still a′replacing aandg′ replacing g) [Φ1−Ψ1]∞≲s⋆· r d∥U∥ n(klogn+ log s⋆+m)! +n−r+1, 18 MRIGANKA BASU ROY CHOWDHURY, SHIRSHENDU GANGULY with probability ⩾1−O(e−m+n−r), for every m⩾0(we used dn−r≤n−r+1above). Recalling our choice ... | https://arxiv.org/abs/2502.17412v1 |
We now prove Theorem 1.1 using as input Theorem 1.2. 3.Application to the multiplicative chaos: Proof of Theorem 1.1 We start by briefly recalling the proof strategy outlined in Section 1.3 in a bit more detail before presenting formal arguments. Given that we will essentially follow the steps from [KK24], to keep thin... | https://arxiv.org/abs/2502.17412v1 |
first moment computation. •Coupling at thick points. Since at level m,the number of thick vertices is polynomially (in 2m) smaller than the number of variables which is of order 2m,Theorem 1.2 may be invoked to couple the Gaussian and non-Gaussian values so that the gap between Sn,aandSn,gis uniformly small (in fact, s... | https://arxiv.org/abs/2502.17412v1 |
I(v)⊆ I(w)). Using these partial sums, one may define a measure eµn,γ,aon[0,1]with a piecewise constant density via eµn,γ,a(dt):=eγeSa(t) Zn,γ,adt,eZn,γ,a:=EeγeSa(t), (3.4) where, by slight abuse of notation, eSa(t) =eSa(v),fort∈I(v). (3.5) As discussed earlier, results from [Jun20] show that the measures eµn,γ,aconver... | https://arxiv.org/abs/2502.17412v1 |
1.2. A related statement appeared in [KK24, Lemma 2.5], which stipulated essentially that |K| ≤ 2n 2.As already alluded to, this was proven as an immediate consequence of a high dimensional CLT result which goes back to [Yur78]. The latter result was covariance agnostic and hence our improvement allowing us to take |K|... | https://arxiv.org/abs/2502.17412v1 |
all the tis are distinct, we have m⩽Pn i=12i·f(i) = 2n+1·f(n). Since the vectors ζkin (3.9) are indexed by k∈[k−, k+],to simplify the index set, define the vectors v0, . . . , v 2n−1−1∈Cmbyvk=ζk′where k′=k+ 2n−1. For notational convenience, we will replace nbyn+ 1so that there are 2nvectors in total and not 2n−1and pro... | https://arxiv.org/abs/2502.17412v1 |
our purposes only their summability will be important. Proof.Let us first estimate the size of Kn,γ. Recalling (3.3), note that, E|Kn,γ|⩽·poly( n)· |Nn−1| · sup vP(eSa(v)⩾(n−1) ((γ−δ) log 2)) + sup vP(eSg(v)⩾(n−1) ((γ−δ) log 2)) , (3.13) where the supremum is over v∈ N n−1, and the poly( n)is a crude upper bound for ... | https://arxiv.org/abs/2502.17412v1 |
2.4] which goes back to [Yur78] was covariance agnostic. Thus, explicitly pinning down the role of Uin the coupling 28 MRIGANKA BASU ROY CHOWDHURY, SHIRSHENDU GANGULY error in Theorem 1.2, and exploiting the DFT structure of Uto obtain a poly( n)bound on ∥U∥in Lemma 3.4, are crucial in obtaining a bound that is effecti... | https://arxiv.org/abs/2502.17412v1 |
=X λk∈(22(n−1),22n]λ−d/4 kakhk(x), Xg(x) =X λk∈(22(n−1),22n]λ−d/4 kgkhk(x), The arguments from here on are particularly simplified by the fact that the eigenfunctions hkare the tensor product of done dimensional eigenfunctions. That is, hk(x1, x2, . . . , x d) =f1(x1)⊗f2(x2)⊗ ··· ⊗ fd(xd) where for any 1≤i≤d,fi(xi) = s... | https://arxiv.org/abs/2502.17412v1 |
one may reduce to the case when Pis strictly positive definite. For any vector v∈Rd, we have vT(Q−P)v=vTP1 2(P−1 2QP−1 2−I)P1 2v INVARIANCE PRINCIPLE FOR GMC 31 so that P⪯Q⇐⇒ I⪯P−1 2QP−1 2(when Pis invertible). In terms of eigenvalues, this is the same as λmin(P−1 2QP−1 2)⩾1, which, by similarity, is equivalent to λmin... | https://arxiv.org/abs/2502.17412v1 |
Bentkus. A lyapunov-type bound in rd. Theory of Probability & Its Applications , 49(2):311– 323, 2005. [Ber41] Andrew C Berry. The accuracy of the gaussian approximation to the sum of independent variates. Trans- actions of the american mathematical society , 49(1):122–136, 1941. [Ber45] Harald Bergström. On the centra... | https://arxiv.org/abs/2502.17412v1 |
[KS91] Ioannis Karatzas and Steven Shreve. Brownian motion and stochastic calculus , volume 113. Springer Science & Business Media, 1991. [Man05] Benoit B Mandelbrot. Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. In Statistical Models and T... | https://arxiv.org/abs/2502.17412v1 |
Stronger Neyman Regret Guarantees for Adaptive Experimental Design Georgy Noarov†, Riccardo Fogliato‡, Martin Bertran‡, Aaron Roth†‡ February 25, 2025 Abstract We study the design of adaptive, sequential experiments for unbiased average treatment effect (ATE) estimation in the design-based potential outcomes setting. O... | https://arxiv.org/abs/2502.17427v1 |
is defined as a deterministic function of the observed population rather than a superpopulation parameter. This distinction ensures robustness to treatment effect heterogeneity and temporal data drift, challenges that can undermine conventional superpopulation-based designs. Our contributions We focus on the design of ... | https://arxiv.org/abs/2502.17427v1 |
relaxed superpopulation setting — and so our method achieves a best-of-both-worlds guarantee, up to logarithmic factors. 2 groups. A key challenge here is to balance the treatment probabilities in a way that balances the efficiency of the ATEs estimates across groups. Our proposed design leverages a variation of the “s... | https://arxiv.org/abs/2502.17427v1 |
paper, we will study both settings: the noncontextual setting in Section 3 and the contextual one in Section 4. Adaptive design In a randomized controlled trial (RCT), the experimenter (randomly) decides whether to apply treatment or control to each unit, and observes the corresponding outcome but not the counterfactua... | https://arxiv.org/abs/2502.17427v1 |
a no-regret design. 3 Efficient Non-Contextual ATE Estimation We now present our first contribution: An adaptive design that achieves eO(logT)Neyman regret under natural assumptions on the outcomes. We begin by discussing the eO(√ T)-Neyman regret design ClipOGD of Dai et al. [2023], and then modifying it to better exp... | https://arxiv.org/abs/2502.17427v1 |
relies on a stricter assumption than the one made by Dai et al. [2023]’s, which we detail below. Assumption 3.1 (Bounds on Potential Outcomes) .There exist positive constants c, Csuch that outcomes {(yt(0), yt(1))}t≥1satisfy for all time horizons T: max t≥1{|yt(0)|,|yt(1)|} ≤C, c ≤min min t≥1 yt(0)2+yt(1)21/2,min... | https://arxiv.org/abs/2502.17427v1 |
population, the regret nonnegativity holds automatically, implying that our adaptive design will necessarily converge to the best nonadaptive design without further assumptions. Corollary 3.5 (Convergence in the Superpopulation Setting) .Suppose that the outcomes are drawn i.i.d. from a superpopulation: (yt(0), yt(1))∼... | https://arxiv.org/abs/2502.17427v1 |
units but also on each subsequence that results from conditioning on units belonging to a group G, simultaneously for all groups G∈ G. 4.1 A New Metric: Multigroup Neyman Regret We introduce multigroup Neyman regret as a strengthening of (vanilla) Neyman regret. Specifically, given any contextual group collection G,G-m... | https://arxiv.org/abs/2502.17427v1 |
generated according to pt,eff. After the outcome is revealed, MGATE updates all group weights, as well as the propensities of groups that were active. We can show that MGATE achieves the following multigroup Neyman regret guarantee. We note that MGATE is anytime valid, meaning that just like our noncontextual design Cl... | https://arxiv.org/abs/2502.17427v1 |
σ=0.1 σ=1 σ=10 100030001000030000 100030001000030000 10003000100003000030100300 0.31.03.010.0 0.11.010.0 RoundNeyman regret Method CLIPOGD0CLIPOGDSCFigure 1: Treatment probabilities and Neyman regret of ClipOGD on Gaussian data for different noise ( σ) levels. As σincreases, ClipOGDSCconverges more slowly. Its regret r... | https://arxiv.org/abs/2502.17427v1 |
paths. We then measure the Neyman regret by averaging the regret across these probabilities obtained at each time step. Hyperparameter choices Throughout the experiments, we use the following hyperparameters. For our method, we set ηt= 2/t, and we set the clipping rate δt= 1/h(t), where the clipping function ish(t) =ex... | https://arxiv.org/abs/2502.17427v1 |
introduced a modification of the ClipOGD algorithm that provably yields vanishing Neyman regret, achieving an anytime-valid eO(logT)Neyman regret, improving upon previous eO(√ T) guarantees. We also extend our framework to incorporate contextual information by introducing a multigroup formulation. Our proposed multigro... | https://arxiv.org/abs/2502.17427v1 |
Introduction to online convex optimization. Foundations and Trends ®in Optimization , 2(3-4):157–325, 2016. Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, et al. Training compute-optimal large lan... | https://arxiv.org/abs/2502.17427v1 |
educational Psychology , 66(5):688, 1974. Vira Semenova and Victor Chernozhukov. Debiased machine learning of conditional average treatment effects and other causal functions. The Econometrics Journal , 24(2):264–289, 2021. Aarohi Srivastava, Abhinav Rastogi, Abhishek Rao, Abu Awal Md Shoeb, Abubakar Abid, Adam Fisch, ... | https://arxiv.org/abs/2502.17427v1 |
Bounds) .Under Assumption 3.1, for every t≥1we have the following bounds in expectation wrt. the design’s randomness: E[|gt|]≤2C2h(t)2,E[g2 t]≤2C4h(t)5. Proof.The bounds follow as shown in Lemma C.5 of Dai et al. [2023], by just expanding out the first and second raw absolute moment of the gradient estimator defined ab... | https://arxiv.org/abs/2502.17427v1 |
4c2t∗X t=1t h(t). Finally, recalling the definition of t∗=hinv(A) =hinv(1 +C/c)and substituting it in, we obtain the desired claim. Finally, with the result of Claim 6 in hand, we observe that (1) the term −c2(T+1)E[(pT+1−p∗)2] is nonpositive and can thus be ignored, (2) the second term on the right hand side is asympt... | https://arxiv.org/abs/2502.17427v1 |
fixed-probability sampling scheme’s variance. And given that our design has a no-regret guarantee with respect to this benchmark, dVBthus also asymptotically approximates the upper bound on our (and any other such) design’s induced IPW estimator variance VT. This is the blueprint of the proof, and we will now briefly r... | https://arxiv.org/abs/2502.17427v1 |
SE regret) .Asleeping experts (SE) algorithm A over domain V⊆Rd, where d≥1is the number of “sleeping experts”, sequentially receives vectors at∈ {0,1}dandℓt∈Rdat rounds t= 1,2, . . .. The vector athas the interpretation that at,i∈ {0,1} (for each i∈[d]) denotes whether expert iis “active” (1) or “inactive” (0) in round... | https://arxiv.org/abs/2502.17427v1 |
2. 7:end for We note that the update for wtin Algorithm 4 is the solution to the original argmax problem in Algorithm 3, with the nonnegative orthant as domain and the rescaled L2-norm as regularizer. Scale-FreeSleepingExperts Now, wewillturnthisjustobtainedscale-freeOLOregretguarantee into a scale-free sleeping expert... | https://arxiv.org/abs/2502.17427v1 |
distribution of egtis determined by all prior history up to and including determining pt; (2) It is an unbiased estimator of f′ t(pt), in that E[egt|Ft−1] =f′ t(pt) =−yt(1)2 p2 t+yt(0)2 (1−pt)2. It is easy to observe that Algorithm 1 conforms to Definition C.5. Algorithm 1 is written as requiring direct access to the s... | https://arxiv.org/abs/2502.17427v1 |
the group-specific ATE Neyman regret minimization algorithm: E[Term 2 ]≤E[RegVarT(AATE)]. The first term will be bounded by the sleeping experts regret of the aggregation algorithm. To continue the analysis, we first collect the properties of the estimated outcomes, losses, and gradients. Namely, we have for any round ... | https://arxiv.org/abs/2502.17427v1 |
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