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To do so, note that if we change a single example ( Xi, Si), then ⟨u, Z n(h)⟩changes by at most n−1supx⟨u, ϕ(x)⟩ ≤ n−1∥u∥2supx∥ϕ(x)∥2. Using homogeneity, for any scalar tthere exists u∈Rdsuch that ⟨u, Z n(h)− E[Zn(h)]⟩ ≥ ∥ u∥2tif and only if there exists u∈Sd−1such that ⟨u, Z n(h)−E[Zn(h)]⟩ ≥t. So if bϕ= supx∈X∥ϕ(x)∥2,... | https://arxiv.org/abs/2503.00220v1 |
≤c"q bϕ(u)α·E[⟨u, ϕ(X)⟩]r dlogn+t n+bϕdlogn+t n# simultaneously for all u∈Bsuch that ⟨u, ϕ(x)⟩ ≥0for all x. If additionally the scores Siare distinct with probability 1, then with the same probability, Eh ⟨u, ϕ(Xn+1)⟩ 1{Yn+1̸∈bC(Xn+1)} −α |Pni ≥ −c"q bϕ(u)α·E[⟨u, ϕ(X)⟩]r dlogn+t n+bϕdlogn+t n# simultaneously for all ... | https://arxiv.org/abs/2503.00220v1 |
VC-dimension calculation, and localized Rademacher complexities [4, 18]. We begin with the form of Talagrand’s empirical process inequality with constants due to Bousquet [7]. Lemma 4.1 (Talagrand’s empirical process inequality) .LetFbe a countable class of functions with Pf= 0 and∥f∥∞≤bforf∈ F. Let Z= supf∈FPnfandσ2=σ... | https://arxiv.org/abs/2503.00220v1 |
assumes functions are mean-zero, but an inspection of the proof shows this is unnecessary); see also the results of [23] and [9, Proof of Proposition 1]. These show that second moments satisfy one-sided concentration bounds with high probability as soon as we have the fourth moment condition E[f4(X, S)]≤b2E[f2(X, S)] f... | https://arxiv.org/abs/2503.00220v1 |
expansion (19) by combining these bounds with inequal- ity (18) and considering that v(bh, u)≤rorv(bh, u)> rwhere r2=O(1)b2 ϕd nlogn d. In the latter, we have v2(bh, u)≤cbϕ(u)αPn⟨ϕ(X), u⟩. We have therefore shown that for any r2≳d nlogn d, with probability at least 1 −Kne−t−e−nr2, for all u∈Bwith⟨u, ϕ(x)⟩ ≥0, (Pn−P)⟨u,... | https://arxiv.org/abs/2503.00220v1 |
bCn(x) = {y|s(x, y)≤bτn}. Setting bτn=Quant (1+1/n)(1−α)(Sn 1), the slightly enlarged quantile, guarantees (1−α) coverage; this follows by letting S(i,n)be the order statistics of Sn 1andS(i,n+1)those of Sn+1 1, and noting that the score Sn+1≤S(k,n)if and only if Sn+1≤S(k,n+1)[25, Lemma 2], so the inflation byn+1 nis n... | https://arxiv.org/abs/2503.00220v1 |
undercovers, especially when the ratio n/d < 20 or so. The naive correction (21) appears to be a bit conservative, while the scaling correction (22) is more effective. 5.1.2 Full conformal versus split-conformal predictions We briefly look at the coverage properties of the full conformalization method (5) from the pa- ... | https://arxiv.org/abs/2503.00220v1 |
to one of computational feasibility. 5.2 Prediction on CIFAR-100 We also perform an exploratory experiment on the CIFAR-100 dataset, a 100-class image classifi- cation dataset consisting of 60,000 training examples and a 10,000 example test set. We use the 22 0 1 2 3 4 marginal Group Index0.000.020.040.060.080.100.12Mi... | https://arxiv.org/abs/2503.00220v1 |
adaptive thresholds of the form bC(x) ={y|s(x, y)≤bh(x)}can indeed provide stronger coverage than non-adaptive thresholds. Moreover, they are much faster to compute with than full conformal methods—in the experiment in Figure 3, the split conformal method was roughly 8000 × faster than the full conformal method. Additi... | https://arxiv.org/abs/2503.00220v1 |
Recalling inequality (16), the second term dominates the first, and so E[Rn(Fr)]≤crr d nlogn difr2≥b2 ϕd nlogn d. Define the random variable Zn(r):= supf∈Fr(Pn−P)f= supf∈Fr|(Pn−P)f|, the equality following by symmetry of Fr. Then Talagrand’s concentration inequality (Lemma 4.1) implies that P Zn(r)≥E[Zn(r)] +q 2(r2+ 2... | https://arxiv.org/abs/2503.00220v1 |
Statistical Science , to appear, 2024. [11] I. Gibbs, J. Cherian, and E. Cand` es. Conformal prediction with conditional guarantees. arXiv:2305.12616 [stat.ME] , 2023. [12] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pat... | https://arxiv.org/abs/2503.00220v1 |
arXiv:2503.00290v1 [econ.EM] 1 Mar 2025Uniform Limit Theory for Network Data Yuya Sasaki∗ Vanderbilt University. Abstract I present a novel uniform law of large numbers (ULLN) for netw ork-dependent data. While Kojevnikov, Marmer, and Song (KMS, 2021) provid e a comprehensive suite of limit theorems and a robust varian... | https://arxiv.org/abs/2503.00290v1 |
task is further complic ated by the intricate dependence structure inherent in network data. The main contribution of this paper is to bridge this gap by establishing a novel ULLN under network dependence. My results build on the KMS framework , which utilizes model restrictions based on conditional ψ-dependence, decay... | https://arxiv.org/abs/2503.00290v1 |
Consider a triangular array {Yn,i}i∈Nnof random vectors in Rv. The following definition introduces the notion of conditional ψ-dependence as provided in KMS. Definition 1 (Conditional ψ-Dependence; KMS, Definition 2.2) . A triangular array {Yn,i}i∈Nnisconditionally ψ-dependent given {Cn}if for each n∈N, there exists a Cn-... | https://arxiv.org/abs/2503.00290v1 |
finite -net approxi- mation off(Yn,i,θ) for allθ∈Θ, as a way to establish the uniform result. 3 The Main Results: Uniform Laws of Large Numbers I now state the first main result of this paper—the uniform law of larg e numbers for network- dependent data. Theorem 1 (Uniform Law of Large Numbers) .If Assumptions 1–6 are sa... | https://arxiv.org/abs/2503.00290v1 |
of large numbers, there is an implici t cost in the background. For instance, if one makes Assumption 7 less restrictive by lowe ringǫ1andǫ2while keeping ǫ0 in Assumption 3′fixed, then the rate of divergence in the bound of the maximal i nequality increases. In particular, E/bracketleftBig max 1≤k≤n/vextendsingle/vexten... | https://arxiv.org/abs/2503.00290v1 |
in probability to a positive definite matrix W, the GMM estimator is defined as ˆθGMM= argmin θ∈ΘQn(θ), whereQn(θ) =¯fn(θ)⊤Wn¯fn(θ). We can define the population criterion by Q(θ) =E/parenleftbig f(Yn,i,θ)/parenrightbig⊤WE/parenleftbig f(Yn,i,θ)/parenrightbig . Suppose that the population moment satisfies the following con... | https://arxiv.org/abs/2503.00290v1 |
Moon (2019). Normal approximation in large n etwork models. arXiv preprint arXiv:1904.11060 . Newey, W. and D. McFadden (1994). Large sample estimation and hy pothesis testing. In R. Engle and D. McFadden (Eds.), Handbook of Econometrics , pp. 2111–2245. Elsevier. 16 Appendix The appendix consists of two sections. Appe... | https://arxiv.org/abs/2503.00290v1 |
that for every θ∈Θ, there exists some θjwith /bardblθ−θj/bardbl ≤δ. For anyθ∈Θ, letθj(θ)∈ {θ1,...,θ J}be an element of the δ-net satisfying /bardblθ−θj(θ)/bardbl ≤δ. Then, we can decompose Sn(θ) as Sn(θ) =Sn(θj(θ))+/parenleftBig Sn(θ)−Sn(θj(θ))/parenrightBig . 20 Triangle inequality yields sup θ∈Θ/vextendsingle/vextend... | https://arxiv.org/abs/2503.00290v1 |
set of assumptions with the basic notations inherited from Section 2 in the main text. Assumption 10 (Zero Mean and Boundedness) . There exists M <∞such thatE[Xn,i|Cn] = 0and|Xn,i| ≤Mfor allia.s. Assumption 11 (Decay Rate) . (i) There exists ψ<∞such that/vextendsingle/vextendsingle/vextendsingleCov/parenleftbig Xn,i,Xn... | https://arxiv.org/abs/2503.00290v1 |
Xn,i1···Xn,ip/vextendsingle/vextendsingle/vextendsingleCn/bracketrightBig/vextendsingle/vextendsingle/vextendsingle≤Mp/parenleftBigg/productdisplay 1≤k<k′≤dE|Xn,ikXn,ik′|/parenrightBigg1/p 27 ≤Mpψd(d−1) 2p/productdisplay 1≤k<k′≤dϑ1/p n,dn(ik,ik′) under Assumptions 10 and 11 (i). The product /productdisplay 1≤k<k′≤dϑ1/p... | https://arxiv.org/abs/2503.00290v1 |
arXiv:2503.00538v3 [math.ST] 9 May 2025Geometric Ergodicity of Gibbs Algorithms for a Normal Model With a Global-Local Shrinkage Prior Yasuyuki Hamura∗ May 12, 2025 Abstract In this paper, we consider Gibbs samplers for a normal linear regres sion model with a global-local shrinkage prior. We show that they produce geo... | https://arxiv.org/abs/2503.00538v3 |
algorithms use rejection sampling, we also discuss approaches to construnct efficient Accept-Reject algorithms. The remainder of the paper is organized as follows. In Sectio n 2.1, our horseshoe model and algorithm are presented. A geometric ergodicity result for the algorithm is stated and proved in Sections 2.2 and 3. ... | https://arxiv.org/abs/2503.00538v3 |
invariant and inverse gamma dis tributions as limiting cases. Also, the half-Cauchy distribution is obtained by setting a=b= 1/2 andc= 1. Polson and Scott (2011) argued that πτ“should have substantial mass near zero”, which correspond s to smaller values ofaandc. What is important for theory in this section is not so m... | https://arxiv.org/abs/2503.00538v3 |
M2>0. 7 Meanwhile, by making the change of variables µk= 1/νkfork= 1,...,p, p/summationdisplay k=1E/bracketleftBig/parenleftBigτ2 νk/parenrightBigε/vextendsingle/vextendsingle/vextendsingle/tildewideλ2/bracketrightBig =p/summationdisplay k=1E/bracketleftBig (τ2)ε/integraldisplay∞ 0µkε/parenleftBig 1+τ2 ˜λ2 k/parenright... | https://arxiv.org/abs/2503.00538v3 |
Γ(ε)/integraldisplay∞ 0wε−1 (1+w)ε+1/2E/bracketleftBig1 (τ2)ε/vextendsingle/vextendsingle/vextendsingle/tildewideλ2/bracketrightBig dw+p/summationdisplay k=11 (˜λ2 k)εΓ(1+ε) Γ(ε)/integraldisplay∞ 0wε−1 (1+w)3/2dw. Ifε<a−p/2, thenE[πτ(τ2)/(τ2)ε+p/2]<∞and we have E/bracketleftBig1 (τ2)ε/vextendsingle/vextendsingle/vexten... | https://arxiv.org/abs/2503.00538v3 |
2E/bracketleftBig˜λ2 k ˜λ2 k+τ2/braceleftBig p/2+a−(a+b)τ2 c+τ2−p/summationdisplay l=1 l/\egatio\slash=kτ2 ˜λ2 l+τ2/bracerightBig/vextendsingle/vextendsingle/vextendsingle/tildewideλ2/bracketrightBig ≥1 2E/bracketleftBig˜λ2 k ˜λ2 k+τ2/vextendsingle/vextendsingle/vextendsingle/tildewideλ2/bracketrightBig E/bracketleftBi... | https://arxiv.org/abs/2503.00538v3 |
(c+τ2)a+b/parenleftBigB∗/productdisplay k=11 ˜λ2∗+τ2/parenrightBig dτ2, which proves part (ii). For part (iii), assume that /braceleftBigg a /∈N, if p∈2N, a /∈N−1/2, ifp∈2N−1. 17 LetA∗andB∗be as in parts (ii) and (iii). Then A∗>0 andB∗> p/2 +a. LetC∗= min{1,B∗−(p/2 +a)}/2. ThenC∗∈(0,1). Fixγ >0. Then by part (ii) and t... | https://arxiv.org/abs/2503.00538v3 |
He et al. (2 022) for an efficient PTN generator.) If πτis truncated gamma, we would use a modified version of their PT N sampler in which the proposal distribution is tuned using a Newton-Rap hson algorithm. The conditional densities of λ1,...,λ pgiven by (4.2) are also not necessarily standard dis- tributions. However, ... | https://arxiv.org/abs/2503.00538v3 |
Next, fixk= 1,...,p. Then Var(˜βk|σ2,τ2,λ)≤σ2 and (E[˜βk|σ2,τ2,λ])2= [(e(p) k)⊤{I(p)+τ2Λ(X⊤X)Λ}−1ΛX⊤y(τ2)1/2]2 ≤τ2(e(p) k)⊤{I(p)+τ2Λ(X⊤X)Λ}−1e(p) ky⊤XΛ{I(p)+τ2Λ(X⊤X)Λ}−1ΛX⊤y ≤τ2y⊤XΛ{τ2Λ(X⊤X)Λ}−1ΛX⊤y =y⊤PXy. Therefore, E[˜βk2/σ2|τ2,λ]≤1+y⊤PXyE[1/σ2|τ2,λ]. 24 Finally, for all ε>0, we have E[(τ2)ε|λ]≤/integraldisplay∞ 0tεπ... | https://arxiv.org/abs/2503.00538v3 |
the Royal Statistical Society. Series C (Applied Statist ics),41, 337–348. [10] Gradshteyn, I.S. and Ryzhik, I.M. (2014). Table of Inte grals, Series, and Products. New York: Academic Press . [11] He, J., Polson, N.G. and Xu, J. (2022). Data augementati on with Polya inverse gamma. arXiv preprint arXiv:1905.12141 . [12... | https://arxiv.org/abs/2503.00538v3 |
Semi-Parametric Batched Global Multi-Armed Bandits with Covariates Sakshi Arya∗ Department of Mathematics, Applied Mathematics, & Statistics, Case Western Reserve University Hyebin Song† Department of Statistics, Pennsylvania State University March 4, 2025 Abstract The multi-armed bandits (MAB) framework is a widely us... | https://arxiv.org/abs/2503.00565v1 |
where arms share a global parameter, making them globally informative [Atan et al., 2015, 2018, Shen et al., 2018]. More specifically, in the GMAB model framework, it is assumed that the expected reward from each arm is a known func- tion of a single global parameter. While this proposal provides an effective framework... | https://arxiv.org/abs/2503.00565v1 |
relevance and challenges in scenarios with a small number of batches ( M≈2,3,4,5), as often seen in clinical trials. In this paper, we explore the batched GMABC problem within a semi-parametric frame- work, aiming to achieve a “best-of-both-worlds” approach to sequential decision-making. Our methodological contribution... | https://arxiv.org/abs/2503.00565v1 |
Arm Elimination (BIDS) algorithm, which integrates single-index guided dynamic binning and successive arm elimination as its core components. Sections 3.1 and 3.2 detail the method- ology for two distinct scenarios: when a pilot estimate is available and when it is not. Section 4 provides the regret analysis for these ... | https://arxiv.org/abs/2503.00565v1 |
perfect foreknowledge of the optimal action at each time step. We make the following assumptions on the reward functions. Assumption 1 (Smoothness) .We assume that the link function f(k):R→Rfor each arm is ( η, L)-smooth, that is, there exists η∈(0,1] and L >0 such that for k∈ {1,2}, |f(k)(u)−f(k)(u′)| ≤L|u−u′|η, holds... | https://arxiv.org/abs/2503.00565v1 |
we propose an algorithm, which we call as Batched single Index Dynamic Binning and Successive arm elimination (BIDS), for the batched GMABC problem that leverages the shared parameter structure across arms and the underlying single-index mod- els. Our approach adopts a successive elimination strategy combined with adap... | https://arxiv.org/abs/2503.00565v1 |
employed in our algorithm can be explained through a tree-based interpretation as follows. Hierarchical partitioning and tree structure We build a tree Tof depth M(recall, Mis the number of batches). Each layer of the tree Tconsists of a partition of X, the support of PX, defined by the direction βand the number of spl... | https://arxiv.org/abs/2503.00565v1 |
we eliminate arms that are “statistically worse than the best arm”. Then, if any active bin still has more than one active arm, this suggests that the current bin is not fine enough for the decision-maker to tell the difference between the two arms. As a result, we split any active bin that still has more than one acti... | https://arxiv.org/abs/2503.00565v1 |
C,i={τC,i(s); 1≤s≤mC,i, smod K≡k} be the set of time points tduring batch iwhere Xtvisits C, and the arm kis pulled fork= 1, . . . , K . Define the average rewards for Cfrom arm k∈ {1,2}during batch i∈ {1, . . . , M }as: ¯Y(k) C,i=1 |τ(k) C,i|X t∈τ(k) C,iY(k) t. (5) Once ¯Y(k) C,ifork∈ {1,2}are obtained, we check wheth... | https://arxiv.org/abs/2503.00565v1 |
X−axis) Interval [−5,−2) [−2,1) [1,4) [4,7]Projection and Binning Figure 1: (a) 3-D representation of the data (toy example) such that ydepends on a linear combination of the covariates. (b) Projecting X∈R2(circles with holes) in the single- index direction (red dotted line with black filled circles as projected points... | https://arxiv.org/abs/2503.00565v1 |
< M then ▷Batch Elimination (at the end of batch i) 10: Rewards during batch i,Yti−1+1, . . . , Y ti, are revealed. 11: Initialize L(i+1)={}. 12: forC∈ L(i)do ▷Iterate over active bins 13: if|IC|= 1then ▷If only one active arm remains in C 14: L(i+1)=L(i+1)∪ {C} 15: Break (Proceed to the next bin C) 16: else|IC|>1 ▷If ... | https://arxiv.org/abs/2503.00565v1 |
implies that |u⊤v|= 0, which means uandvare orthogonal. Equivalently, we can express this in terms of the sine principal angle distance sin ∠u, v∈[0,1], where sin ∠u, v= 0 implies that u, vare identical up to sign and sin ∠u, v= 1 implies uandvare orthogonal. 17 Assumption 4. The initial vector βsatisfies sin∠β, β 0≤C0... | https://arxiv.org/abs/2503.00565v1 |
(9)and (11), the following bound on the expected regret RT(π) = E[RT(π)]holds for sufficiently large T: RT(π)≤C2Mlog(T)T1−γ 1−γM, where γ=η(1+α) 2η+1, where C2is a constant depending on model parameters such as α, η, D 0, L,cX, cX, andRX, but not on the sample size T. Corollary 1 shows that when the number of batches i... | https://arxiv.org/abs/2503.00565v1 |
vector from a Single-Index Regression (SIR) method used in Algorithm 2. Specifically, we require that the SIR algorithm used in Algorithm 2 produces an estimate that satisfies a parametric error bound up to a log term with high probability when applied to an i.i.d dataset of size nk. Assumption 5. Letk∈ {1,2, . . . , K... | https://arxiv.org/abs/2503.00565v1 |
γ=η(1+α) 2η+1, where C6depends on the single index parameter βand other constants such as α, η, D 0, L, R X,cX, cX. Proof. We know from (15) that, RT(π)≤tinit+RT−tinit(π;β). Define Eβto be the event that the inequality (13) holds for all k∈ {1, . . . , K }, which holds with probability at least 1 −KC 4(tinit/2K)−ϕunder... | https://arxiv.org/abs/2503.00565v1 |
with Dynamic Binning (BaSEDB) algorithm of Jiang and Ma [2024]. We consider both the cases discussed in Sections 4.1 and 4.2: 1) when the pilot direction is available under varying degrees of accuracy, and 2) when the pilot direction is unknown and estimated using the initial tinitamount of data, under varying signal-t... | https://arxiv.org/abs/2503.00565v1 |
points choices according to (9) and (10), and in the second case with unknown pilot directions, the initial batch size is set to T2/3, and the remaining time points are partitioned according to the same rules. In addition, in the latter case, Algorithm 2 requires specifying an SIR algorithm and arm weights. For the SIR... | https://arxiv.org/abs/2503.00565v1 |
algorithm with pilot directions of varying accuracies, compared to BaSEDB algo- rithm. As the perturbation level increases, the performance of the BIDS algorithm with the perturbed pilot estimate declines. However, it consistently outperforms the nonpara- metric batched bandit algorithm (BaSEDB), even under high pertur... | https://arxiv.org/abs/2503.00565v1 |
moderately large dimensions. 29 6 Application to Real Data We compare the performance of the batched single-index and batched nonparametric BaSEDB algorithm on three publicly available real datasets: a) Rice classification [Cinar and Koklu, 2019]: Classifying rice into two common varieties in Turkey, namely, Cammeo and... | https://arxiv.org/abs/2503.00565v1 |
1278 t[4] == 2534 t[5] == 3809t[1] == 243 t[2] == 1172 t[3] == 2206 t[4] == 3252 t[5] == 3809 0.20.30.40.5 0 1000 2000 3000 4000 tFraction of incorrect pullsRice Classification t[1] == 404 t[2] == 902 t[3] == 2188 t[4] == 4045 t[5] == 6000 t[6] == 8143t[1] == 404 t[2] == 1333 t[3] == 2712 t[4] == 4370 t[5] == 6255 t[6]... | https://arxiv.org/abs/2503.00565v1 |
tinit= 243, 404, and 607 ( ≈T(2/3)) each, respectively. We display the estimate βi(with standard errors over the 60 replications in the parenthesis) for each i= 1, . . . , d , for d= 7,5,14, for the three datasets, respectively. We can infer about the relevance of a variable by the absolute value of the corresponding e... | https://arxiv.org/abs/2503.00565v1 |
covariate space is determined by the single-index direction. We provided rigorous theoretical regret analysis under two scenarios: (1) when the single-index direction is known a priori and (2) when it is estimated from initial data. In the former case, the regret rate matches the minimax optimal rate for nonparamet- ri... | https://arxiv.org/abs/2503.00565v1 |
is bounded below and above by constants cX S1 Category Notation Description Problem setupT Total time horizon K Number of arms M Number of batches X Covariate space in Rd G Partition of {1, . . . , T }inMbatches {t0, t1, . . . , t M} Batch end points RT(π) Cumulative regret of π RT(π) Expected cumulative regret of π ∠u... | https://arxiv.org/abs/2503.00565v1 |
process, respectively. Note |I′ C|>1 will trigger splitting Cinto its children sets. Define IC= k∈ {1,2}: sup x∈C{f(∗)(x⊤β0)−f(k)(x⊤β0)} ≤c0|C|η T , (S-2) IC= k∈ {1,2}: sup x∈C{f(∗)(x⊤β0)−f(k)(x⊤β0)} ≤c1|C|η T , (S-3) forc0= 4L0+ 1 with L0=L(2C0RX+ 1)η,c1= 8c0γ1/2 Xwhere γX=cX/cX, and f(∗)(x⊤β0) = max k∈{1,2}f(k)(x... | https://arxiv.org/abs/2503.00565v1 |
C∈Bi∩JTrb T(C)1(Sc C∩ GC∩ E) +X C∈BM∩JTrb T(C)1(GC). S6 Let, for i= 1, . . . , M −1, Ui:=X C∈Bi∩JTrl T(C)1(SC∩ GC), V i:=X C∈Bi∩JTrb T(C)1(Sc C∩ GC∩ E), andWM=:P C∈BM∩JTrb T(C)1(GC) so that RT(π)1(E)≤M−1X i=1(Ui+Vi) +WM. (S-10) Next, we bound these three terms, namely, Ui, ViandWMseparately. Controlling Ui.Let us fix s... | https://arxiv.org/abs/2503.00565v1 |
for 1 ≤i≤M−1, E[Ui]≤(ti−ti−1)D0{c1wη i−1}1+α. (S-12) S9 Controlling Vi.Similarly, choose some 1 ≤i≤M−1 and bin C∈ B i∩ J T. We have C=CA(β) for some A∈ A i. We have from definition of rb T(C), E[rb T(C)1(GC∩ Sc C∩ E)] =E"TX t=1(g∗(Xt)−g(πt(Xt))(Xt))1(Xt∈C)1(C∈ J t)1(GC∩ Sc C∩ E)# =E TX t=ti−1+1(g∗(Xt)−g(πt(Xt))(Xt))1... | https://arxiv.org/abs/2503.00565v1 |
is not split into its children sets after the batch elimination at the end of batch i, and 3.Cis born at the beginning of batch i, and is split into its children sets after the batch elimination at the end of batch i. In case 1, Cis never born, i.e., C /∈ L tfor all 1 ≤t≤T, as a set C∈ B ican be born only at batch iby ... | https://arxiv.org/abs/2503.00565v1 |
let µ=E[X]denote the sum’s expected value. Then for any δ >0, P(|X−µ| ≥δµ)≤2e−δ2µ/3. More details on multiplicative Chernoff bound and its extensions can be found in Kusz- maul and Qi [2021]. Next, we use the multiplicative Chernoff bound to provide a concen- tration result on the number of covariates falling in a bin ... | https://arxiv.org/abs/2503.00565v1 |
¯g(k) C−g(k)(x) ≤1 PX(C)Z y∈CL{23/2C0RXT−ξ/(2η+1)+wi}ηdPX(y) ≤L{23/2C0RXT−ξ/(2η+1)+wi}η. From (10), we note that wi≍T−1−γi 1−γM1 2η+1. Therefore for ξ≥1,there exists T0<∞such S18 thatT−ξ/(2η+1)≤wiforT≥T0. For such T, ¯g(k) C−g(k)(x) ≤sup x,y∈C g(k)(y)−g(k)(x) ≤L(23/2C0RX+ 1)ηwη i=L0wη i. (S-23) Lemma S-5. LetC∈ ∪M−1 l=... | https://arxiv.org/abs/2503.00565v1 |
the second inequality is due to Lemma S-4, and the third inequality is due to the S21 choice of x0in (S-26). Applying Lemma S-4 again, ¯g(k2) C> g(k1)(x0) +c1|A|η−L0|A|η >{¯g(k1) C− |¯g(k1) C−g(k1)(x0)|}+c1|A|η−L0|A|η >¯g(k1) C+ (c1−2L0)|A|η >¯g(k1) C+3 2U(mC,i, T, C ), where for the last inequality we use (S-24). On t... | https://arxiv.org/abs/2503.00565v1 |
, X t), as for any t∈N,{τC(s)> t}={Pt n=11{Xn∈C}< s}and therefore {τC(s)> t}isFX t-measurable. First, we compute E[Y(k) τC(s)]. First note that 1 =P∞ t=s1{τC(s) =t}almost surely and {τC(s) =t} =[ (i1,...,is−1)⊆{1,...,t−1} (j1,...,jt−s)⊆{1,...,t−1}\(i1,...,is−1){Xi1∈C, . . . , X is−1∈C, X j1∈Cc, . . . , X jn−s∈Cc}\ {Xt∈... | https://arxiv.org/abs/2503.00565v1 |
nj−1, ij−ij−1−1)) ×P(Xnj∈C, Y(k) nj∈Bj) =mY j=1 ∞X nj=nj−1+(ij−ij−1)P(EC(nj−1, nj−1, ij−ij−1−1))P(X1∈C, Y(k) 1∈Bj) =mY j=1P(X1∈C, Y(k) 1∈Bj) P(X1∈C)(S-32) S26 where for the last equality we use the fact that for any j∈ {1, . . . , m }, ∞X nj=nj−1+(ij−ij−1)P(EC(nj−1, nj−1, ij−ij−1−1)) =∞X nj=nj−1+(ij−ij−1)nj−n... | https://arxiv.org/abs/2503.00565v1 |
versions and provides non-asymptotic bounds for estimating a matrix whose column space lies within the effective dimension reduction (e.d.r) space. Using this bound and the Davis- Kahan inequality, we will derive a non-asymptotic bound for the initial vector that satisfies Assumption 5. SADE algorithm We briefly descri... | https://arxiv.org/abs/2503.00565v1 |
n, with probability not less than 1 −δ: ˆV1,cov− V 1,cov ∗⩽d√ d 195τ2 η+ 2τ2 ℓ √nr log24d2 δ +8L2τ2 y+ 16τητyL√ d+ 157τ2 η+ 2τ2 ℓ d√ d nlog232d2n δ. (S-35) Non-asymptotic bound for the estimated initial vector Now, combining the non- asymptotic bound for V1,covand Davis-Kahan Theorem, we present the non-asymptotic ... | https://arxiv.org/abs/2503.00565v1 |
2018] described in Algorithm 3 for each arm and then using Algorithm 2 to construct the average index estimator. We consider varying level of model noise σand compare the performance of the proposed Algorithm 1 with the nonparametric analogue, i.e., the BaSEDB algorithm of Jiang and Ma [2024]. The average regret over 2... | https://arxiv.org/abs/2503.00565v1 |
2 (Uniform): Unknown pilot with model se: σ t[1] == 2154 t[2] == 7231 t[3] == 28360 t[4] == 71402 t[5] == 100000t[1] == 2154 t[2] == 20360 t[3] == 45711 t[4] == 72899 t[5] == 100000 0.050.060.070.080.09 0 25000 50000 75000 100000 tAverage regret sin(θ) 0.000.250.500.751.00 Method BaSEDB BIDSSetting 2 (Uniform): Known p... | https://arxiv.org/abs/2503.00565v1 |
tRice Classification t[1] == 394 t[2] == 668 t[3] == 1429 t[4] == 2685 t[5] == 3809t[1] == 394 t[2] == 1262 t[3] == 2228 t[4] == 3205 t[5] == 3809 0.20.30.40.5 1000 2000 3000 4000 tRice Classificationt[1] == 90 t[2] == 588 t[3] == 1874 t[4] == 3731 t[5] == 5686 t[6] == 8143t[1] == 90 t[2] == 1019 t[3] == 2398 t[4] == 4... | https://arxiv.org/abs/2503.00565v1 |
eeg recordings: speed-up gains using signal epochs and mutual information measure. In Proceedings of the 23rd International Database Applications & Engineering Symposium , pages 1–6, 2019. Onur Atan, Cem Tekin, and Mihaela Van der Schaar. Global multi-armed bandits with h¨ older continuity. In Artificial Intelligence a... | https://arxiv.org/abs/2503.00565v1 |
rates of consistency for k-nn regression. In Pro- ceedings of the AAAI Conference on Artificial Intelligence , volume 33, pages 3999–4006, 2019. Rong Jiang and Cong Ma. Batched nonparametric contextual bandits. arXiv preprint arXiv:2402.17732 , 2024. Tianyuan Jin, Jing Tang, Pan Xu, Keke Huang, Xiaokui Xiao, and Quanqu... | https://arxiv.org/abs/2503.00565v1 |
Eye State. UCI Machine Learning Repository, 2013. Oliver R¨ osler and David Suendermann. A first step towards eye state prediction using eeg. Proc. of the AIHLS , 1:1–4, 2013. Cong Shen, Ruida Zhou, Cem Tekin, and Mihaela van der Schaar. Generalized global bandit and its application in cellular coverage optimization. I... | https://arxiv.org/abs/2503.00565v1 |
Asymptotic Theory of Eigenvectors for Latent Embeddings with Generalized Laplacian Matrices∗ Jianqing Fan1, Yingying Fan2, Jinchi Lv2, Fan Yang3, and Diwen Yu3 Princeton University1, University of Southern California2, and Tsinghua University3 March 1, 2025 Abstract Laplacian matrices are commonly employed in many real... | https://arxiv.org/abs/2503.00640v1 |
manifold representations, often much lower than the ambient embedding dimensionality of each node. The Laplacian matrices for network data have been widely used to construct latent embeddings of graphs, where the nodes of the graph are represented in a latent sub- space spanned by the corresponding leading eigenvectors... | https://arxiv.org/abs/2503.00640v1 |
statistical inference for the DCMM model encounters additional complexities due to the presence of matrix Θ, whose entries can vary wildly in magnitude; see e.g., the related discussions in Fan et al. (2022b, 2024); Bhattacharya et al. (2023). To deal with such an issue, notice that under certain normalization, Λ:=E[L]... | https://arxiv.org/abs/2503.00640v1 |
are above a certain threshold. In particular, we will derive both the law of large numbers (LLN) and central limit theorems (CLTs) for the spiked sample eigenval- ues and eigenvector components. Our results extend significantly the previous works Fan et al. (2022a, 2024) to the context of the generalized Laplacian matr... | https://arxiv.org/abs/2503.00640v1 |
as well as components and general projections of spiked eigenvectors of the generalized Laplacian matrix using these local laws. The desired LLN and CLTs are the consequence of these new expansions. In establishing such local laws and deriving the limiting distributions, one of the most significant challenges is to han... | https://arxiv.org/abs/2503.00640v1 |
sample correlation matrix model. The rest of the paper is organized as follows. Section 2 introduces the model setting. We suggest the new framework of the asymptotic theory of eigenvectors for latent embeddings with generalized Laplacian matrices (ATE-GL) and present the main results in Section 3. Section 4 details th... | https://arxiv.org/abs/2503.00640v1 |
a signal-plus-noise decomposition eX=H+W, (6) where H=EeX= (Hij)1≤i,j≤nis a symmetric deterministic signal matrix and W= (Wij)1≤i,j≤nis a symmetric random noise matrix with centered and independent upper tri- angular entries. Further, assume that the signal part His of low rank K≥1. In particular, we allow Ktodiverge s... | https://arxiv.org/abs/2503.00640v1 |
∼stands for the asymptotic order. Then the means and variances for the entries of eXare typically of order θ, which leads to the assumptions in (13) and (14). For such setting, the last bound in (13) follows from the second bound in (13) and the fact that when |Wij| ≤C, E|Wij|p≤Cp−2E|Wij|2≤Cp−1θ (15) 8 for each p≥3. Ob... | https://arxiv.org/abs/2503.00640v1 |
≥ ··· ≥ | bδn|and|δ1| ≥ ··· ≥ | δK|>0, and bvi’s and vi’s are the corresponding eigenvectors. Given the empirical and population eigen-decompositions in (20) above, let us define the diagonal matrices of spiked eigenvalues b∆:= diag( bδ1,···,bδK) and ∆:= diag( δ1,···, δK), (21) as well as the corresponding spiked eigen... | https://arxiv.org/abs/2503.00640v1 |
(iii) (Eigengap) There exists some constant ϵ0>0such that min 1≤k≤K0|δk| |δk+1|>1 +ϵ0, (24) where we do not require eigengaps for smaller eigenvalues |δk|withK0+ 1≤k≤K. (iv) (Low-rankness of signals) The rank KofHsatisfies that Kξq1−4α |δK0|β1+2αn+ξ qβ2n+∥V∥max ≪q (25) withVgiven in (22). The lower bound on qin Condi... | https://arxiv.org/abs/2503.00640v1 |
−k+VT −keΥ(x)V−kVT −keΥ(x)vk= 0 (29) over x∈eIk, where eΥ(·) is defined in (27). Using similar arguments as in the proof of Lemma 3 in Fan et al. (2022a) and Section A.2 of Fan et al. (2024), we can establish the 12 following lemma, which provides the existence, uniqueness, and asymptotic properties of the population q... | https://arxiv.org/abs/2503.00640v1 |
1 above reveals that the population quantity tkis indeed the first-order asymp- totic limit of the empirical spiked eigenvalue bδk. In view of (26), we have that for x∈eIk, fMi(x) =−x−1+O(|x|−3) with 1 ≤i≤n; see Lemma 5 in Section B.1 of the Supplemen- tary Material for more details. Combining this fact with (29), we s... | https://arxiv.org/abs/2503.00640v1 |
more precise estimate for bvk(i) in the theorem below, which will allow us to derive the central limit theorem as n→ ∞ . 15 Theorem 3. Assume that Definition 1 and Assumption 1 are satisfied, and Keψn(δk)βn≲1,∥V∥max≪q1−4α |δk|β1+2αn+ξ qβ2n(44) for each 1≤k≤K0. Then for each 1≤k≤K0and1≤i≤n, it holds w.h.p. that bvk(i) =... | https://arxiv.org/abs/2503.00640v1 |
we omit the details for the latter here for simplicity. Corollary 1. Under the conditions of Theorem 3, if ∥vk∥∞→0and ∥V∥max √ Kq1−4α |δk|β2αn+Kξ q!q1−4α |δk|β1+2αn+ξ qβ2n +ξq1−4α √n|δk|β2αnq1−4α |δk|β2αn+ξ qβn ≪βα nσk,i,(49) we have (Lα ibvk(i)−Λα ivk(i))/σk,id−→ N (0,1)asn→ ∞ for each 1≤k≤K0. In particular, (49) ... | https://arxiv.org/abs/2503.00640v1 |
n sij−τi nδj isiivk(i)vk(j) Λ1+α iΛα j,(58) where Σ a:=P i,j∈[n]sijand tr(Σ) =P i∈[n]sii. In practice, Akcan be estimated as bAk, by replacing all parameters by their counterparts, see the bias correction idea and (73) at the end of this section for more details. We now examine the higher-order asymptotic expansions ... | https://arxiv.org/abs/2503.00640v1 |
in (A.154) and(A.155) , respectively (see Section C.9 of the Sup- plementary Material), it holds that vT k(L/Λ)−αbvk−vT k(L/Λ)−αvk−Evk,k svk,kd−→ N (0,1) (65) asn→ ∞ , where we choose the direction of bvksuch that bvT kvk>0and Evk,k:=α2 2EvT kL−Λ Λ2 vk−1 2t2 kEvT kW2vk+ (δ2 kvT keΥ′ k(tk)vk)−1/2 −1 +1 2vT k(t2 keΥ′(t... | https://arxiv.org/abs/2503.00640v1 |
by bvk, and tkcan be estimated by bδk. The estimation of sij=E|Wij|2 is provided by the bias correction idea from Fan et al. (2022b), as we discuss below. A naive estimator of sijiscW2 0,ij, with cW0= (cW0,ij) :=eX−LαP k∈[bK]bδkbvkbvT k Lα andbKgiven by (69). However, this estimator is not accurate enough in practice... | https://arxiv.org/abs/2503.00640v1 |
Green functions) of relevant random matrices defined as G(z) := W−z(L/Λ)2α−1andR(z) := W−zI−1(74) where W:=Λ−αWΛ−αandz∈C. We next focus on the equation governing the behavior 24 of the empirical spiked eigenvalue bδk, observing that det(X−bδkI) = 0 ⇐⇒ det(Λ−αeXΛ−α−bδk(L/Λ)2α) = 0 ⇐⇒ det(G−1(bδk) +V∆VT) = 0 ⇐⇒ det(∆... | https://arxiv.org/abs/2503.00640v1 |
Theorem 13 (see Section B.1 of the Supplementary Material). Then we can derive the local laws of Gfrom those of Rby controlling the difference G−R. However, the presence of correlations between random matrices eXandLposes a significant challenge in extending the local laws of R(z) to those of G(z), particularly for The... | https://arxiv.org/abs/2503.00640v1 |
the projection of the empirical spiked eigenvector in different regimes, and our higher-order asymptotic expansion confirms and quantifies this phenomenon. We also want to highlight an interesting observation regarding the results of the empirical spiked eigenvectors. Instead of directly considering eigenvector bvkof r... | https://arxiv.org/abs/2503.00640v1 |
should not exhibit any low-rank structure and be close to a centered Wigner matrix with independent entries modulo the symmetry. Then by subsampling the residual matrix entries, a test statistic can be constructed as the summation of the subsampled entries with self-normalization. They proved that under H0, the test st... | https://arxiv.org/abs/2503.00640v1 |
This observation can provide a concrete expression for the CI. Then, nodes with∥ri∥2 2falling into the CI can be the candidate estimates for pure nodes. 5.3 Confidence intervals for network parameters Recently, Ke and Wang (2024) and Jiang and Fan (2024) proposed methods that can achieve the optimal estimation of vario... | https://arxiv.org/abs/2503.00640v1 |
j) :i̸=j,1≤i, j≤n}sharing similar membership profiles can be constructed based on the eigenvectors of the Laplacian matrix X. 6 Simulation study In this section, we conduct a simulation study to verify the asymptotic distributions of the empirical spiked eigenvalues and spiked eigenvectors for the generalized Laplacian... | https://arxiv.org/abs/2503.00640v1 |
≤k≤3, respectively, where each distribution curve is centered by the corresponding asymptotic limit tkgiven in Lemma 1. It can be seen from Figures 1–3 that the distributions of the empirical spiked eigenvalues bδk’s corrected by Ak’s are indeed close to the target asymptotic distributions established in Corollary 2. I... | https://arxiv.org/abs/2503.00640v1 |
1DensityFigure 1: The kernel density estimate (KDE) for the distribution of the empirical spiked eigenvalue bδkcorrected by Akfor the generalized Laplacian matrix Xwith k= 1 across different values of αbased on 500 replications for simulation example in Section 6 with θ= 0.9. The generalized (regularized) Laplacian mat... | https://arxiv.org/abs/2503.00640v1 |
blue curves represent the KDEs for the rescaled empirical spiked eigenvector component, whereas the red curves stand for the target normal density. Both curves are centered with the asymptotic limit vk(i). 36 −0.006 −0.002 0.002 0.006050150250350alpha = 0.25, k = 2Density −0.006 −0.002 0.002 0.0060100200300alpha = 0.5,... | https://arxiv.org/abs/2503.00640v1 |
di+τ¯d+λ:i∈[n] without the rescaling population parameters q andβn. α θ Empirical Eigenvalue Asymptotic Eigenvalue Empirical SD Asymptotic SD 0.25 0.1 4.1352 4.1090 0.0230 0.0233 0.5 8.7895 8.7882 0.0202 0.0201 0.9 11.7246 11.7248 0.0160 0.0162 0.5 0.1 0.4450 0.4427 0.0024 0.0025 0.5 0.4238 0.4237 0.0010 0.0010 0.9 0.... | https://arxiv.org/abs/2503.00640v1 |
-0.01166 0.00052 0.00126 0.9 -0.01169 -0.01166 0.00088 0.00089 Table 6: The means and standard deviations (SDs) of the empirical spiked eigenvector component bvk(i) (rescaled by Lα i/Λα i) for the generalized Laplacian matrix Xwith k= 3 andi= 1 as well as their asymptotic counterparts across different settings of ( α, ... | https://arxiv.org/abs/2503.00640v1 |
Number 118. Cambridge University Press. Arias-Castro, E. and N. Verzelen (2014). Community detection in dense random networks. The Annals of Statistics 42 (3), 940–969. Bai, Z. and J. W. Silverstein (2006). Spectral Analysis of Large Dimensional Random Ma- trices , Volume 20. Springer. 42 Baik, J., G. Ben Arous, and S.... | https://arxiv.org/abs/2503.00640v1 |
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