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effects. Journal of the American Statistical Association , (just-accepted):1–25, 2025. [26] LarryHan,YigeLi,BijanNiknam,andJoséRZubizarreta. Privacy-preserving,communication-efficient, and target-flexible hospital quality measurement. The Annals of Applied Statistics , 18(2):1337–1359, 2024. [27] Larry Han, Zhu Shen, a... | https://arxiv.org/abs/2501.18798v2 |
Ronghui Xu. Doubly robust estimation under covariate-induced de- pendent left truncation. Biometrika , page asae005, 2024. [48] Ted Westling, Alex Luedtke, Peter B Gilbert, and Marco Carone. Inference for treatment-specific survival curves using machine learning. Journal of the American Statistical Association , pages ... | https://arxiv.org/abs/2501.18798v2 |
sites. We let γ(k) = DT(k) =DC(k) =δT(k) =δC(k) = 0fork= 0,1, . . . , 4. •Covariate Shift : Covariates X1,X2, and X3vary across sites. We let γ(k) =kandDT(k) = DC(k) =δT(k) =δC(k) = 0, fork= 0,1, . . . , 4. •Outcome Shift : Conditional outcome distribution varies across sites. We assign γ(k) = 0, DT(k) =δT(k) =k, and D... | https://arxiv.org/abs/2501.18798v2 |
0.110.140.170.110.140.170.140.170.20.060.080.090.060.080.090.070.090.1 0.10.130.150.10.130.150.110.160.180.050.070.080.050.070.080.050.070.090.140.170.20.140.170.190.140.170.20.070.080.090.070.080.090.080.10.12 0.110.160.180.110.160.180.110.160.180.050.070.080.060.070.080.060.090.10.120.140.160.120.140.160.140.170.20.0... | https://arxiv.org/abs/2501.18798v2 |
(n= 340 )OA (n= 297 )BP (n= 428 )US (n= 470 ) Age (year) at baseline 25.9 (4.72) 26.6 (5.28) 25.4 (4.78) 25.2 (3.94) 26.1 (3.79) Standardized risk score 0.0 (1.00) 0.02 (0.92) -0.02 (0.98) 0.75 (0.67) -0.68 (0.73) Weight at baseline (kg) 72.5 (16.35) 67.6 (14.77) 65.1 (13.64) 71.1 (12.84) 81.8 (17.5) HIV diagnosis by w... | https://arxiv.org/abs/2501.18798v2 |
clearer visualization of temporal trends in this specific example. From Figure 11, we observe that for each region, the FED and FED (BOOT) methods yield results similar to the TGT estimator, while also recovering some interval estimations at earlier time points. This finding is consistent with the observations made in ... | https://arxiv.org/abs/2501.18798v2 |
Appendix D.3 by replacing the fixed ηt,awith the estimated value bηt,a. 31 Algorithm 2 Double/debiased machine learning (DML) algorithm for nuisance function estima- tions and influence function calculations in Algorithm 1 at a given time point and treatment. 1:Input: Observed multi-source right-censored data O={Oi= (X... | https://arxiv.org/abs/2501.18798v2 |
function under submodel Pϵ[4]. Thus, to find the EIF, we begin by writing the equation 0 =∂ ∂ϵθ0(t, a;Pϵ) ϵ=0=∂ ∂ϵEPϵ{¯Sϵ(t|a,X)|R= 0} ϵ=0 =E{[¯S(t|a,X)−θ0(t, a)]˙ℓX|R=0|R= 0}+EZ∂ ∂ϵ¯Sϵ(t|a,x) ϵ=0µ(dx) R= 0 ,(3) where µ(·)denotes the distribution of Xinduced by Pand, for any sets of variables VandW,˙ℓV|W denotes the ... | https://arxiv.org/abs/2501.18798v2 |
and ¯G∞(t|a,x)≥1/ηwith probability tending to 1. Condition D.3. Define ¯rn,t,a, 1= max mP|{b¯πm(a|X)−¯π(a|X)} · {b¯Sm(t|a,X)−¯S(t|a,X)}|, ¯rn,t,a, 2= max mP|{bq0 m(X)−q0(X)} · {b¯Sm(t|a,X)−¯S(t|a,X)}|,and ¯rn,t,a, 3= max mP b¯Sm(t|a,X)Zt 0(¯G(u|a,X) b¯Gm(u|a,X)−1) ¯S b¯Sm−1! (du|a,X) . Then, it holds that all ¯rn,t,a, ... | https://arxiv.org/abs/2501.18798v2 |
balls need not be in F. An (ε,∥ · ∥)bracket is a set of the form {f∈ F :l(x)≤f(x)≤ u(x)for all x∈ X}such that ∥u−l∥ ≤εand is denoted [l, u]. Here landuneed not be elements of F. The bracketing number N[](ε,F,∥·∥)is then defined as the minimal number of (ε,∥·∥)brackets needed to cover F. It is well known that N(ε,F,∥ · ... | https://arxiv.org/abs/2501.18798v2 |
have L2(Q)covering number bounded by 2/ε2for any probability measure Q. Therefore, supQN(εη3,Mt, L2(Q))≤4/ε4. By Jensen’s inequality, ∥ht− hs∥L2(Q)≤ ∥mt−ms∥L2(µ∗×Q)for any Q, which implies that supQN(εη3,F3 S,p0,q0,G,t,a, L2(Q))≤ supQN(εη3,Mt, L2(µ∗×Q))≤4/ε4for all Q. Therefore, we have shown that the three classes hav... | https://arxiv.org/abs/2501.18798v2 |
2.1.14 in Van der Vaart and Wellner [46], there is therefore a constant ¯Cnot not depending on normsuch that E" sup g∈Gn,m,t,a|Gm ng|# ≤¯CP" sup u∈[0,t]{φCCOD n,m,u,a (O)−φCCOD ∞,u,a(O)}2#2 ≤¯CC(η)6X j=1¯Aj,n,m,t,a , where the second inequality follows notation and results of Lemma D.1. We therefore have that 1 MMX m=1... | https://arxiv.org/abs/2501.18798v2 |
Gm n bφCCOD t,a−φCCOD t,a =op(n−1/2). Therefore, we have that sup u∈[0,t] 1 MMX m=1Mnm nPm n(bφCCOD n,m,t,a −θ0(t, a)) ≤sup u∈[0,t]2η2{¯rn,u,a, 1+ ¯rn,u,a, 2+ ¯rn,u,a, 3}, which is op(n−1/2)by Condition D.5. Thus, supu∈[0,t] bθCCOD n (u, a)−θ0(u, a)−Pn(φ∗CCOD u,a ) = op(n−1/2). Since {φ∗CCOD u,a :u∈[0, t]}is a unifor... | https://arxiv.org/abs/2501.18798v2 |
= 1) Sk(Y− |A,X) Sk(Y|A,X)Dk(Y|A,X) A=a,X=x, R=k =Zt 0Sk(y− |a,x)Nk 1(dy|a,x) Sk(y|a,x)Dk(y|a,x), 50 and E{Hk(t∧Y, A, X)|A=a,X=x, R=k} =ZZt I(u≤y)Sk(u− |a,x)Nk 1(du|a,x) Sk(u|a,x)Dk(u|a,x)2P(dy|a,x, k) =Zt 0P(Y≥u|A=a,X=x, R=k)Sk(u− |a,x)Nk 1(du|a,x) Sk(u|a,x)Dk(u|a,x)2P(dy|a,x, k) =Zt 0Sk(u− |a,x)Nk 1(du|a,x) Sk(u|a,x... | https://arxiv.org/abs/2501.18798v2 |
D.7, we first introduce and prove several lemmata in the next section. 53 D.2.3 Lemmata for the source-site estimator To establish the RAL related results of bθk n(t, a), we start from considering the difference bθk n(t, a)− θ0(t, a). Recall that Pm nis the empirical distribution corresponding to the mth validation set... | https://arxiv.org/abs/2501.18798v2 |
the observed data. Proof.The class FS,π,ω,p0,pk,G,t 0,a0is uniformly bounded by η(1 + 2 η3)because of the assumed upper bounds of 1/p0,1/pk, ω,1/πand1/G. Therefore, the envelop function can be taken as F=η(1 + 2 η3). Define the following two functions ftandhtpointwise as ft(x, r, a 0, δ, y) =I(r=k, a=a0, y≤t, δ= 1)ω(x)... | https://arxiv.org/abs/2501.18798v2 |
∞−Λk)(dy|a,X) . (8) 58 D.2.4 Proof of Theorem D.7 Proof.By (7) with πk ∞=πk,ωk,0 ∞=ωk,0,Gk ∞=Gk, and Sk ∞=Sk, bθk,0 n(t, a)−θ0(t, a) =Pn[φ∗k,0 t,a] +1 MMX m=1Mn1/2 m nGm nh bφk,0 n,m,t,a −φk,0 t,ai +1 MMX m=1Mnm nPh bφk,0 t,a−θ0(t, a)i . By Conditions D.6 and D.7, the second summand on the right-hand-side is op(n−1/2)... | https://arxiv.org/abs/2501.18798v2 |
60 The space S∗ t,ais both time- and treatment-varying, indicating that a source site may not consistently be useful or unhelpful across different time points or treatments. However, it offers the advantageofincreasedflexibilityandadaptivity, allowingformoreeffectiveborrowingofinformation at different points along the ... | https://arxiv.org/abs/2501.18798v2 |
weights to the target and none to the source. In contrast, the estimator that leverages the proposed adaptive ensemble approach is denoted as bθfed n(t, a;bηt,a). Here bηt,acan recover the optimal weights ¯ηt,athat are associated with the minimum asymptotic variance. Consequently, the variance of bθfed n(t, a;bηt,a) is... | https://arxiv.org/abs/2501.18798v2 |
Transfer Learning for Nonparametric Contextual Dynamic Pricing Fan Wang1, Feiyu Jiang2, Zifeng Zhao3, and Yi Yu1 1Department of Statistics, University of Warwick 2School of Management, Fudan University 3Mendoza College of Business, University of Notre Dame February 3, 2025 Abstract Dynamic pricing strategies are crucia... | https://arxiv.org/abs/2501.18836v1 |
leverage information to achieve improved decision-making. For instance, historical data from other platforms or existing markets can help sellers optimize pricing strategies more efficiently when entering new environments. This naturally falls in the territory of transfer learning (e.g. Pan and Yang, 2009), where datas... | https://arxiv.org/abs/2501.18836v1 |
with respect to the ℓ∞- norm. Let B(s, r),BX(s, r) and BP(s, r) denote the balls in Z,XandP, respectively, centred atswith radius r. For any ball B, letr(B) denote its radius and c(B) its centre. For n∈Z+, let [n] ={1, . . . , n }. For any distribution P, let supp( P) be its support. The remainder of the paper is organ... | https://arxiv.org/abs/2501.18836v1 |
function led by the changes in covariates and prices. This is commonly used in the literature of online learning with nonparametric reward function (e.g. Slivkins, 2011; Chen and Gallego, 2021; Chen et al., 2023), facilitating theoretical analysis of the estimation and learning processes. Assumption 2 below is a regula... | https://arxiv.org/abs/2501.18836v1 |
detailed in Algorithm 1. TLDP borrows the idea of contextual zooming from Slivkins (2011) proposed for MAB problems with only target data, and addresses the unique challenge of leveraging source data to refine the exploration of the covariate-price space in the target domain. At a high level, TLDP is an upper confidenc... | https://arxiv.org/abs/2501.18836v1 |
Ipre t(B′) +CI∥c(B)−c(B′)∥∞ , (8) where CI>0 is a constant and the pre-index Ipre t(B) is given by Ipre t(B) =vt(B) +CIr(B) +conf t(B),with vt(B) =ret(B) nt(B). (9) 6 The index It(B) is adapted from its MAB counterpart proposed in Slivkins (2011) to balance the estimated reward vt(B), the radius of the ball r(B), the u... | https://arxiv.org/abs/2501.18836v1 |
global exploration coefficient in Definition 2, one can instead define a scale-dependent exploration coefficient κr∈[0,1]for any radius r∈(0,1/2]as κr= inf x∈supp( PX), r′∈[r,1/2], p∈[r′,1−r′]µ [p−r′, p+r′]×BX(x, r′) 2r′·PX(BX(x, r′)). Note that κris non-decreasing in rand satisfies that κ= inf r∈(0,1/2]κr. It quanti... | https://arxiv.org/abs/2501.18836v1 |
(e.g. Luo et al., 2022; Xu and Wang, 2022; Fan et al., 2024) or additional shape constraints beyond Lipschitz continuity (e.g. Chen and Gallego, 2020), with detailed comparisons in Appendix A. We conclude with a minimax lower bound for the scenario where only target data is available. Corollary 3. LetI(CLip, cQ)denote ... | https://arxiv.org/abs/2501.18836v1 |
exploration radius ˜ ras an input and the constant CI to construct the index defined in (8). Specifically, we let CI= 1 and ˜ rbe defined in (12) with Cr= 1/4. To show the sensitivity of the choice of the constants, additional simulation studies are conducted for varying CIandCr. For the ABE algorithm, we set M= 0.1, t... | https://arxiv.org/abs/2501.18836v1 |
4 ,3 4,3 4,3 4,3 4o . 11 InScenario 1 , the reward function is linear in covariates and quadratic in price, while in Scenario 2 , it is fully nonparametric. The simulation results for Configuration 1 ofScenario 1 are presented in Figure 1 with additional results provided in Appendix D. Some key observations are high... | https://arxiv.org/abs/2501.18836v1 |
reward metric is defined as the product of the consumer’s decision (whether the loan is accepted) and the computed price, capturing the total revenue generated for the lender. For this analysis, five covariates identified as significant in prior studies (e.g. Luo et al., 2024; Zhao et al., 2024) are included: the loan ... | https://arxiv.org/abs/2501.18836v1 |
leverages source data information to guide pricing for the target data. We show that TLDP achieves optimal regret by establishing a matching minimax lower bound. Our work offers several interesting directions for future research. First, the optimality of TLDP 13 depends on prior knowledge of the transfer exponent γand ... | https://arxiv.org/abs/2501.18836v1 |
Matthew Wiener. Classification and regression by randomforest. R News , 2(3): 18–22, 2002. URL https://CRAN.R-project.org/doc/Rnews/ . Yiyun Luo, Will Wei Sun, and Yufeng Liu. Contextual dynamic pricing with unknown noise: Explore-then-ucb strategy and improved regrets. Advances in Neural Information Processing Systems... | https://arxiv.org/abs/2501.18836v1 |
with those from relevant studies on dynamic pricing and contextual bandits that do not incorporate transfer learning: •Slivkins (2011) studied contextual bandits with similarity information, where both contexts and actions are embedded in a metric space equipped with a distance function. The regret bounds in Slivkins (... | https://arxiv.org/abs/2501.18836v1 |
B, It(Bsel t)≥It(B) 17 =CIr(B) + min B′∈At Ipre t(B′) +CI∥c(B)−c(B′)∥∞ ≥CIr(B) + min B′∈At f(B′) +CI∥c(B)−c(B′)∥∞ ≥CIr(B) +f(B)≥f∗(Xt), (15) where the first inequality follows from the selection rule in Algorithm 1 (Step 4 therein), the first equality follows from (8), the second inequality follows from (14), and the... | https://arxiv.org/abs/2501.18836v1 |
Bwas selected for the target data. By Lemma 6, we have that |Fr| ≤NPack r(Z)≤Nr/2(Z)≤2 rd+1 , (24) where NPack r(Z) and Nr(Z) denote r-packing number and r-covering number of Z, respectively, and the second inequality follows from the fact that NPack 2r(Z)≤Nr(Z). Then, note that (II) =EnQX t=1n f∗(Xt)−f(Xt, pt)o 1{E... | https://arxiv.org/abs/2501.18836v1 |
(Et 1)c ≤18n−1 Q˜r−3log (κnp)d+3 d+3+γ , (38) which implies that (I)≤nQX t=1P{(Et 1)c} ≤18˜r−3log (κnp)d+3 d+3+γ . (39) Step 3.2: Bound for (II). By similar arguments as (25), we have (II)≤C1nQ˜r+EX r:˜r<r≤1X B∈FrX t∈SQ(B)C1r 1{Et 1} . (40) When nQ<(κnP)d+3 d+3+γ, similar to (26), we have |SQ(B)| ≤TQ B≤ω(B)≤log (κ... | https://arxiv.org/abs/2501.18836v1 |
Then, we have that 8n−2 QTQ B+ 2{nQ∨(κnP)d+3 d+3+γ}−2≤8n−2 Qω(B) + 2n−2 Q≤8n−2 Qr(B)−2log(nQ) + 10 n−2 Q ≤8n−2 Q˜r−2log(nQ) + 10 n−2 Q ≤18n−2 Q˜r−2log(nQ), (49) where the first inequality follows from (10), the second inequality follows from (11), and the third inequality follows from r(B)≥˜r. Then combining (48) and (... | https://arxiv.org/abs/2501.18836v1 |
term in (56). Note that for any δ >0 and T′>0, by Azuma– Hoeffding inequality (e.g. Corollary 2.20 in Wainwright, 2019), we have that P ∃t∈[T′]: n+tX i=1Xi > δ ≤T′X t=12 exp −δ2 2(n+t) ≤2T′exp −δ2 2(n+T′) . (57) Note that P ∃t∈[T]: n+tX i=1Xi >p 2(n+t) log(1 /δ1) ≤⌊log2(T)⌋X j=0P ∃2j≤t≤2j+1: n+tX i=1Xi >p 2(n+... | https://arxiv.org/abs/2501.18836v1 |
1−CEκnP˜rd+γ+1 28 ≤CE 8 2 d+3+γ (κnP)2 d+3+γlogd+1+γ d+3+γ (κnp)d+3 d+3+γ + 1 −8d+1+γ d+3+γC2 d+3+γ E(κnP)2 d+3+γlogd+1+γ d+3+γ (κnp)d+3 d+3+γ ≤1−21/2C2 d+3+γ E(κnP)2 d+3+γlogd+1+γ d+3+γ (κnp)d+3 d+3+γ <0, where the first inequality follows from (11) and the event EB, the second inequality follows from r(B)≥˜rand ... | https://arxiv.org/abs/2501.18836v1 |
constructed as follow. First, define the function ϕ:R+7→[0,1] by ϕ(z) = 1, if 0≤z <1/4, 2−4z,if 1/4≤z <1/2, 0, otherwise . Next, define the function φ:Z 7→ [0,1/4] via φ(x, p) =Cφ˜rϕ ∥(x⊤, p)⊤∥∞/˜r , where Cφ= (CLip∧1)/4. For any ω∈Ωm, the reward function fω:Z 7→ [0,1] is defined as follows fω(x, p) = 1 /2 +mX ... | https://arxiv.org/abs/2501.18836v1 |
where the first inequality follows from (69) and the final equality holds beacuse κ−1−˜rκ 1−˜r=κ−1 1−˜r≤0. As a result, the exploration coefficient defined in Definition 2 for the constructed source covariate- price pair distribution equals κ. Lipschitz condition. By the construction in (70), it suffices to show that t... | https://arxiv.org/abs/2501.18836v1 |
π, for any u∈[⌊1/˜r⌋], let Ou=nQX t=11{pπ t∈[p∗ u−˜r/2, p∗ u+ ˜r/2]}. (77) Since MX i=1⌊1/˜r⌋X u=1Eω(i)(Ou)≤MnQ, it follows that for at least ⌊M/2⌋elements in {ω(i)}M i=1, there must exist at least one index u∈[⌊1/˜r⌋] such that Eω(i)(Ou)≤2nQ ⌊1/˜r⌋. (78) Therefore, we can choose one such uand without loss of generalit... | https://arxiv.org/abs/2501.18836v1 |
pP t θfω(0) Yi x, pP t)# pX(x)dx ≤κnP˜rX j∈[m]:ω(i) j̸=ω(0) jPX BX(x∗ j,˜r/4) KL Bernoulli(1 /2 +Cφ˜r),Bernoulli(1 /2−Cφ˜r) ≤32C2 φnPκ˜r3X j∈[m]:ω(i) j̸=ω(0) jPX BX(x∗ i,˜r/4) ≤32C2 φcQcγnPκm˜rd+γ+3=CPκnP˜rd+γ+3m, (82) where 35 •the second equality follows from (68), (72) and (73), •the first inequality follow... | https://arxiv.org/abs/2501.18836v1 |
Results for Configuration 1 in Scenario 2. Panel (A) and (B): varying source data size nP and target data size nQ, respectively. Panel (C) varying the transfer exponent γ(top axis) and the exploration coefficient κ(bottom axis). Panel (D): varying the index constant CI(top axis) and the exploration radius constant Cr(b... | https://arxiv.org/abs/2501.18836v1 |
Understanding Generalization in Physics Informed Models through Affine Variety Dimensions Takeshi Koshizuka1and Issei Sato1 1Department of Computer Science, The University of Tokyo 1{koshizuka-takeshi938444, sato }@g.ecc.u-tokyo.ac.jp ABSTRACT In recent years, physics-informed machine learning has gained significant at... | https://arxiv.org/abs/2501.18879v1 |
affine variety. In Section 4, we discuss the dimension of affine variety especially in the context of nonlinear operators and introduce methods for their approximate calculation. Section 5 provides experimental evidence supporting our theoretical claims, demonstrating the practical advantages of incorporating physical ... | https://arxiv.org/abs/2501.18879v1 |
converges at a rate surpassing the Sobolev minimax rate. Doum `eche et al. (2024a) quantified the generalization capacity of the physics-informed estimator for general linear PDEs using the concept of effective dimension (Caponnetto & De Vito, 2007), a well-known metric in kernel method analysis. The effects of incorpo... | https://arxiv.org/abs/2501.18879v1 |
connected components K, the more complex the topology of V. Moreover, the dimension at which we slice the variety is also important. If the slice (affine plane) is large enough in dimension, i.e., the codimension is small ( < dV), then any intersection of the slice with Vis limited to at most Kconnected pieces. Otherwi... | https://arxiv.org/abs/2501.18879v1 |
The problem Eq. (2) is reduced to the physics-informed linear regression (PILR) given by ˆβ= arg min β∈V(D,B,T)1 n∥y−Φβ∥2 2+λn∥β∥2 2, (3) V(D,B,T):= β:⟨D β⊤ϕ , ψk⟩µk= 0,∀(ψk, µk)∈ T, ϕj∈ B ,(4) where y= [y1, y2, . . . , y n]⊤∈Rnis the target vector, Φ= [ϕ(x1),ϕ(x2), . . . ,ϕ(xn)]⊤∈Rn×dis the design matrix, and ∥ · ... | https://arxiv.org/abs/2501.18879v1 |
(informal) .LetV(D,B,T)be the (K, dV)-regular affine variety defined in Eq. (4). Suppose that the basis function is bounded by a constant, the minimum eigenvalue of the design matrix is restricted, and the stability condition for the estimator holds. For δ∈(0,1), with probability 1−δ, the minimax risk for PILR defined ... | https://arxiv.org/abs/2501.18879v1 |
the rank–nullity theorem, dV=d−rankD, indicating that the higher the rank of the matrix D, the better the minimax risk of regression. We show that our theory is consistent with existing theories. The effect of incorporating physical structure, represented by linear differential equations, on generalization has been ana... | https://arxiv.org/abs/2501.18879v1 |
chains V0⊂V1⊂. . .⊂Vdof distinct nonempty subvarieties of V. Definition 4.2. The degree of the denominator of the Hilbert series of the affine variety V. Definition 4.3. The maximal dimension of the tangent vector spaces at the non-singular points U⊆Vof the variety. dV= max β∈U d−rank ∇⊤p1(β) ... ∇⊤pK(β) ,... | https://arxiv.org/abs/2501.18879v1 |
and project the obtained solutions onto the basis Bto sample β∗∈ V. For the linear operator, the dimension does not depend on the weight β, and the rank of the matrix Ddiscussed in Section 3.4 precisely gives the dimension dV. 5 E XPERIMENTS We compared the performance of ridge regression (RR) and physics-informed line... | https://arxiv.org/abs/2501.18879v1 |
diffusion equation D[u] = 0 with diffusion coefficient αand periodic boundary conditions is given by: D[u] =∂ ∂tu−α∂2 ∂x2u (x, t)∈[−Ξ,Ξ]×[0, T] u(x,0) = u0(x) x∈[−Ξ,Ξ] u(−Ξ, t) =u(Ξ, t),∂u ∂x(−Ξ, t) =∂u ∂x(Ξ, t). The solution to the problem is analytically given by: u(x, t) =jmaxX j=0[Ajcos (ωjx) +Bjsin (ωjx)]e−αω2 jt,... | https://arxiv.org/abs/2501.18879v1 |
correspond to the following one-dimensional piecewise constant functions of size ntfor the Euler method, i.e., ϕτ(t) = 1 fort∈[tτ, tτ+1)and0otherwise . The test functions are constant functions that output 1, and the measure used is the Dirac measure δtτcentered at the collocation points. The results are shown in Table... | https://arxiv.org/abs/2501.18879v1 |
the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. REFERENCES Tara Akhound-Sadegh, Laurence Perreault-Levasseur, Johannes Brandstetter, Max Welling, and Siamak Ravanbakhsh. Lie point symmetry and physics-informed networks.... | https://arxiv.org/abs/2501.18879v1 |
Kawaguchi. When do extended physics- informed neural networks (xpinns) improve generalization? SIAM Journal on Scientific Computing , 44(5): A3158–A3182, 2022. Ameya D Jagtap, Ehsan Kharazmi, and George Em Karniadakis. Conservative physics-informed neural networks on discrete domains for conservation laws: Applications... | https://arxiv.org/abs/2501.18879v1 |
, 375:113603, 2021. Yifan Zhang and Joe Kileel. Covering number of real algebraic varieties and beyond: Improved bounds and applications. arXiv e-prints , pp. arXiv–2311, 2023. 13 A M ATHEMATICAL BACKGROUND ON AFFINE VARIETIES In this section, we provide a formal definition of several concepts related to affine varieti... | https://arxiv.org/abs/2501.18879v1 |
of linear subspaces within V, which corresponds to the dimension of Vas a linear space. When we focus on the local structure, the following equivalent definition is obtained: Definition 4.3. The maximal dimension of the tangent vector spaces at the non-singular points U⊆Vof the variety. dV= max β∈U d−rank ∇⊤p1(β)... | https://arxiv.org/abs/2501.18879v1 |
theorem, the Krull dimension of the projective variety Pmatches the order of the Hilbert series at the pole t= 1, which is one of the most important results in commutative algebra. Therefore, the dimension of the affine variety is defined using the Hilbert series, as follows: Definition 4.2. The degree of the denominat... | https://arxiv.org/abs/2501.18879v1 |
=Eh ϵ⊤Φˆβi ≤q E[∥ϵ⊤Φ∥2 2]s E ˆβ 2 2 ≤σvuutdX j=1nX i=1|ϕj(xi)|2R =σ√nMR(16) Here, the third inequality follows from the Cauchy-Schwarz inequality, and the fourth inequality is derived from the fact that |ϵ⊤Φj|2/(σ∥Φj∥2)2follows a chi-squared distribution with 1degrees of freedom and ˆβ∈ VR. By combining Eq. (14), Eq.... | https://arxiv.org/abs/2501.18879v1 |
N(ξ, ν)≲X λ∈σ(CM−1C)1 1 +λ−1≤X λ∈σ(M−1)1 1 +λ−1, (20) where M:=ξI+νD⊤GD∈R|B|×|B|andC∈R|B|×|B|is the matrix of the inner products of the basis functions, i.e., Cj,j′=⟨ϕj, ϕj′⟩µfor all ϕj, ϕj′∈ B. Since the matrix D⊤GD is positive semi-definite, the eigenvalues of the matrix Min ascending order σj(·) are given by σj(M) =... | https://arxiv.org/abs/2501.18879v1 |
dt= 2, dx∈ {10,15,20,25}are the sets of the number of basis functions, and (xk, tk)∈Ωis uniformly sampled from data with K= 50×500. D.2 E XPERIMENTS ON NUMERICAL SOLUTION In the experiments in Section 5.2, we numerically simulate the Bernoulli equation using the explicit Euler method and the diffusion equation using th... | https://arxiv.org/abs/2501.18879v1 |
PROPORTIONAL ASYMPTOTICS OF PIECEWISE EXPONENTIAL PROPORTIONAL HAZARDS MODELS A P REPRINT Emanuele Massa Physics of Machine Learning and Complex Systems Radboud Universiteit Nijmegen, The Netherlands emanuele.massa@donders.ru.nl February 3, 2025 ABSTRACT We study the flexible piecewise exponential model in a high dimen... | https://arxiv.org/abs/2501.18995v1 |
is associated with the outcome, then even a properly tuned Lasso penalization cannot restore consistency (for all the parameters simultaneously). Hence, the standard theoretical analysis via the Lasso theory is not readily appli-arXiv:2501.18995v1 [math.ST] 31 Jan 2025 Proportional asymptotics of piecewise exponential ... | https://arxiv.org/abs/2501.18995v1 |
4 referring to the appendices for the technical details. We illustrate the agreement of theory and simulations in section 5 via numerical experiments, where we quantify the prediction error by means of the concordance-index and an oracle version of the integrated brier score. Concluding remarks are in section 6. 2 SETT... | https://arxiv.org/abs/2501.18995v1 |
ridge-like regularizer. To keep the setting simple we consider a simple uniform ridge regularization (although smoothness penalties based on finite differences might be easily considered in the present setting), i.e. the model is fitted to the data {(∆1, T1,X1), . . . , (∆n, Tn,Xn)}, generated according to (1) and (2),... | https://arxiv.org/abs/2501.18995v1 |
equations v2ζ=ET,Z0,Qh ˆξ−wZ0−vQ 2i (17) w(1 +ητ/ϕ ) = ET,Z0,Qh Z0ˆξi (18) v(1−ζ+ητ/ϕ ) = ET,Z0,Qh Qˆξi (19) τ=vp ζ (20) ωk=1 ηαEh ∆ψk(T)i −W01 ηαEh eˆξψk(T)i expn1 ηαEh ∆Ψk(T)io , k= 1, . . . , ℓ, (21) where W0is the (real branch of) Lambert W - function [40], defined as the (real) solution of the equation W0(x) exp... | https://arxiv.org/abs/2501.18995v1 |
“artificially” restrict the saddle point problem (26) onto compact, convex sets. Intuition suggests that if a saddle point exists and the set is sufficiently large, then there is not going to be any difference between the bounded and unbounded problem. Hence, from now on the min over β,ξand the max over ϕoperations are... | https://arxiv.org/abs/2501.18995v1 |
D. 5 NUMERICAL EXPERIMENTS In the following we simulate the model under study and compare various quantities, such as goodness of fit and prediction metrics, against the theory for different values of the regularizer ηwhich controls the amount of ridge shrinking on ˆβn. The data simulations are carried out as follows. ... | https://arxiv.org/abs/2501.18995v1 |
. (40) This quantity measures the integrated Mean Squared Error of the estimated survival function with respect to the true one. The “ideal” in underscore is due to the fact that the quantity above is what any estimator of the Integrated Brier Score (IBS) aims ideally to estimate. The ideal IBS (40) can only be compute... | https://arxiv.org/abs/2501.18995v1 |
required to rigorously establish more recent heuristic results for the semi-parametric Cox model [9]. The proof technique used here relies completely on the assumption of Gaussian covariates, since it hinges on the Convex Gaussian Min Max theorem [29]. What is intriguing is that the replica methods requires only that t... | https://arxiv.org/abs/2501.18995v1 |
Shengchun and B Nan. Non-asymptotic oracle inequalities for the high-dimensional cox regression via lasso. Statistica Sinica , 2014. [15] R Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological) , 58(1):267–288, 1996. [16] P Bühlmann and S van d... | https://arxiv.org/abs/2501.18995v1 |
data under the proportional hazards model. International Statistical Review / Revue Internationale de Statistique , 43(1):45–57, 1975. [36] FE Harrell. Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis . Graduate Texts in Mathematics. Springer, 2001. [37] H K... | https://arxiv.org/abs/2501.18995v1 |
that ∆ 1−log ∆ + log Λ( T|ω) ≤1 + log Λ( T|ω) ≤1 + loglX k=1exp(ωk)(τk+1−τk) <∞. (49) Since∥ω∥is bounded by a large constant by assumption. 11 Proportional asymptotics of piecewise exponential proportional hazards models A P REPRINT Proposition 2 (FINITE V ARIANCE) .The random function (ω, w, v, τ )→ M g(.,ω,∆,T)(wZ0... | https://arxiv.org/abs/2501.18995v1 |
The proposition follows from differentiation and definition of proximal mapping. Proposition 7 (DERIV ATIVES OF THE EXPECTED MOREAU ENVELOPE 1) . ∂ ∂wEh Mg(.,ω,∆,T)(wZ0+vQ, τ )i =1 τEh Z0 ∆τ−W0 τΛ(T|ω)e∆τ+xi (66) ∂ ∂vEh Mg(.,ω,∆,T)(wZ0+vQ, τ )i =1 τEh Q ∆τ−W0 τΛ(T|ω)e∆τ+xi (67) ∂ ∂τEh Mg(.,ω,∆,T)(wZ0+vQ, τ )i =... | https://arxiv.org/abs/2501.18995v1 |
POINT OF THE AO In this section we show that min ω,w,v≥0max 0≤ϕ≤λmaxmin τ>0Ln(ω, w, v, ϕ, τ )P− − − − − → n,p→∞min ω,w,v≥0max 0≤ϕ≤λmaxmin τ>0L(ω, w, v, ϕ, τ ) (84) The idea is to use the so-called convexity lemma , this guarantees that point-wise convergence of convex functions over compact sets implies uniform converg... | https://arxiv.org/abs/2501.18995v1 |
holds by definition that Lω,w,v(ϕ)≤lim τ→0+L(ω, w, v, ϕ, τ ) (92) where lim τ→0+L(ω, w, v, ϕ, τ ) = lim τ→0+Eh Mg(.,ω,∆,T) wZ0+vQ, τ/ϕ −∆ log λ(T|ω)i −ϕvp ζ+1 2α∥ω∥2= =Eh g wZ0+vQ,ω,∆, T −∆ log λ(T|ω)i −ϕvp ζ+1 2α∥ω∥2. (93) Ifv >0, then limϕ→∞Lω,w,v(ϕ) =−∞, since the expectation of gexists finite for fixed ω, w, v ... | https://arxiv.org/abs/2501.18995v1 |
convex functions. The same is true for L, furthermore, LnP− → L pointwise. Let us denote Sω,w,v=Sω×[0, Cβ]×[0, Cβ]. The function L(ω, w, v)is strongly convex in w, v and strictly convex in ω(it is sufficient to compute the second derivative, which is always non-negative on Sω). In particular, L(ω, w, v)is strongly conv... | https://arxiv.org/abs/2501.18995v1 |
ASYMPTOTIC OPTIMALITY THEORY OF CONFIDENCE INTERVALS OF THE MEAN Vikas Deep Kellogg School of Management Northwestern University Evanston, IL 60201 vikas.deep@kellogg.northwestern.eduAchal Bassamboo Kellogg School of Management Northwestern University Evanston, IL 60201 a-bassamboo@kellogg.northwestern.edu Sandeep June... | https://arxiv.org/abs/2501.19126v1 |
regime: Iflimδ→0Nδ log(1/δ)→0, the limiting width of the CI is the length of the support of the mean for any stable CI construction policies. This implies that no learning is possible as the sample size is not sufficient. 2)Sufficient learning regime: Iflimδ→0Nδ log(1/δ)→kfork∈(0,∞), we have a sharp characterization of... | https://arxiv.org/abs/2501.19126v1 |
minimum limiting width of the CI (main setting of the paper) is the sum of the limiting minimum half-widths of the two types of one-sided CIs (greater than some threshold and less than some threshold). 1.3 Random sampling cost setting The above results assume that the cost of each sample is fixed and equal to one. Howe... | https://arxiv.org/abs/2501.19126v1 |
width depends solely on the mean of the cost distribution. 2 Literature review There is a vast amount of literature on constructing non-asymptotic CIs that proceeds by inverting finite-sample con- centration inequalities. For the probability distributions with bounded support, one can use various concentration inequali... | https://arxiv.org/abs/2501.19126v1 |
for construction of CI LetΠCIdenote the collection of policies for constructing CIs. Let [bµπ L(N, δ),bµπ R(N, δ)]denote the estimated CI after observing X1, X2, . . . , X Nunder a policy πfor any given δ∈(0,1). For any policy π∈ΠCImust satisfy the following for any given δ∈(0,1), ∀n∈N:Pν(µ∈[bµπ L(n, δ),bµπ R(n, δ)])≥1... | https://arxiv.org/abs/2501.19126v1 |
4 Main results on the minimum limiting width of CI in different regimes Recall that ν, with mean µ, represents the true distribution from which samples are generated. In other words, νis the true underlying but unknown environment. To start the analysis of the minimum limiting width, we define an alternate environment ... | https://arxiv.org/abs/2501.19126v1 |
Section 5) has zero limiting width in this regime. Hence, we denote this regime as the complete learning regime. Additionally, in the complete learning regime, we characterize the rate at which the CI width converges to zero. Under certain technical assumptions on CI construction policies, we establish the fastest achi... | https://arxiv.org/abs/2501.19126v1 |
L(µ, k),where µ∗ L(µ, k)< µ uniquely solves d(µ, µ∗ L(µ, k)) =1 k. The above results can similarly be applied to one-sided CIs where the goal is to ensure the upper bound of the interval is below a threshold by replacing all occurrences of subscripts LwithRin the definitions and results. Remark 1. In the Appendix, we s... | https://arxiv.org/abs/2501.19126v1 |
L(µ)] =µ−µ. Further, limδ→0bµπ L(τδ, δ)p→µandlimδ→0bµπ R(τδ, δ)p→µ. b)Sufficient learning regime : Iflimδ→0Cδ log(1/δ)→kfork∈(0,∞), then [µπ R(µ)−µπ L(µ)]≥µ∗ R(µ, k,c)−µ∗ L(µ, k,c), (7) where, µ∗ L(µ, k,c)< µ andµ∗ R(µ, k,c)> µ uniquely solve the following system of equations, d(µ, µ∗ R(µ, k,c)) =d(µ, µ∗ L(µ, k,c)) =c ... | https://arxiv.org/abs/2501.19126v1 |
outcome distribution ν∈Pandx∈R, let, KLinf(ν,P, x) =infκ∈P:m(κ)≥xKL(ν, κ),ifx≥m(ν) infκ∈P:m(κ)≤xKL(ν, κ),ifx < m (ν).(9) KLinf(ν,P, x)is the minimum amongst the KL divergences between a given distribution νand all distributions in the same family Pwhich have higher mean than xifx≥m(ν). It is similarly defined for x < ... | https://arxiv.org/abs/2501.19126v1 |
results are shown in Figure 2. As the sample size Nincreases, we observe that the width of the confidence interval approaches our asymptotic lower bound. At last, we evaluate the numerical performance of our policy ˆπ1when the distribution νis Bernoulli with mean parameter 0.5and the cost distribution is Unif [0,2]. We... | https://arxiv.org/abs/2501.19126v1 |
Xu, L. (2021). A generalized catoni’s m-estimator under finite α-th moment assumption withα∈(1,2).Electronic Journal of Statistics , 15(2):5523–5544. Chick, S. E. and Inoue, K. (2001). New two-stage and sequential procedures for selecting the best simulated system. Operations Research , 49(5):732–743. Garivier, A. and ... | https://arxiv.org/abs/2501.19126v1 |
for for ˜ν∈K2(µπ R(ν)). Since π∈Πs CI, hence using the definition of CI, we get, P˜ν(Eδ)≤δ. Now observe that, Pν(Eδ)≥Pν{˜µ <bµπ L(Nδ, δ)}. Taking the limit of δ→0on both sides and using (13), we get, lim δ→0Pν(Eδ) = 1 . Hence using the definition of ϕ(·), we get, lim inf δ→0Ψ(Pν(Eδ),P˜ν(Eδ)) log(1/δ)≥1, (15) Hence usin... | https://arxiv.org/abs/2501.19126v1 |
the proof. □ 11.3 Results and proofs for the rate of convergence analysis in complete learning regime. We first introduce a set of policies, denoted by Πsr CI⊆Πs CIfor which the analysis is valid. For π∈Πsr CI, we assume that following holds as limδ→0Nδ log(1/δ)=∞: 1. lim δ→0[ˆµπ Nδ−bµπ L(Nδ, δ)]2 log(1/δ) Nδp→θπ L(µ)a... | https://arxiv.org/abs/2501.19126v1 |
=d(ˆµπ1 Nδ,bµπ1 R(Nδ, δ)) =log(2/δ) Nδ. (21) First of all observe that, ˆµπ1 Nδ, i.e., sample average satisfies the central limit theorem. Hence we have, lim δ→0√Nδ(µ−ˆµπ1 Nδ) log(1/δ)p→0. Now, to prove that π1∈Πsr CIand the (b) part of the Theorem, we do Taylor series expansion of d(ˆµπ1 Nδ,bµπ1 L(Nδ, δ)) andd(ˆµπ1 Nδ... | https://arxiv.org/abs/2501.19126v1 |
be re-written as, d(ˆµτδ,bµˆπ1 L(τδ, δ)) =d(ˆµτδ,bµˆπ1 R(τδ, δ)) =log(2/δ) CδCδ τδ. Taking δ→0and using the joint-continuity of d(µ, x)in(µ, x)and (22), we get that π1∈ˆΠs CIand part (b), (c) of this theorem holds. This completes the proof. □ 17 APREPRINT - FEBRUARY 3, 2025 11.6 Proofs of results in Section 8. Proof of... | https://arxiv.org/abs/2501.19126v1 |
Fast exact recovery of noisy matrix from few entries: the infinity norm approach BaoLinh Tran1, Van Vu1 1Department of Mathematics, Yale University Abstract The matrix recovery (completion) problem, a central problem in data science and theoretical computer science, is to recover a matrix Afrom a relatively small sampl... | https://arxiv.org/abs/2501.19224v2 |
. . . . . . . . . . . . . . . . . . . . . . . 5 1.5.1 Nuclear norm minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5.2 Modified alternating projections . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5.3 Low rank approximation with Gradient descent . . . . . . . . . . . . . . . . . 7 1.5.... | https://arxiv.org/abs/2501.19224v2 |
. . . . . . . . . . . . . . 29 4.3 Proof of the bound on generic series . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.4 Bounding each term in the generic series . . . . . . . . . . . . . . . . . . . . . . . . . 36 A Proof of the full matrix completion theorem 40 B Proofs of technical lemmas 43 B.1 Proof of... | https://arxiv.org/abs/2501.19224v2 |
each s∈[r], let As=Ps i=1σiuivT ibe the best rank- sapproximation of A. Define Bs analogously for any matrix B. •When discussing A, we denote N:= max {m, n}. •Thecoherence parameter ofUis given by µ(U) := max i∈[m]m r∥eT iU∥2=m∥U∥2,∞ r, (2) where the 2-to- ∞norm of a matrix Mis given by ∥M∥2,∞:= sup {∥Mu∥∞:∥u∥2= 1}, wh... | https://arxiv.org/abs/2501.19224v2 |
end, we says that a matrix Ahave a finite precision ε0, if its entries are integer multiples of a parameter ε0>0. For instance, if all entries have two decimal places, then ε=.01. In many pratical problems, such as compelting/rating a recommendation system, the parameter ε0is actually quite large. For instance, in the ... | https://arxiv.org/abs/2501.19224v2 |
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