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1)≤ |S|ln 1 + e−kc/2≤Ne−kc, so|S| ≤Ne−kc/2(e+ 1)< Ne−kc/2+2. Part Three: By Definition B.1, let network g=F−kc/2,kc/2, we show that, with high probability, E(x,y)∼Dyg(x)has a lower bound. Firstly, by part two, there areP (x,y)∈Dtryg(x)≥N(kc(1−e−kc/2+2)/2−kce−kc/2+2/2) = Nkc(1−2e−kc/2+2)/2. Then, let H={yF(x) :F(x)∈Hσ W... | https://arxiv.org/abs/2503.04111v1 |
D (n)andN≥VC(Hσ W0(n))(1−4ϵ−δ), with probability 1−δofDtr, we haveAD(F)≥1−ϵfor all F ∈arg min f∈HW(n)P (x,y)∈DtrL(f(x), y). We will derive contradictions on the basis of this conclusion. Part 1: Find some points and values. For a simple expression, let k=VC(Hσ W0(n)), and{ui}k i=1bekpoints that can be shattered by VC(H... | https://arxiv.org/abs/2503.04111v1 |
have at most 2|len(Z)|different Sss(D, Z)forD ∈ D S. Because when D1andD2satisfy y(D1)i=y(D2)ifor any i∈len(Z), we have Dss(D1, Z) =Dss(D2, Z), and2|len(Z)|situations of label of uiwhere i∈len(Z), so there exist at most 2|len(Z)|different Sss(D, Z). Then, by part 3.1, for an Sss(D, Z), at mostP[2kϵ] i=0Ci k−|len(Z)|ofD... | https://arxiv.org/abs/2503.04111v1 |
to see that |Π(S)| ≤2Nbecause Sgn (Wxi+b)∈ {− 1,1}. So, when N≤n+ 1, it is obviously correct. When N > n + 1. Consider that the VC-dim of the linear space is n+ 1, and Π(S) ={(Sgn(Wxi+ b))n i=1:W∈Rn, b∈R}is the growth function of linear space under dataset S. So by Theorem 1 of (Sauer, 1972), we have |Π(S)| ≤Pn+1 i=0Ci... | https://arxiv.org/abs/2503.04111v1 |
i+0.5,Fgives the same label to xu1(n), xu+11(n), . . . , x u+k1(n), which means that Fcan classify at most [(k+1)+1 2]samples in them. Similar to when Pi+0.5≤xu< xu+1<···< xu+k< Pi+1. Letqi=|{j:Pi/2≤xj< Pi/2+0.5}|where i∈[2M]. Consider that each sample in Sis appeared in aPi1(n)→Pi+0.51(n)orPi+0.51(n)→Pi+11(n), soP2M i... | https://arxiv.org/abs/2503.04111v1 |
when N≥4ln(δ/2) ln(0.5+1/n), with probability at least 1−(0.5 + 1 /n)ln(δ/2) ln(0.5+1/n)/(0.5 + 1/n) = 1−δ/(1 + 2 /n)≥1−δ. This is what we want. Part Four, we prove the result. LetDtr∼ DN n. For any W, letF ∈ arg min f∈Hσ W(n)P (x,y)∈DtrL(f(x), y)andF= PW i=1σ(Wix+bi) +c. Firstly, with probability 1−δ, there are four s... | https://arxiv.org/abs/2503.04111v1 |
∈HW(n), we can write F=P⌈W W0⌉−1 i=0PW0 j=1ReLU( WiW0+jx+biW0+j) +c, which is a representation of the sum of ⌈W W0⌉small networks with width of W0. So by part one and by the assumption in the theorem, with probability 1−δofDtr∼ DN, there is a Dr∈R(Dtr, ϵ) such thatP (x,y)∈DryF1(x)≤2N0c1for all F1∈HW0(n). Then we haveP ... | https://arxiv.org/abs/2503.04111v1 |
probability at least 1−e−2N/2002ofDtr, there are at least N/200points in s3. Using also the Hoeffding inequality andP(x,y)∼D(y= 1) = 99 /200, we know that with probability at least 1−e−2N(99/200−98/200)2 ofDtr, there are at least 98/200Npoints with label 1 in Dtr; similar, with probability at least 1−e−2N(101/200−98/20... | https://arxiv.org/abs/2503.04111v1 |
we know that, with probability 1−δofDtr, there are |P (x,y)∈DtrL(F(x), y)/N− E(x,y)∼D[L(F(x), y)]| ≥4(n+1)(√ 5 log(4)+√2log2n)√ 98N/200+ 6(n+ 2)q ln(2/δ) 2N). So, with probability 1−3e−2N/2002−δ,Dtrsatisfies the above two conditions. For such a Dtr, assume that F ∈ arg minf∈HW(n)P (x,y)∈DtrL(f(x), y), andFmust give the... | https://arxiv.org/abs/2503.04111v1 |
network FDtrmentioned in part two. Firstly, we show that FDtr(x)∈arg min F∈H W(n)P (x,y)∈DtrL(F(x), y). Because FDtr(x) =Ft(xt) =z1when (x,1)∈ D trandFDtr(x) =Ft(xt) =z−1when (x,−1)∈ D tr. SoL(FDtr(x), y)reaches the minimum value for any (x, y)∈ Dtr, which implies FDtr∈arg min F∈H W(n). Secondly, there are AD(FDtr(x)) ... | https://arxiv.org/abs/2503.04111v1 |
andF0(1) =a. Byϕ(a)+ϕ(−b) =Lb(F0(0),−1)+Lb(F0(1),1)< Lb(F(0),−1)+Lb(F(1),1) = ϕ(W(n+1)+ 1)+ϕ(−W−1), andϕis a decreasing concave function, there must be W(n+1)+1 −a <−b+W+1, which implies |a−b|> Wn . Consider |a−b|=|PW i=1aiReLU( bi)−PW i=1aiReLU( Wi1+bi)| ≤ |PW i=1ai1Wi| ≤Wn. This is a contradiction to |a−b|> Wn which ... | https://arxiv.org/abs/2503.04111v1 |
4, then yF(x′)>0for all ||x′−x||∞≤c 2D0+1WD0−1 0n. As shown in Lemma K.2, there are yF(x′)≥[W W0]D0−1cWD−D0 4−WL−1n||x−x′||∞. So when ||x−x′||∞≤c[W W0]L0−1 4nWL0−1≤c 2D0+1WD0−1 0n, there are yF(x′)>0. Part four . Let r=c 2D0+1WD0−1 0n. we can divide [0,1]ninto1 (r/2)ndisjoint cubes that have side length r/2. Then by pa... | https://arxiv.org/abs/2503.04111v1 |
accuracy, which is consistent with Theorem 4.3 where the number of data (network size) is located on the denominator. (2) When the number of data is fixed, increasing the network size has a limitation effect on improving accuracy, as shown in Figure 1, which is consistent with Theorem 4.3, because the number of data ca... | https://arxiv.org/abs/2503.04111v1 |
Published as a conference paper at ICLR 2025 PROVABLE ROBUST OVERFITTING MITIGATION IN WASSERSTEIN DISTRIBUTIONALLY ROBUST OPTI- MIZATION Shuang Liu1, 2, Yihan Wang1, 2, Yifan Zhu1, 2, Yibo Miao1, 2, Xiao-Shan Gao1, 2, 3∗ 1State Key Laboratory of Mathematical Sciences Academy of Mathematics and Systems Science, Chinese... | https://arxiv.org/abs/2503.04315v1 |
to Figure 2). A primary factor contributing to overfitting behavior is the inherent statistical error arising from finite sampling of train data (Van Parys et al., 2021; Bennouna & Van Parys, 2022). This error, defined as the discrepancy between the empirical distribution of the training data and the true underlying da... | https://arxiv.org/abs/2503.04315v1 |
Schmidt et al. (2018) attributed robust overfitting to sample complexity theory and suggested that more training data are required for adversarial robust generalization. This assertion is supported by empirical results in subsequent studies (Schmidt et al., 2018; Zhai et al., 2019). Recent works also proposed various s... | https://arxiv.org/abs/2503.04315v1 |
(LP) metric and the total variation metric (TV). The Levy-Prokhorov metric is theoretically important because weak convergence of probabil- ity distribution is equivalent to the convergence in the Levy-Prokhorov metric. In Appendix A.1, we establish the relationship among these probability discrepancies. 4 STATISTICALL... | https://arxiv.org/abs/2503.04315v1 |
data distribution, and Dnbe the observed empirical distribution sampled i.i.d. from D. Then for all ε >0, letδ= (ε diam(Z)+1)p, we have P ∀θ∈Θ,ED[L(θ, z)]≤ Lε,γ(θ,Dn) ≥1−e−γn4 δm(Z,δ) (5) where m(Z, δ) := min {k≥0 :∃ξ1,···, ξk∈ Z,s.t.∪k i=1B(ξi, δ)⊇ Z} is the internal covering number of the support set Z. 4 Publish... | https://arxiv.org/abs/2503.04315v1 |
on additional assump- tions about the loss function, such as Lipschitz continuity and boundedness, which are not required in our framework. Furthermore, the form of the generalization bounds in our framework is fun- damentally different from those of WDRO. Our bound provides a stronger guarantee, where the probability ... | https://arxiv.org/abs/2503.04315v1 |
of Stackelberg equilibrium is considered for the first time for WDRO approaches. It can be ob- served that the Stackelberg equilibrium requires weaker conditions to exist than the Nash equilib- rium, which gives the smallest statistically robust loss among all decision models can be consid- ered optimal robust in certa... | https://arxiv.org/abs/2503.04315v1 |
large positive number to prevent label conversion. Now we revisit the statistically robust loss: Lε,γ(θ,Dn) = sup D′∈U(Dn)ED′[L(θ, z)] = sup {sup{ED′[L(θ, z)] :D′,KL (Q,D′)≤γ}:Q,Wp(Dn,Q)≤ε}. Intuitively, we start with the empirical distribution Dnand identify the most adversarial sample distribution that satisfies the ... | https://arxiv.org/abs/2503.04315v1 |
Unless otherwise specified, we set γ= 0.1to its default value. The ablation study on γis provided in latter part. We repeated all the experiments three times for different seeds. More training details are provided in Appendix A.6. Figure 2: Comparison of SR-WDRO against other robust training methods on CIFAR10 (ε= 8/25... | https://arxiv.org/abs/2503.04315v1 |
±0.07 6.83 ±0.55 UDR-AT 85.99 ±0.15 49.01 ±0.13 55.82±0.19 6.81±0.31 HR 84.89 ±0.16 48.45 ±0.31 52.45 ±0.27 4.00 ±0.34 Ours 84.52 ±0.27 51.26±0.42 53.87±0.06 2.61±0.47 Robustness for smaller ε.Theorem 5 shows that to ensure robustness on the test set, it is generally necessary to employ a larger attack budget during tr... | https://arxiv.org/abs/2503.04315v1 |
overhead. Due to the intractability of Equations (4) and (8), a better approxi- mation is welcomed to solve SR-WDRO to mitigate the compromise of accuracy and computational cost. Furthermore, we mainly focus on supervised learning in this paper, expanding SR-WDRO to broader tasks including unsupervised learning, regres... | https://arxiv.org/abs/2503.04315v1 |
31, 2018. Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXiv:1706.06083 , 2017. Andrey Malinin, Neil Band, German Chesnokov, Yarin Gal, Mark JF Gales, Alexey Noskov, Andrey Ploskonosov, Liudmila Pr... | https://arxiv.org/abs/2503.04315v1 |
gen- eralization. In Advances in Neural Information Processing Systems , volume 33, pp. 2958–2969, 2020. Chaojian Yu, Bo Han, Li Shen, Jun Yu, Chen Gong, Mingming Gong, and Tongliang Liu. Under- standing robust overfitting of adversarial training and beyond. In International Conference on Machine Learning , pp. 25595–2... | https://arxiv.org/abs/2503.04315v1 |
function. As d(z, z′)≤diam(Z)·I(z̸= 14 Published as a conference paper at ICLR 2025 z′), we have: Wp(µ, ν) = inf π∈Π(µ,ν) E(z,z′)∼π[dp(z, z′)] 1 p ≤ inf π∈Π(µ,ν) E(z,z′)∼π[diam( Z)p·I(z̸=z′)] 1 p = diam( Z) inf π∈Πµ,ν) E(z,z′)∼π[I(z, z′)] 1 p = diam( Z)·TV(µ, ν)1 p. And the second part is by Pinsker’s inequality tha... | https://arxiv.org/abs/2503.04315v1 |
introduce the Levy-Prokhorov (LP) metric given by Bennouna et al. (2023), LPεis defined as: LPε(D,D′) := infZ 1 (d(z, z′)> ε) dπ(z, z′) :π∈Π (D,D′) . (11) Proof of the Theorem 5 We restate the theorem below. Let Dbe the true data distribution, Dnthe empirical distribution sampled i.i.d. from D, and δ= (σ diam(Z)+1)pf... | https://arxiv.org/abs/2503.04315v1 |
of player 1and player 2beX andY, respectively. Their utility functions are denoted by u1(x, y)andu2(x, y), where x∈Xis the strategy of player 1, and y∈Yis the strategy of player 2. In a zero-sum game, u1(x, y) = −u2(x, y). Definition 19 (Nash Equilibrium) .A Nash equilibrium is a pair of strategies (x∗, y∗)such that ne... | https://arxiv.org/abs/2503.04315v1 |
clear that: sup D′∈U(Dn)ED′[L(θ, z)]≤ sup D′∈BWp(Dn, ε+γa·dp min(Z))ED′[L(θ, z)]. The supremum on the right side of the above inequality is finite by Yue et al. (2022, Theorem 2), which applies Assumption 8 (iii) and Dnhas a finite p-th moment. Thus the supremum have a finite upper bound. It remains to be shown that su... | https://arxiv.org/abs/2503.04315v1 |
weakly upper semi-continuous in D′. This follows directly from the proof of Proposition 9. This analysis establishes that all the conditions of Sion’s minimax theorem are satisfied. Consequently, the infimum and supremum in Equation (7) can be interchanged. It remains to show that both maxima in (7) are reached. Howeve... | https://arxiv.org/abs/2503.04315v1 |
and best robust accuracy is the smallest in our method, indicating that our approach most effectively mitigates overfitting even for different attack budgets. Table 4: Robust performance of different methods on CIFAR-10using a ResNet-18 for l∞with budget 6/255, 10/255 and 12/255. We use PGD-10 as the attack to evaluate... | https://arxiv.org/abs/2503.04315v1 |
training budgets 10/255,12/255while keeping other parameters of PGD attack the same. The results shown in Table 6 and Table 7 demonstrate that our proposed SR-WDRO achieves the best robustness under various adversarial attacks. Computation cost We evaluate the computational efficiency of our proposed SR-WDRO method aga... | https://arxiv.org/abs/2503.04315v1 |
LEARNING CAUSAL RESPONSE REPRESENTATIONS THROUGH DIRECT EFFECT ANALYSIS Homer Durand, Gherardo Varando, Gustau Camps-Valls Image and Processing Lab Universitat de Valencia Valencia homer.durand@uv.es gherardo.varando@uv.es gustau.camps@uv.es ABSTRACT We propose a novel approach for learning causal response representati... | https://arxiv.org/abs/2503.04358v1 |
relevant information about the intervention’s effect. Additionally, it can help disentangle the direction in which the outcome is affected by the intervention from the direction where the distribution remains unchanged. We will demonstrate that this has important implications in different application domains with a foc... | https://arxiv.org/abs/2503.04358v1 |
effect subspace (DES). Our work focuses on recovering this reduced space, with its basis ordered by the variance in Yexplained by X, while controlling for Z, analogous to how Principal Component Analysis (PCA) identifies directions of maximum variance in a random vector. We now summarise conditional independence testin... | https://arxiv.org/abs/2503.04358v1 |
direction in which Ymaximises a conditional independence statistic can effectively uncover the subspace of Ymost caused by X. This example is illustrated in Fig. 1, where we observe that projection along Σ−1bimproves the separability of distributions under different interventions (here X= 0andX= 1). A natural basis for... | https://arxiv.org/abs/2503.04358v1 |
specifically for the DE, to know, a subspace that retains all relevant information about the DE. 3 Learning Framework Our goal is to identify the components of Ythat are most caused by X, conditional on Z, assuming all confounders C⊆Zare observed and the causal relationship X→Yis known. Specifically, we aim to find a s... | https://arxiv.org/abs/2503.04358v1 |
commonly used for variable selection [Hocking, 1976] or causal discovery [Nogueira et al., 2022]. When the conditioning set Zconsists of the past values of Y, the empirical version of TFcorresponds to the 4 statistic of the well-known Granger causality test [Granger, 1969]. In this context, the maximisation of TFwith r... | https://arxiv.org/abs/2503.04358v1 |
:Rr→Rd,b∈Rdand with Ny∼ N(0,Σ). Again, the relationship between Xand Zis left undefined as applying do(X)breaks any statistical dependencies that existed in the observational setting. We denote by Σψ(z)the covariance of ψ(Z). For the remainder of this section, we assume that the intervention ϕ(x) is bounded. Concretely... | https://arxiv.org/abs/2503.04358v1 |
eigenvalue of Σor of∥b∥2, independently. All of these conditions are related to the observation that as the dimensionality increases, Yx’s distribution contains ’more signal’ relative to its noise level. This phenomenon occurs, for example, when the sources of noise are limited and the resolution of the observations is... | https://arxiv.org/abs/2503.04358v1 |
on the p-values of λD. Proposition 4.7 (Upper bound on ΛDunder conditional independence) .Under similar assumptions as in Prop 4.6 we have under the null hypothesis H0:X⊥ ⊥Y|ZthatP(ΛD≥λD|H0)≤P(ΛF≥λD|H0). Testing is straightforward by rejecting the null hypothesis if (d fd/d fn )λ1deviates sufficiently from F(d fn, d fd... | https://arxiv.org/abs/2503.04358v1 |
Gaussian noise, where fa(Z) =Zand set p=r=q= 1. Our analysis compares tests based on the optimisation of TFandTD against four common conditional independence (CI) tests: partial CCA [Rao, 1969], the Generalised Covariance Measure (GCM) [Shah and Peters, 2018a], Fisher’s Z test [Kalisch and Bühlman, 2007], and the Kerne... | https://arxiv.org/abs/2503.04358v1 |
across different locations, although both TDand Detrending struggle in highly variable regions. Overall, our approach provides a principled framework for disentangling forced and internal climate variability. Climate change attribution. In this experiment, we examine the direct effects of external forcing and investiga... | https://arxiv.org/abs/2503.04358v1 |
of equality between sets of coefficients in two linear regressions. Econometrica: Journal of the Econometric Society , pages 591–605, 1960. G. Danabasoglu, J.-F. Lamarque, J. Bacmeister, D. Bailey, A. DuVivier, J. Edwards, L. Emmons, J. Fasullo, R. Garcia, A. Gettelman, et al. The community earth system model version 2... | https://arxiv.org/abs/2503.04358v1 |
Advances in Neural Information Processing Systems , pages 6446–6456, 2017. N. A. Macmillan. Signal detection theory. Stevens’ handbook of experimental psychology: Methodology in experimental psychology , 4:43–90, 2002. R. V . Mises and H. Pollaczek-Geiringer. Praktische verfahren der gleichungsauflösung . ZAMM - Journa... | https://arxiv.org/abs/2503.04358v1 |
test and application in causal discovery. Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI) , pages 804–813, 2012. 12 Contents 1 Introduction 1 2 Preliminary 2 2.1 Introductory example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 R... | https://arxiv.org/abs/2503.04358v1 |
. 15 A.4 Noise term behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 B Proofs 16 B.1 Auxiliary lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 B.2 Signal-to-Noise optimality . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2503.04358v1 |
the interventional distributions p(Y|do(X=x))andp(Y|do(X=x+δx)), improving their separability under small interventions. This follows from the well established connection between Fisher information and the Kullback–Leibler divergence. Proposition A.1. LetP(Y|x)be a probability distribution over Yparameterised by x∈Rd. ... | https://arxiv.org/abs/2503.04358v1 |
case where q= 1, assuming that the dimensionality of the direct effect of XonYis rank one. In a manner analogous to the power iteration method [Mises and Pollaczek- Geiringer, 1929], we can extract additional components by employing a deflation technique. We now provide an algorithm for this approach. Algorithm 1: Powe... | https://arxiv.org/abs/2503.04358v1 |
theoretical results. As this will be useful for most of the theoretical development, we first get a result for the first eigenvector of each optimisation problem. B.1 Auxiliary lemma Lemma B.1. LetwS,wF, andwDdenote the first eigenvectors associated with the optimisation problems in Eq. (3), Eq.(4), and Eq. (5), respec... | https://arxiv.org/abs/2503.04358v1 |
SNR, if either wSorwFhas an SNR that grows to infinity, then the SNR of wD will also tend to infinity. B.4 Equivalence of Signal-to-Noise ratio and Fisher information Proposition B.5 (Equivalence between Fisher Information and SNR) .Consider a SCM as described in (6), and let the intervention function be ϕ(x) =v⊤x, whe... | https://arxiv.org/abs/2503.04358v1 |
sketch. As the conditioning set in the computation of root squared errors R2 noiseis larger than of R2 full, the empirical residuals R2 noiseare always larger than R2 fullthus we have that R2 res−R2 full R2 noise≤R2 res−R2 full R2 full. Hence, we have that P(ΛD≥λD|H0)≤P(ΛF≥λD|H0)withH0:X⊥ ⊥Y|Z. Note that by using this ... | https://arxiv.org/abs/2503.04358v1 |
Eh ∥Σ−ˆΣ∥opi ≤Nfullp C2p κ2(n) +C2κ2(n) +C4√nusing assumptions (2)and(5). Finally, applying the Davis-Kahan theorem, we have: E ∥w1−wS∥2 2 ≤√ 2Eh ∥Σres−ˆΣres∥Fi +Eh ∥Σfull−ˆΣfull∥Fi δM(30) ≤√ 2Nfull√C2p κ2(n) +C2κ2(n) +C4√n+Nres√C1p κ1(n) +C1κ1(n) +C3√n δM(31) =o(p κ1(n) +p κ2(n)), (32) assuming that κ1(n)andκ2(n)dec... | https://arxiv.org/abs/2503.04358v1 |
recover the signal when the noise variance increases rapidly and the number of samples falls below the outcome dimension. We also conducted an experiment where XandZwere generated as independent variables to highlight the potential advantages of our learning algorithms. In this case, we observed that partial CCA (pCCA)... | https://arxiv.org/abs/2503.04358v1 |
the vector b, denoted as b⊥, to recover the internal variability component ˆYinternal . This projection isolates the portion of the temperature field that is not correlated with the external forcing, allowing us to separate the forced and internal components effectively. Finally, we compute the climate-forced response ... | https://arxiv.org/abs/2503.04358v1 |
exhibits high variability (e.g., at d= 30 ord= 493 , typically located in polar regions), both methods struggle to fully capture this variability. This may be due to the smoothing of GMT in the external forcing, but this observation warrants further investigation, as these regions of high variability may also reflect m... | https://arxiv.org/abs/2503.04358v1 |
Estimation of relative risk, odds ratio and their logarithms with guaranteed accuracy and controlled sample size ratio Luis Mendo* *Information Processing and Telecommunications Center, Universidad Polit ´ecnica de Madrid, Avenida Complutense, 30, Madrid, 28040, Spain. Abstract Given two populations from which independ... | https://arxiv.org/abs/2503.04876v1 |
the other population can be taken as needed. In many use cases, it may be required that the two sample sizes be similar, or that they approximately satisfy a given ratio. Another possible sampling procedure is group sam- pling , whereby samples are collected in groups or batches of l1andl2samples from the two populatio... | https://arxiv.org/abs/2503.04876v1 |
number is denoted as Hn=n ∑ k=11 k. (3) Matrices are written in boldface letters, and Q⊺represents the transpose of a matrix Q. The regularized incomplete beta function is defined as I(x;u,v) =1 B(u,v)Zx 0tu−1(1−t)v−1dt, (4) where B (u,v)is the beta function; and from Abramowitz and Stegun (1970, equations (6.1.15), (6... | https://arxiv.org/abs/2503.04876v1 |
the second-stage IBS parameters s1ands2are computed such that (16) and (17) are satisfied. The results of the second stage, i.e. N1andN2, are then used to produce the final estimate ˆθ. The rationale is as follows. According to (15), E[M1+N1] E[M2+N2]=r1+s1 (r2+s2)θ. (18) Each of the IBS procedures in the second stage ... | https://arxiv.org/abs/2503.04876v1 |
µ12=1. (30) This will be referred to as error function . The parameters µ1,µ2,µ12are introduced for con- venience; this way the expression of e(s1,s2)for other estimators will be the same as (29), only with different values of these parameters. Then, requiring e(s1,s2)≤A (31) guarantees that condition (16) holds; in fa... | https://arxiv.org/abs/2503.04876v1 |
of s1and s2as functions of Xand 1 /X. In fact, for certain values of A,δ1andδ2the approximations (37) and (38) are exact. This will be analyzed in Subsection 2.3. Substituting (37) and (38) into (15), and introducing an additional term ξto account for the effect of rounding s1ands2, the average sample sizes are approxi... | https://arxiv.org/abs/2503.04876v1 |
s2=s1+δ1 γX−δ1−µ1+µ2, (65) D= (γX(A(δ1+µ1)+1)−A(δ1−µ1)+1)2 −4A(γX((Aµ1+1)(δ1+µ1)+µ1−µ12)−(Aµ1+1)δ1).(66) Substituting (39)–(42) and (60)–(62) into (54) and (55) yields E[M1+N1]≈1 A+r1+µ1+ξ1 p1+λ p2 , (67) E[M2+N2]≈1 A+r1+µ1+ξ1 λp1+1 p2 . (68) It is convenient to use a normalized version of the average sample si... | https://arxiv.org/abs/2503.04876v1 |
(78) and using (39)–(42) and (60)–(62) yields E[M1+N1]φ<1 A+r1+µ1+11√ λθ+√ λθ√ λ, (79) E[M2+N2]φ<1 A+r1+µ1+11√ λθ+√ λθ1√ λ. (80) These bounds hold for ξ=1 and for any r1such that c(A,r1,1)≥0. Considering the restriction (23), the best (i.e. lowest) upper bounds are obtained by choosing r1as r1=min{r=3,4,5,...|c... | https://arxiv.org/abs/2503.04876v1 |
evaluated by means of Monte Carlo simulations in the following. For each combination of parameters a simulation is run consisting of 106 realizations of the estimator. The empirical MSE and average numbers of samples are com- puted from the simulation (i.e. expectation is replaced by sample mean), and then they are com... | https://arxiv.org/abs/2503.04876v1 |
locations, produced by the rounding applied to s1ands2(see for example the right- most region for E [M2+N2]in Figure 4c with r1=3, which in that region corresponds to the bold dots). These are only observable in the simulation results, because the theoretical curves do not explicitly model the rounding of s1ands2; and ... | https://arxiv.org/abs/2503.04876v1 |
.5 and 77 .9 respectively. The change of the rounded value of s2, from being 2 to being 3 with probabilities near 1, causes the rightmost vertical gap in Figure 5a, from a relative MSE approximately equal to 0 .50 down to 0 .34. In contrast, for small Aboth s1ands2are large and the effect of rounding is less marked, be... | https://arxiv.org/abs/2503.04876v1 |
sample size, and the regularity conditions are different. For estimators based on independent observations from twopopulations, as considered in this paper, a variation of the Cram ´er–Rao bound can be applied in the fixed-size case (Kay, 1993, chapter 3). However, to the author’s knowledge, there is no analogue of Wol... | https://arxiv.org/abs/2503.04876v1 |
are also considerably smaller than their bounds, as discussed. On the other hand, the bound is quite tight for small or moderate values of Aandφ, which is precisely when it is most important to characterize ηsepaccurately, as sample sizes are large in that case. It is easily seen that lim A→0Ar2 1=1 for Aandr1related b... | https://arxiv.org/abs/2503.04876v1 |
are approximately given by (70) and (71). Regarding E [|∆|], it is shown in Appendix A.1 that, for φsmall, E[|∆|]≈E r1+s1 l1p1−r2+s2 l2p2 . (97) 21 Define ˜∆=r1+s1 l1p1−r2+s2 l2p2. (98) Using approximations (37) and (38) for s1,s2and including the rounding term ξas in Sub- section 2.2 (this has a very small effect o... | https://arxiv.org/abs/2503.04876v1 |
ηgr=∂ζ ∂p12p1(1−p1) l1+∂ζ ∂p22p2(1−p2) l2 E[G]Var[ˆζ]. (109) 23 10-310-210-1100 A101102103(a)l1=1,l2=1,θ=1,φ=0.01 (p1=0.01,p2=0.01) 10-310-210-1100 A101102103(b)l1=1,l2=1,θ=16,φ=0.01 (p1=0.04,p2=0.0025) 10-310-210-1100 A101102103 (c)l1=2,l2=5,θ=16,φ=0.01 (p1=0.04,p2=0.0025) 10-310-210-1100 A101102103(d)l1=2,l2=5,θ=... | https://arxiv.org/abs/2503.04876v1 |
=E[E[ˆΘ2|s1,s2]]−Θ2<E e(s1,s2)+Θ2 −Θ2<A. (116) That is, condition (31) ensures that (112) holds for any p1,p2∈(0,1). As in the RR case, s1ands2are obtained from the equation system formed by (20) and (33), except with µ1,µ2,µ12given by (115). The solution is (64)–(66). The values of s1and s2should then be rounded whi... | https://arxiv.org/abs/2503.04876v1 |
slightly different because µ1 andr1are), and the efficiency approaches 1 when Aandφtend to 0. Figure 12 compares this bound with simulation results, considering only two specific cases for brevity. The efficiency values from simulation are computed as explained in Subsection 2.4. Results are similar to those for RR (Fi... | https://arxiv.org/abs/2503.04876v1 |
the RR and LRR estimators. Thefirstdifference is that in the second stage twoIBS procedures are used for each popula- tion. Given s1,s2(which will be computed from the results of the first stage), IBS is applied to population 1 to obtain s1successes, which requires N′ 1samples. Then IBS is applied again to population 1... | https://arxiv.org/abs/2503.04876v1 |
(s2−2)p2 2 1−¯p2 s2−2p2 . Therefore, since the observations are independent, ˆψ=(s1−1)(s2−1)N′′ 1N′ 2 (s1−2)(s2−2)(N′ 1−1)(N′′ 2−1)(125) is an unbiased estimator of ψ, and for s1,s2≥3 Eˆψ2|s1,s2 ψ2=p2 2(1−p1)2 p2 1(1−p2)2 Var(s1−1)N′′ 1 (s1−2)(N′ 1−1) s1 +p2 1 (1−p1)2 · Var(s2−1)N′ 2 (s2−2)(N′′ 2−1) s2 +(1−p... | https://arxiv.org/abs/2503.04876v1 |
parameter ¯ pi, obtained by transforming samples with parameter pi,i=1,2. Specifically, from (37), (38), (119), (120), (129), (132) and (133), and introducing the term ξ to account for the effect of rounding s1ands2, E[M1+N1] =3r1/2+s1−2p1 ¯p1≈3r1/2+s1 ¯p1≈a1+3r1/2+ξ ¯p1+b1r2 (r1−1)¯p2, (134) and analogously E[M2+N2] =... | https://arxiv.org/abs/2503.04876v1 |
comparing (27) and (28), and these do not tend to 1 for large s1,s2. (In LRR the factors are more cumbersome, see Mendo (2025, theorem 2); but for large s1,s2 they tend to the same values 1 −p1, 1−p2as in RR.) Figure 15 shows simulation results for E [Mi+Ni]¯φ,i=1,2. The values are very similar to those for E [Mi+Ni]φi... | https://arxiv.org/abs/2503.04876v1 |
Average sample sizes: (145), (146); approximate ratio λ; Efficiency with separate sampling: (148); approaches 1 for Asmall Average number of groups: (150), (151); Efficiency with group sampling: (152). where µ1=2. This is similar to the bound (92) for RR and LRR, the only differences being the 3/2 factor in (148), aris... | https://arxiv.org/abs/2503.04876v1 |
method described in Mendo (2025, section 3), which consists of two IBS procedures with the same parameter. Specifically, to estimate log (pi/(1−pi)),i=1,2, IBS is applied until sisuccesses are obtained, which requires a random number N′ iof observations from population i, and then IBS is applied until sifailures are ob... | https://arxiv.org/abs/2503.04876v1 |
OR, but with µ1=5/4 and with a possibly different computed value for r1. Simulation results are omitted. The efficiency with group sampling is expressed by (152), as for OR. The results, plotted in Figure 19, are not the same as in that case because of the differences in µ1andr1, which affect E [G]and thus the efficien... | https://arxiv.org/abs/2503.04876v1 |
applied to the two populations (second stage), which requires N1andN2samples, where s1ands2are obtained from a previous pair of IBS processes with fixed parameters r1andr2(first stage). Then, (s1−1)(s2−1) (N1−1)(N2−1) is an unbiased estimator of p1p2, and by means of (14) a target relative error can be guaran- teed. Th... | https://arxiv.org/abs/2503.04876v1 |
and (37), and including the rounding term ξwhen computing E [s1], the second and third summands in (162) can be expressed as E VarN1 l1 s1 φ2l1l2=E[s1(1−p1)] λθ≈a1+ξ+b1E[X] λθ, (165) Var EN1 l1 s1 φ2l1l2=Var[s1] λθ≈b2 1Var[X] λθ. (166) Substituting (52) and (160) into (165) and (166), and using (39), (41), (57)... | https://arxiv.org/abs/2503.04876v1 |
the other hand, using (173) and computing Var [¯Mi]from (13), Var[E[Mi|¯Mi]] =(3−4 ¯pi)2 4(1−¯pi)2Var[¯Mi] =(3−4 ¯pi)2ri 4 ¯p2 i(1−¯pi). (175) 42 Substituting (174) and (175) into (171), Var[Mi] =(14 ¯p2 i−23 ¯pi+9)ri 4 ¯p2 i(1−¯pi)=(9−14 ¯pi)ri 4 ¯p2 i, (176) and therefore, taking into account (139), VarM1 l1 ¯φ2l1l... | https://arxiv.org/abs/2503.04876v1 |
Lehmann EL, Casella G (1998) Theory of Point Estimation, 2nd edn. Springer Mendo L (2025) Estimating odds and log odds with guaranteed accuracy. Statistical Papers 66(1):1–17. https://doi.org/10.1007/s00362-024-01639-w Mendo L, Hernando JM (2006) A simple sequential stopping rule for Monte Carlo simulation. IEEE Transa... | https://arxiv.org/abs/2503.04876v1 |
Kernel-based causal estimators for functional causal effects Yordan P. Raykov1, Hengrui Luo2, Justin D. Strait3, and Wasiur R. KhudaBukhsh4 1School of Mathematical Sciences, Horizon Digital Economy Institute, University of Nottingham, Nottingham, UK 2Department of Statistics, Rice University, USA; Lawrence Berkeley Nat... | https://arxiv.org/abs/2503.05024v3 |
et al., 2013). Causal inference methods for more complex outcomes have gained traction only recently. For example, Lin et al. (2023) considers distributional differences in generalized Wasserstein space for scalar potential outcomes, while Kurisu et al. (2024) generalizes treatment effect estimation to outcomes in geod... | https://arxiv.org/abs/2503.05024v3 |
study—a passivemonitoringstudyinvolving50participants2—weestimatetheeffectsoflevodopatherapyand disease status (PD vs. age-matched non-PD controls). Our higher-dimensional estimators uncover significant disease-related impacts on recorded digital outcomes that often remain undetectable using standard aggregated measure... | https://arxiv.org/abs/2503.05024v3 |
on how the baseline covariates Vinfluence both treatment and outcome (Hernán and Robins, 2010; Rosenbaum and Rubin, 1983; Robins et al., 1994). Specifically, Vmust satisfy either the back-door criterion or thefront-door criterion with respect to (X, Y)(see Appendix A for details). IfVsatisfies the back-door criterion w... | https://arxiv.org/abs/2503.05024v3 |
is the Doubly Robust (DR) estimator, which augments IPW with outcome models: ˆφDR−IPW=1 nnX i=1Xi(Yi−ˆm1(Vi)) ˆπ(Vi)+ ˆm1(Vi)−(1−Xi)(Yi−ˆm0(Vi)) 1−ˆπ(Vi)+ ˆm0(Vi) ,(7) where ˆm1(Vi) =E[Y|X= 1,V=Vi]andˆm0(Vi) =E[Y|X= 0,V=Vi]are estimated outcome regression models. The DR-IPW estimator is consistent if at least one of ... | https://arxiv.org/abs/2503.05024v3 |
x∈ {0,1}the population Fréchet 2-mean3of the potential outcome distribution ηxis: F(ηx) = arg min f∈FZ Fϕ2(f, g)dηx(g) = arg min f∈FZhZ Fϕ2(f, g)dηx g|V=vi dPV(v).(8) When (F, ϕ)has non-positive curvature (e.g. any Hilbert space with its induced norm) or when f7→ϕ2(f,·)is strictly convex, the minimiser in (8) exists ... | https://arxiv.org/abs/2503.05024v3 |
weighted estimator for the average treatment effect to functional or metric-valued outcomes. Let bπbe any uniformly consistent estimator of πx, and define stabilised weights bωx,i=1{Xi=x} xbπ(Vi)+(1−x) [1−bπ(Vi)].The empirical Fréchet mean ˆFn= arg min f∈FnX i=1bωx,iϕ2(f,Yi) (14) converges in probability (a.s. if the w... | https://arxiv.org/abs/2503.05024v3 |
properties unless additional regularity is imposed on the functions. For this reason, we consider Sobolev spaces of functions (Adams and Fournier, 2003). LetWk,p([0,1],R)denote the subspace of Lp([0,1])containing functions fsuch that the function fand its weak derivatives up to order khave a finite Lpnorm. When equippe... | https://arxiv.org/abs/2503.05024v3 |
Hence, for any fixed n, ϕ(ˆFn,T,ˆFn,∞)a.s.− − − − → T→∞0. (B) By Chen’s weighted strong law of large numbers (Wooldridge, 2007), bηIPW x,n⇒ηxalmost surely inW2. The same contraction property gives ϕ(ˆFn,∞, F(ηx)) =ϕ F(bηIPW x,n), F(ηx) ≤W2 bηIPW x,n, ηxa.s.− − − → n→∞0. (C) For every nandTthe ordinary triangle ineq... | https://arxiv.org/abs/2503.05024v3 |
the residuals between the effect ∆ = (∆(1) , . . . , ∆(T))Tand the point-wise estimator for effect ˆ∆∈RTare asymptotically normal: √n ˆ∆−∆d− → N (0,K), 12 where K=E ∆∆T reflects the covariance structure of the expected effect. For non-zero population effect∥∆∥2>0, we further get √n ˆφdATE−φdATE =√n ∥ˆ∆∥2− ∥∆∥2d−... | https://arxiv.org/abs/2503.05024v3 |
the plain L2norm in the SRSF domain: dFR(f1, f2) = qf1−qf2 L2([0,1]). Assuming the Fisher-Rao geometry, the set of phase-equivalence classes carries the metric (F/Γ, dFR), d FR [f1],[f2] = inf γ∈Γ qf1−qf2◦γ L2([0,1]). The quotient space F/Γisno longer a Hilbert (nor even a complete) space : orbits may accu- mulate wi... | https://arxiv.org/abs/2503.05024v3 |
both finite-dimensional ( RT) and infinite-dimensional (e.g., L2([0,1])orWk,2) outcome representations under a common RKHS-based causal estimation pipeline. Phase-shifts via alignment assumption. Throughout, we assume that for outcomes with flex- ibility of domain warping (e.g. equipped with Fisher-Rao metric ϕ), each ... | https://arxiv.org/abs/2503.05024v3 |
(Y(u1),···,Y(uT))∈RT. Following this, we adopt the convention: Y= Y1(u1)···Y1(uT) ...... Yn(u1)···Yn(uT) ∈Rn×T,vec(Y) = Y1(u1) ... Y1(uT) ... Yn(u1) ... Yn(uT) ∈RnT×1. Although this effectively treats each functional curve Yias aT-dimensional vector (which can have drawbacks; see Ramsay ... | https://arxiv.org/abs/2503.05024v3 |
be evaluated at every t∈ Twithout any additional interpolation scheme. LetYidenoteanobservedoutcomecurve. Afterinterpolationweregardeachcurveasanelement of theseparable Hilbert space HY:=Wk,2 [0,1],R with k≥1, equipped with the Sobolev inner product ⟨·,·⟩k,2. We define K: (X × H V)×(X × H V)−→ L (HY), 17 where L(HY)d... | https://arxiv.org/abs/2503.05024v3 |
(26)), generalizing those estimators to the fully functional setting. Remark 10 (Fisher-Rao kernel) .When phase variability remains after the global slice S, one can inject additional warp-invariance directly into the operator-valued kernel by replacing the output factor IHYin (29) with a Fisher–Rao similarity: kFR(f, ... | https://arxiv.org/abs/2503.05024v3 |
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