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operator with respect to the random v ar iable seq uence 𝑿 = ( 𝑋1 , 𝑋2 , … , 𝑋𝑚 ) , a seq uence of independent random v ar iables, eac h of whic h is g enerated 20 b y the identical unif or m dis tr ibution Uni ( 𝒳 ) . N ote that since 𝒳 and 𝒴 are finite sets, 𝑓𝑞 is a computable map f or all 𝑞 ∈ [ 𝑄 ] . The... | https://arxiv.org/abs/2502.12187v2 |
≔ | 𝒳 \ { 𝑥1 , 𝑥2 , … , 𝑥𝑚 } | satisfies 𝑅 ≥ | 𝒳 | − 𝑚 ≥1 2| 𝒳 | b y the definition of 𝑚 . Hence, w e ha v e HPUni ( 𝒳 ) , 𝑓𝑞( ℎ𝑞 ) =1 | 𝒳 |∑ 𝜉 ∈ 𝒳𝟙 ( ℎ𝑞 ( 𝜉 ) ≠ 𝑓𝑞 ( 𝜉 ) ) ≥1 | 𝒳 |∑𝑅 𝑟 = 1𝟙 ( ℎ𝑞 ( 𝜉𝑟 ) ≠ 𝑓𝑞 ( 𝜉𝑟 ) ) ≥1 2 𝑅∑𝑅 𝑟 = 1𝟙 ( ℎ𝑞 ( 𝜉𝑟 ) ≠ 𝑓𝑞 ( 𝜉𝑟 ) ) .(16) Thus, usin... | https://arxiv.org/abs/2502.12187v2 |
, 𝑥2 , … , 𝑥𝑚 ) . 1. The size (cardinality) is giv en b y | 𝒬𝒚 | = | ℱ𝒚 | = | 𝒴 | = 𝑝 . It is because all the maps in ℱ𝒚 retur n the same v alue 𝑦𝑗 f or the in put 𝜉𝑗 f or eac h of 𝑗 ∈ [ 𝑛 ] \ { 𝑟∗} but the y retur n dis tinct v alues in 𝒴 f or the in put 𝜉𝑟∗ . 2. F or an y 𝑞 , 𝑞′∈ 𝒬𝒚 , the maps ... | https://arxiv.org/abs/2502.12187v2 |
positiv e r eal number and le t 𝑍 be a r andom v ariable taking a v alue in [ 0 , 𝑐 ] and assume its expectation is giv en by 𝔼 𝑍 = 𝜇 ∈ ℝ . Then, f or any r eam number 𝑎 ∈ ( 0 , 𝑐 ) , t he f ollo wing ineq uality holds: Pr ( 𝑍 > 𝑎 ) ≥𝜇 − 𝑎 𝑐 − 𝑎. (21) Pr oof. N oting that 𝑍 ≤ 𝑎 is eq uiv alent to 𝑐 − 𝑍... | https://arxiv.org/abs/2502.12187v2 |
arXiv:2502.12326v1 [math.ST] 17 Feb 2025Stability Bounds for Smooth Optimal Transport Maps and their Statistical Implications Sivaraman Balakrishnan†and Tudor Manole⋄ †Department of Statistics and Data Science Machine Learning Department Carnegie Mellon University siva@stat.cmu.edu ⋄Statistics and Data Science Center M... | https://arxiv.org/abs/2502.12326v1 |
al. ,2017,Deb and Sen ,2023,Ghosal and Sen ,2022,Gunsilius ,2022,Manole et al. ,2024], and dual estimators , which are based on a characterization (see Theorems 1and2) of the optimal transport map as the gradient of a convex function ϕ0which solves the so-called semi-dual optimization problem: ϕ0∈argmin ϕ∈L1(P)/integra... | https://arxiv.org/abs/2502.12326v1 |
assumption that the sampling dis tributions were supported on thed-dimensional flat torus. Using our new stability bound we pro vide an unconditional result with a much simpler proof, completely bypassing the regular ity issues. We provide another illustration of the generality of our stability bounds by us ing them to ... | https://arxiv.org/abs/2502.12326v1 |
the set of couplings of PandQ, i.e. Π(P,Q) ={π∈ P2(Ω×Ω) :π(·×Ω) =P,π(Ω×·) =Q}. In contrast to the Monge problem ( 1), the Kantorovich problem ( 6) is always feasible, and is a linear program. From ( 6) we can derive the following intuitive optimality property : forany pair of random variables Uwith distribution P, andV... | https://arxiv.org/abs/2502.12326v1 |
=/integraldisplay /⌊a∇d⌊l·/⌊a∇d⌊l2 2dP+/integraldisplay /⌊a∇d⌊l·/⌊a∇d⌊l2 2dQ−2 inf ϕ∈L1(P)SP,Q(ϕ), where the semi-dual functional SP,Qis defined in ( 4). The semi-dual optimization problem ( 3) can be seen as a dual optimization problem to Kantorovich’s l inear program [ Villani ,2003]. The semi-dual functional is centr... | https://arxiv.org/abs/2502.12326v1 |
ondition, which endows the semi- dual optimization problem ( 3) with a desirable growth property (see ( 5)), is that there exists a 6 Brenier potential ϕ0: Ω→RfromPtoQin the sense of condition A0, which is additionally smooth and strongly convex: A1(α)The function ϕ0is convex, continuously differentiable over Ω, and sat... | https://arxiv.org/abs/2502.12326v1 |
he distributions under consideration were supported on the flat d-dimensional torus and that the target of inference was the t orus OT map (the OT map with respect to a modified Euclidean metric) . In contrast, building upon our improved stability bound we obtain sharp results fo r the usual plugin estimators of the OT m... | https://arxiv.org/abs/2502.12326v1 |
each of our stability bounds as direct consequences of a single unified result. We n ow briefly discuss a few highlights of the proof of Theorem 3before providing some historical context for our work. Proof Highlights for Theorem 3:We give a full proof of this result in Appendix C, but discuss some interesting aspects of... | https://arxiv.org/abs/2502.12326v1 |
from Pto/hatwideQ, andT0=∇ϕ0is the OT map from Pto Q, with the potential ϕ0satisfying condition A1(α), then: /⌊a∇d⌊lT0−/hatwideT/⌊a∇d⌊l2 L2(P)/lessorsimilarEX∼P/⌊a∇d⌊l/hatwideT(X)−X/⌊a∇d⌊l2 2−EX∼P/⌊a∇d⌊lT0(X)−X/⌊a∇d⌊l2 2. 10 This result suggests that the excess transport cost of a sub- optimal transport map grows in pr... | https://arxiv.org/abs/2502.12326v1 |
smoot h. Beyond the statistical applications that we have in mind, bou nds of type ( 17) are highly sought after, as the quantity /⌊a∇d⌊l/hatwideT−T0/⌊a∇d⌊lL2(P)is itself a metric between /hatwideQandQ, which can be viewed as a proxy for the Wasserstein distance. This quant ity is sometimes known as the linearized Wass... | https://arxiv.org/abs/2502.12326v1 |
Geer ,2000]. Such arguments cannot directly be translated to our setti ng since/hatwidePncannot be taken to be the empirical measure, however for some choices of this estimator, such as linear smoothers, the above empirical pr ocess can in principle still be controlled using known bounds on suprema of smoothed empi ric... | https://arxiv.org/abs/2502.12326v1 |
which satisfies conditions A1(/tildewideα)–A2(/tildewideβ)for some /tildewideα,/tildewideβ >0. To the best of our knowledge, however, there is no known quantitative relation between th e resulting parameters /tildewideα,/tildewideβand the original problem parameters M,d,γ,s , thus we prefer to formulate the above assump... | https://arxiv.org/abs/2502.12326v1 |
over its support. In what follows, we s ignificantly weaken these assumptions, showing that the nearest-neighbor estimator is minimax optimal in estimating bi-Lipschitz optimal transport maps, subject only to mild m oment constraints. 14 Concretely, given i.i.d. samples X1,...,X n∼PandY1,...,Y m∼Q, let/hatwideπnmbe (the... | https://arxiv.org/abs/2502.12326v1 |
Schwarz inequality. This result shows that, up to a logarith mic factor, the nearest neighbor estimator achieves the convergence rate (n∧m)−2/dfor estimating a bi-Lipschitz optimal transport map T0, and is therefore minimax optimal [ Hütter and Rigollet ,2021]. Our result merely assumes that the measures admit 4+ǫmomen... | https://arxiv.org/abs/2502.12326v1 |
methods are not known to be minimax optimal in the same level of generality as plugin or dual estimators, they typically e njoy more favorable computational properties. Our results relied on smoothness and strong convexity assum ptions on the underlying Brenier potentials. These assumptions can be relaxed in vari ous w... | https://arxiv.org/abs/2502.12326v1 |
Chernozhukov, A. Galichon, M. Hallin, and M. Henry. Monge –Kantorovich depth, quantiles, ranks and signs. The Annals of Statistics , 45:223–256, 2017. S. Chewi, J. Niles-Weed, and P. Rigollet. Statistical optim al transport. Ecole d’Eté de Prob- abilités de Saint-Flour XLIX. arXiv:2407.18163 , 2024. N. Deb and B. Sen. ... | https://arxiv.org/abs/2502.12326v1 |
smooth optimal transport maps. The Annals of Statistics , 49:1166–1194, 2021. L. V. Kantorovich. On the translocation of masses. In Dokl. Akad. Nauk. USSR (NS) , vol- ume 37, pages 199–201, 1942. 19 L. V. Kantorovich. On a problem of Monge. In CR (Doklady) Acad. Sci. URSS (NS) , volume 3, pages 225–226, 1948. M. Ledoux... | https://arxiv.org/abs/2502.12326v1 |
Birkhäuser, 2015. J. Segers. Graphical and uniform consistency of estimated o ptimal transport plans. arXiv preprint arXiv:2208.02508 , 2022. A. Vacher and F.-X. Vialard. Convex transport potential sel ection with semi-dual criterion. Advances in Neural Information Processing Systems 36 , 2022. A. Vacher, B. Muzellec, ... | https://arxiv.org/abs/2502.12326v1 |
PandQ. Then let us define the random variables X∼P, Y∼Q, U 1,U2,U3∼/hatwideP, V 1,V2,V3∼/hatwideQ, with the following joint distributions: (U1,Y)∼π/hatwideP,Q,(X,V1)∼πP,/hatwideQ (X,U2)∼πP,/hatwideP,(Y,V2)∼πQ,/hatwideQ (X,Y)∼πP,Q,(U3,V3)∼π/hatwideP,/hatwideQ. These joint distributions are summarized in the following fi g... | https://arxiv.org/abs/2502.12326v1 |
r-covering number of the set S⊆Rdwith respect to the Euclidean distance. We partition Rdinto the sets: S0=B0,1, Sj=B0,2j\B0,2j−1, j= 1,2,.... Recalling that we denote by nn (x)the nearest neighbor of xin a sample X1,...,X n, our goal is to bound: E[e2] =EX,X1,...,Xn∼P/⌊a∇d⌊lX−nn(X)/⌊a∇d⌊l2 2=/integraldisplay∞ 0P(/⌊a∇d⌊... | https://arxiv.org/abs/2502.12326v1 |
Unsupervised optimal deep transfer learning for classification under general conditional shift Junjun Lang1and Yukun Liu∗1 1KLATASDS-MOE, School of Statistics, East China Normal University, Shanghai 200062, China Abstract Classifiers trained solely on labeled source data may yield misleading results when applied to unl... | https://arxiv.org/abs/2502.12729v1 |
1, YP 1),···,(XP nP, YP nP)i.i.d∼P(X,Y), XQ 1,···, XQ nQi.i.d∼QX, (1) where QXis the distribution function or probability measure of Xin the target domain. Because the labels of the target data are unavailable, it is challenging, if not impossible, to learn a classifier with theoretical guarantee for the target data ba... | https://arxiv.org/abs/2502.12729v1 |
source domain using deep neural networks (DNNs). Our contributions are outlined as follows: 1. We introduce a novel distribution shift assumption, namely the GCS model presented in (2), which encompasses the traditional label shift assumption as a special case. The GCS model not only provides greater flexibility compar... | https://arxiv.org/abs/2502.12729v1 |
transfer learning for classification under label shift, the estimation of the label distribution (or class proportions) of the target data, as well as the distribution shift function between the target and source domains, represents a pivotal task. Broadly speaking, existing methodologies for this purpose can be catego... | https://arxiv.org/abs/2502.12729v1 |
is the deep neural network (DNN), which has driven substantial advancements over the past few decades (Szegedy et al. , 2015; Sarikaya et al. , 2014; Miao and Miao, 2018). Mathematically, a DNN is a composite function comprising simplistic functions layered in multiple tiers, imparting it with the remarkable ability to... | https://arxiv.org/abs/2502.12729v1 |
in S= (X × Y )nP× XnQ, and use H={˜h:X 7→ Y} to denote the set of all classifiers on X. For any classifier ˜h∈ H, its excess-risk is defined as EQ(˜h) =Q(Y̸=˜h(X))−Q Y̸=f∗ Q(X) , (3) where f∗ Q(X) = argmaxl∈YηQ,l(X) is the Bayes classifier for the target data. Let∥ · ∥ denote the standard Euclidean norm. We employ λm... | https://arxiv.org/abs/2502.12729v1 |
underlying problem are presumed to belong to a H¨ older class, such as Ha d(D, M) (Cai and Wei, 2021; Maity et al. , 2022; Reeve et al. , 2021). Under this assumption, the minimax optimal rate of convergence for the classifier on the target data, in terms of excess-risk, hinges on the smoothness index aand the dimensio... | https://arxiv.org/abs/2502.12729v1 |
maximum likelihood estimator, named ˆπQ. The classifier is constructed by combining ˆηPandˆπQthrough Bayes’ formula. A notable advantage of our method is its innovative circumvention of the necessity to estimate the unknown function h(·) in the GCS model (2). 3.1 DNN estimator for ηP Since ηPrepresents the regression o... | https://arxiv.org/abs/2502.12729v1 |
the complexity and flexibility of the neural networks in F(K, s,p, D), as well as those inherent in the conditional probabilities bηP(x). These hyperparameters can be selected using cross-validation or held-out validation in practical applications. 3.2 Pseudo maximum likelihood estimator of πQ In the second step, we pr... | https://arxiv.org/abs/2502.12729v1 |
PMLE ˆπQofπQ. To facilitate the ensuing discussion, we introduce additional notations. For any two sequences ( anP)nPand ( bnP)nP, we denote anP≲bnPif there exists a positive constant Csuch that anP≤CbnPfor all nP. The notation anP≍bnP signifies that both anP≲bnPandbnP≲anP. We define ξ0(x) = (α0 1+ϕ0 1(x),···, α0 k−1+ ... | https://arxiv.org/abs/2502.12729v1 |
substantial, the term γnPlog2nPbecomes the dominant factor, whereas for smaller nQ, the term n−1/2 Qprevails. This finding presents a more intuitive and streamline form compared to the result in Iyer et al. (2014) [e.g.,Theorem 1]. Theorem 2 indicates that the convergence rate of ˆ πQ,lis no faster than n−1/2 P. Becaus... | https://arxiv.org/abs/2502.12729v1 |
which, up to a logarithm factor, coincides with the optimal minimax rate in Corollary 1 of Stone (1985) for additive (mean) regression. •When ϕ1,jis a linear function of x(j)forj∈[d], the GAM reduces to the single index model (Ichimura, 1993), and the convergence rate contribution of the source data is of the order n−θ... | https://arxiv.org/abs/2502.12729v1 |
immediately deduce the minimax optimal rate for all classifiers and establish the minimax optimality of the proposed classifier ˆf. Corollary 1. Assume the conditions in Theorems 3 and 4. Then, the minimax optimal rate of convergence for all classifiers from StoHisγnP+n−1/2 Qup to a logarithm factor log2(nP), and the p... | https://arxiv.org/abs/2502.12729v1 |
Specifically, ˆPX|Y=yrepresents the kernel-density estimator of PX|Y=yfory= 1,2, and ˜ πQ,1be the PMLE of πQ,1obtained by replacing the estimates of PX|Y=1/PX|Y=2with the ratio of their kernel-density estimators. Maity-PC estimates ηQ,1(x) by ˜ηQ,1(x) =˜πQ,1ˆPX|Y(x|1) ˜πQ,1ˆPX|Y(x|1) + (1 −˜πQ,1)ˆPX|Y(x|2). •Saerens: T... | https://arxiv.org/abs/2502.12729v1 |
label shift and 23 GCS assumptions are met. In these scenarios, the conditional distributions of X(i)given Yare discrete and continuous, respectively. However, in Scenario III, the label shift assumption is violated, while our GCS assumption remains valid with h(x) = exp(2 + β⊤x), where β= (−1,−1,−1,−1)⊤. Let p(t|y) re... | https://arxiv.org/abs/2502.12729v1 |
0.5); Lower panel: the unbalanced case ( πQ,1= 0.25) The class proportions πQ,1are crucial for the Bayes classifier in the target population. Among the classification methods being compared, three estimators for πQ,1were utilized: (1) DNN: The proposed PMLE ˆ πQ,1, where the density ratio PX|Y=1/PX|Y=2is estimated usin... | https://arxiv.org/abs/2502.12729v1 |
the three estimators of πQ,1under comparison with nQ= 400. Upper panel: the balanced case ( πQ,1= 0.5); Lower panel: the unbalanced case ( πQ,1= 0.25) Alzheimer’s Disease Dataset of Kharoua (2024). This dataset encapsulates comprehensive health information for 2,149 patients, encompassing demographic specifics, lifesty... | https://arxiv.org/abs/2502.12729v1 |
Section 5 to predict the diagnosis results of the patients in the target domain. The samples in the target domain are randomly partitioned into training and validation sets, with the training set consisting of 100 p% of the total target data, where p= 0.5, 0.6, or 0 .7. Although the target data here are all labeled, we... | https://arxiv.org/abs/2502.12729v1 |
histogram-based visual comparison method is employed to justify the GCS assumption; however, it lacks the rigor of quantitative analysis. This thereby raises the question of how to construct a valid hypothesis test for checking the GCS assumption based on labelled source data and unlabelled target data. Intuitively, a ... | https://arxiv.org/abs/2502.12729v1 |
functions. The Annals of Statistics ,35(6), 2589 – 2619. Huang, J.-T., Li, J., Yu, D., Deng, L., and Gong, Y. (2013). Cross-language knowledge transfer using multilingual deep neural network with shared hidden layers. In 2013 IEEE international conference on acoustics, speech and signal processing , pages 7304–7308. IE... | https://arxiv.org/abs/2502.12729v1 |
Journal of Advanced Computer Science and Applications , 9(10). Pedamonti, D. (2018). Comparison of non-linear activation functions for deep neural networks on mnist classification task. Preprint, arXiv:1804.02763 . Qin, J. (1998). Inferences for case-control and semiparametric two-sample density ratio models. Biometrik... | https://arxiv.org/abs/2502.12729v1 |
Existence of Direct Density Ratio Estimators Erika Banzato∗, Mathias Drton†, Kian Saraf-Poor‡, and Hongjian Shi§ Abstract Many two-sample problems call for a comparison of two distributions from an exponential family. Density ratio estimation methods provide ways to solve such problems through direct estimation of the ... | https://arxiv.org/abs/2502.12738v1 |
al. (2017) but is restricted to the Gaussian setting. For more general settings involving exponential families, a framework to solve two-sample prob- lems emerges from the literature on density ratio estimation (Sugiyama, Suzuki and Kanamori, 2012). This is particularly promising for differential network analysis with ... | https://arxiv.org/abs/2502.12738v1 |
measure of the exponential family. IfPhas density p(x) =f(x;θ(p)) and Qhas density q(x) =f(x;θ(q)), both with respect to the dominating measure ν, then the parameter of interest is the difference vector ∆=θ(p)−θ(q). (2.2) The difference vector emerges when considering the density ratio p(x) q(x)≡f(x;θ(p)) f(x;θ(q))=Z(θ... | https://arxiv.org/abs/2502.12738v1 |
¯txandTyare obtained from the samples XandY, respectively. To ease the notation, we write ¯tx= (¯tx 1, . . . , ¯tx k)⊤with each entry defined as ¯tx v:=1 nxnxX i=1tv(Xi), v = 1, . . . , k, (3.1) and write t(Yj)≡ty j= (ty 1j, . . . , ty kj)⊤with each entry given by ty vj:=tv(Yj), v = 1, . . . , k, j = 1, . . . , n y. (3... | https://arxiv.org/abs/2502.12738v1 |
global minimum at ∆0is also a local minimum since Rk, the domain of ∆, is an open set. According to the interior extremum theorem, when the gradient ∇ℓKL(∆) exists, it must be the zero vector at a local minimum. Here, the v-th component of the gradient ∇ℓKL(∆) is given by ∂ℓKL(∆) ∂∆v=−¯tx v+Pny j=1expn ∆⊤ty jo ty vj Pn... | https://arxiv.org/abs/2502.12738v1 |
Lemma 1 concludes that f∗attains a strictly positive minimum on Sd−1∩U: δ:= min ∆∈Sd−1∩Uf∗(∆)>0. (3.10) We write ˜∆:=∆/∥∆∥for∆∈Uif∆̸=0. Then for ∆∈U, when ∥∆∥ → ∞ , lim ∥∆∥→∞ℓKL(∆) = lim ∥∆∥→∞log 1 nynyX j=1exp(∆⊤uj) = lim ∥∆∥→∞log 1 nynyX j=1exp ∥∆∥˜∆⊤uj ≥lim ∥∆∥→∞log1 nymax 1≤j≤nyexp ∥∆∥˜∆⊤uj = lim ∥∆... | https://arxiv.org/abs/2502.12738v1 |
the ∥ · ∥ #-norm distance of ¯txto the polytope spanned by the ty 1, . . . ,ty ny. Proof of Theorem 2. In view of Theorem 1, assertion (i) follows immediately. The second part of assertion (ii) is also obvious, while the first part follows from Lemma 2 easily. It remains to show assertion (iii). First, we give an alter... | https://arxiv.org/abs/2502.12738v1 |
two cases have different dominating measures but share the form of the statistics with tuv(xu, xv). We, thus, present simulation results focused on Gaussian models. Gaussian graphical models are based on multivariate normal distributions with density f(x;Θ) =det(Θ)1/2 (2π)m/2exp −1 2x⊤Θx ,x∈Rm, where Θ∈Rm×mis the pre... | https://arxiv.org/abs/2502.12738v1 |
4, we observe that the increased magnitude of the difference parameter θ∗ 1leads to an increase in λ#. In Figure 4, even for d= 20, there is a non-negligible probability that λLiuis smaller than λ#. In this case, one would have to find a larger regularization parameter to infer differences between two Gaussian graphica... | https://arxiv.org/abs/2502.12738v1 |
0.4, θ∗ 1=−0.4. 14 10 20 40 800.0 0.4 0.8 d (number of changed edges)λ#m = 144, np = 248 0.00 0.00 0.01 0.99 λLiu = 0.354 10 20 40 800.0 0.4 0.8 d (number of changed edges)λ#m = 144, np = 497 0.00 0.00 1.00 1.00 λLiu = 0.25 10 20 40 800.0 0.4 0.8 d (number of changed edges)λ#m = 144, np = 745 0.00 0.17 1.00 1.00 λLiu =... | https://arxiv.org/abs/2502.12738v1 |
of changed edges)λ#m = 400, np = 300 0.43 0.61 0.56 0.71 λLiu = 0.353 10 20 40 800246 d (number of changed edges)λ#m = 400, np = 599 0.49 0.59 0.72 0.91 λLiu = 0.25 10 20 40 800246 d (number of changed edges)λ#m = 400, np = 899 0.66 0.73 0.87 0.96 λLiu = 0.204 10 20 40 800246 d (number of changed edges)λ#m = 400, np = ... | https://arxiv.org/abs/2502.12738v1 |
asnp/logmis required to be larger than the rate of d2. This could also be attributed to the fact that, in our simulations, nqis not sufficiently large — specifically, nq= 0.01n2 pwas used in some experiments in Liu et al. (2017), as suggested by Theorem 1. However, this implies that nqwould need to exceed 10 ,000 when ... | https://arxiv.org/abs/2502.12738v1 |
University Press, Cambridge, second edition. Kim, B., Liu, S., and Kolar, M. (2021). Two-sample inference for high-dimensional Markov networks. J. R. Stat. Soc. Ser. B. Stat. Methodol. , 83(5):939–962. Liu, S., Quinn, J. A., Gutmann, M. U., Suzuki, T., and Sugiyama, M. (2014). Direct learning of sparse changes in Marko... | https://arxiv.org/abs/2502.12738v1 |
Simpson’s Paradox with Any Given Number of Factors Guisheng Dai1and Weizhen Wang2∗ 1School of Mathematics, Beijing Normal University, Beijing 100875, P. R. China 2Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, U.S.A February 19, 2025 Abstract Simpson’s Paradox is a well-known pheno... | https://arxiv.org/abs/2502.12864v1 |
A), as well as ( B1, ..., B m) for some integer m∈[0, n]. For example, Xis the status of a patient after seeing a doctor in a hospital: X1(cured) and X0(not cured); Ais the factor of two hospitals: A1(a local clinic) and A0(a national hospital); B1is the health condition of patient: B1,1(severe) and B1,0(not severe); B... | https://arxiv.org/abs/2502.12864v1 |
1 only, i.e., (1) and (2), the direction of the inequali- ties changes once, and this represents the classic Simpson’s Paradox. When considering m= 0,1, . . . , n for a general n > 1, i.e., (1) through (5), the direction of the inequalities changes n times. This phenomenon may occur in observational studies where the a... | https://arxiv.org/abs/2502.12864v1 |
four positive constants x1, x2, y1,andy2, letθxandθybe the angles of vectors (x2, x1)and(y2, y1)in the R2plane, respectively. Then the following three are equivalent: i)x1 x1+x2>y1 y1+y2; ii)tan(θx)>tan(θy); iii)θx> θy. Proof . Note tan( θx) =x1/x2, tan( θy) =y1/y2, and both θxandθyare in (0 , π/2). The equivalence bet... | https://arxiv.org/abs/2502.12864v1 |
d2=d0−d1.(17) Then, (10) and (12) hold and a1 a1+b1<c1 c1+d1,a2 a2+b2<c2 c2+d2. ii) If a0/(a0+b0)< c 0/(c0+d0), let As= tan(arctan(a0 b0) 2), Bs= tan(3π 8+arctan(c0 d0) 4), a1=As(a0−Bsb0) As−Bs, b1=a0−Bsb0 As−Bs, a2=a0−a1, b2=b0−b1; Cs= tan(arctan(a0 b0) 4), Ds= tan(π 4+arctan(c0 d0) 2), c1=C... | https://arxiv.org/abs/2502.12864v1 |
2, b(1) 2, c(1) 2, d(1) 2), respectively. Then, we obtain (a(2) 1, b(2) 1, a(2) 2, b(2) 2) and ( c(2) 1, d(2) 1, c(2) 2, d(2) 2) for the second paradox and ( a(3) 1, b(3) 1, a(3) 2, b(3) 2) and 12 A1 P(X1|A1) 0.81 A1¯B1 A1¯B0 P(X1¯Bi|A1) 0.0263 0.7737 P(X1|A1¯Bi) 0.200520.89043 A1¯B1,1 A1¯B1,0 A1¯B0,1 A1¯B0,0 P(X1¯Bi,j... | https://arxiv.org/abs/2502.12864v1 |
J., Hammel, E. A. and O’Connell, J. W. (1975). Sex bias in graduate admissions: data from Berkeley. Science 187(4175) : 398–404. [2] Blyth, C. R. (1972). On Simpson’s Paradox and the sure-thing principle. Journal of the American Statistical Association .67(338): 364–366. [3] Charig, C. R., Webb, D. R., Payne, S. R., an... | https://arxiv.org/abs/2502.12864v1 |
arXiv:2502.12912v1 [stat.OT] 18 Feb 2025A S IMPLIFIED AND NUMERICALLY STABLE APPROACH TO THE BG/NBD C HURN PREDICTION MODEL Dylan Zammit Gaming Innovation Group dylan.zammit@gig.comChristopher Zerafa Gaming Innovation Group christopher.zerafaa@gig.com February 19, 2025 ABSTRACT This study extends the BG/NBD churn proba... | https://arxiv.org/abs/2502.12912v1 |
patterns, such as the iGaming sector. In these cont exts, these definitions of churn become impractical, as customers may naturally have long periods of inactivity bet ween major sporting events or seasons. FEBRUARY 19, 2025 To address these limitations, a more practical definition of churn is required. One such approach... | https://arxiv.org/abs/2502.12912v1 |
paper well. This would provide a new modelling perspective of churn prediction in the industry, overcomin g problems such as seasonal and event-driven activity inher - ent in the iGaming industry. This statistical model also pro vides a more explainable implementation, which is often required by businesses to enable in... | https://arxiv.org/abs/2502.12912v1 |
an alternate definition of what it means for a customer to churn, providing a simplified expression of the special case of Equation 34 in [3]. Moreover, we offer a simple transformation involving exponentials and logarithms, providing numerical stability and tractabili ty. This expression was implemented in the open-sour... | https://arxiv.org/abs/2502.12912v1 |
Asymptotic Optimism of Random-Design Linear and Kernel Regression Models Hengrui Luo1and Yunzhang Zhu2 1Department of Statistics, Rice University; Lawrence Berkeley National Laboratory. hrluo@lbl.gov;hrluo@rice.edu 2Amazon∗; Department of Statistics, the Ohio State University. ryzhux@gmail.com Abstract We derived the c... | https://arxiv.org/abs/2502.12999v2 |
for Simulations 24 B Simulation Monte-Carlo Sample Sizes 31 C Proof of Proposition 1 34 D Proof of Proposition 2 36 E Calculation for (3.23) 37 F Proof of Theorem 3 38 G Proof of Corollary 5 44 H Proof of Corollary 7 44 I Proof of Corollary 8 45 J Computational Examples using Corollary 8 47 K Proof of Theorem 9 48 L Pr... | https://arxiv.org/abs/2502.12999v2 |
VC dimension (e.g., neural networks (NN), supported vector machines (Vapnik, 1999)), the minimal length principle measures (e.g., encoders, decoders (Rissanen, 2007)) and the degree of freedom for classical statistical models (e.g., linear and ANOVA models (Ravishanker et al., 2002)). However, there is not a well-accep... | https://arxiv.org/abs/2502.12999v2 |
is independently regenerated to reflect the randomness in response. The random-X prediction error can be written as: ErrRX:=EyEx∗,y∗|X,y1 nnX i=1∥yi,∗−ˆµ(xi,∗)∥2 2(1.4) ErrRX≈1 NX yconditioned on XEx∗,y∗|X,y1 nnX i=1ℓ(yi,∗,ˆµ(xi,∗)). (1.5) In this setting, the error Err RXinvestigates the input locations where the mode... | https://arxiv.org/abs/2502.12999v2 |
on the loss function does not hold for the NN, otherwise we would expect the single-descent instead of the double-descent. In linear regression models, Hastie et al. (2020) pointed out that when the training dataset is not fixed, in an asymptotic setting, the double-descent phenomena even exists for linear regression m... | https://arxiv.org/abs/2502.12999v2 |
2 and link it to predictive model complexity measure. Detailed examples and our main results concerning linear models are presented in sections 3, followed by discussions in section 4. 2 Optimism Measures of Model Complexity After identifying the first descent phenomena caused by the variance-bias trade-off (Belkin et ... | https://arxiv.org/abs/2502.12999v2 |
first two terms and have following expression, which is independent of signal µ: OptRX=σ2 ϵ ∥H∥2 2−1 ntrace HTH +1 ntrace (2H) (2.3) =σ2 ϵ ∥H∥2 2−1 ntrace (H) . (2.4) This is the closed form expression when the model fitting procedure can be described as a linear projection method with a certain choice of basis f... | https://arxiv.org/abs/2502.12999v2 |
optimism Opt RX≥0. The trained model ˆµtrainis defined in the same functional space Fn, which is independent of 9 {xi, yi}n i=1and{x∗,i, y∗,i}n i=1but may depend on sample size n: ˆµtrain= arg min f∈FnTX= arg min f∈Fn1 nnX i=1ℓ(f(xi), yi), (3.2) For the optimism defined for ˆµtrainwe have EXOptRX≥0. Proof.See Appendix ... | https://arxiv.org/abs/2502.12999v2 |
regression we have a formula for the scaled optimism in (3.4): Corollary 7. When x∗∼N(0,1)andX∼N(0,1)we have a special form of (3.4)using an independent standard normal random variable Z: nEXOptRX 2σ2 ϵZ∼N(0,1)≍3 (EZµ(Z))2+EZ2µ(Z)2−2EZ3µ(Z)·EZµ(Z) σ2 ϵ+ 1 + Op1 n1/2 . (3.8) 11 Proof.See Appendix H. We can further wri... | https://arxiv.org/abs/2502.12999v2 |
λ x∗y∗− x∗xT ∗+λI Σ−1 λη 2 2+Op1 n3/2 . EXErrTX=1 nEX y−Xˆβ 2 2 =Ex∗ y∗−xT ∗Σ−1 λη2+ηTΣ−1 λη−ηTΣ−1 λΣΣ−1 λη −1 nEx∗ Σ−1/2 x∗y∗− x∗xT ∗+λI Σ−1 λη 2 2+Op1 n3/2 . The expected random optimism for the least squares estimator is EXOptRX=1 nEh Σ−1 λΣΣ−1 λ+Σ−1 λ x∗y∗− x∗xT ∗+λI Σ−1 λη 2 2i +ηTΣ−1 λ(Σ−Σλ)Σ... | https://arxiv.org/abs/2502.12999v2 |
ϕ:Rd→Rq,K(X,X) = JK(xi,xj)Kn i,j=1=ΦTΦ∈Rq×qis the Gram matrix of the kernel K:Rd×Rd→R,K(x∗,X)is the 1×nkernelized vector (K(x∗,x1),···, K(x∗,xn))andλis the regularization parameter. The following assumption holds if the feature mapping ϕis Lipschitz bounded and Assumption A2 holds. Assumptions A3. Letˆηϕ=1 nΦTy(X) =1 n... | https://arxiv.org/abs/2502.12999v2 |
a model complexity measure, the NN may have a very low complexity measure value because the NN usually generalize well even when trained on one set but tested on another. In the subsequent simulation experiments, we set the N= 100andntrain=ntest= 1000 unless otherwise is stated. We consider the following signal functio... | https://arxiv.org/abs/2502.12999v2 |
scaled expected optimism shown in Figure 3.2), this would depend on the specific form of the signal function 17 and how it interacts with the xvariables in the model. If the signal function does not accurately capture the true relationship between the variables for certain values of k, then the model would be mis-speci... | https://arxiv.org/abs/2502.12999v2 |
shapes of signals. In each panel, the x-axis is the changing k, y-axis is the (scaled) optimism computed from NMC= 10 ,000. The model NTK_0means kernel regression using (3.19) (See Algorithm 5 in Appendix A) with no regularization; Ridge_λmeanslinearridgeregressionwithdifferentregularizationparamters λ. Figure 3.3: Dif... | https://arxiv.org/abs/2502.12999v2 |
be used to study more com- plex models. By analyzing the asymptotic expressions for the optimism, we may gain more insights into the factors that drive the double descent phenomenon and understand how different models behave in the underparameterized, interpolation threshold, and overparam- eterized regions. Our paper ... | https://arxiv.org/abs/2502.12999v2 |
Li, and Ruosong Wang. Fine-grained analysis of optimization and generalization for overparameterized two-layer neural networks. In International Conference on Machine Learning , pages 322–332. PMLR, 2019. Mikhail Belkin, Daniel J Hsu, and Partha Mitra. Overfitting or perfect fitting? risk bounds for classification and ... | https://arxiv.org/abs/2502.12999v2 |
Degrees of Freedom in Linear Regression. arXiv:2106.15682 [math, stat] , June 2021. URL http://arxiv.org/ abs/2106.15682 . Hengrui Luo, Giovanni Nattino, and Matthew T Pratola. Sparse additive gaussian process regression. Journal of Machine Learning Research , 23(61):1–34, 2022. Hengrui Luo, Younghyun Cho, James W Demm... | https://arxiv.org/abs/2502.12999v2 |
in systematic ways to estimate both the training performance and the generalization error. Algorithm 2, referred to as the “hold-out” generalization, creates a single split of the dataset into a training portion and a test portion, trains the model on the training set over a specified number of epochs, and then evaluat... | https://arxiv.org/abs/2502.12999v2 |
mean train and test losses. We use it to study how different levels of noise in the training data (as shown in Figure B.1) and different signal complexities (controlled by parameter k) affect the models’ learning process (as shown in Figure B.2) andtheirabilitytogeneralizefromtrainingdatatounseentestdata(asshowninFigur... | https://arxiv.org/abs/2502.12999v2 |
the average Ltrainand the average Ltestover all runs, as well as any variability mea- sures. 28 Algorithm 4 3-layer NN construction in python using pytorch. The network consists of linear input layer, with ReLU of 50 outputs; hidden layer with ReLU of 50 outputs; output layer with ReLU of 1 output. •class SimpleNN(nn.M... | https://arxiv.org/abs/2502.12999v2 |
relationship between the variables is not linear or does not follow the specified signal function, then the model would be mis-specified. We choose the MC sample size to be 10000, which seems to guarantee the accuracy of estimated model optimism for the signal we considered. Trends in optimism versus epoches. Figure B.... | https://arxiv.org/abs/2502.12999v2 |
that 1 n∥µ(X)−Hµ(X)∥2 2=1 n µ(X)Tµ(X)−2µ(X)THµ(X) +µ(X)THTHµ(X) Note that HTH=H =1 nµ(X)T(I+HTH)µ(X)−2 ntrace µ(X)THµ(X) =1 nµ(X)T(I+HTH)µ(X)−2 ntrace βTXTHXβ (C.4) and use the fact that EXTAX =trace (AVarX) + (EX)TAEX, 2 nnX i=1 EyT ihT iy −2 ntrace µ(X)THµ(X) =2 nnX i=1 EyT ihT iy −2 ntrace EyTHEy =1 nt... | https://arxiv.org/abs/2502.12999v2 |
y∗,i)! . For the above proof to hold, we emphasize that in (D.1) the functional space Fnmust be the same and independent of training and testing dataset, although they can vary with the sample size n=ntest. E Calculation for (3.23) When k <0.5, the calculation follows as (EXxµ(x))2= EXx·0.5−k 0.5max (0 , x)2 =Z∞ 0(1... | https://arxiv.org/abs/2502.12999v2 |
y∗−xT ∗Σ−1η2 xT ∗Σ−1x∗ +Op1 n2 . (F.9) 40 For training error, we recall the definition of hat matrix H=X XTX−1XT= XˆΣ−1XTandHTH=H, and take yet another arbitrary pair (x∗, y∗)as an indepen- dent copy of X,y: 1 nEX y−Xˆβ 2 2 =1 nEXyT(I−H)y =1 nEX yTy−n·ˆηTˆΣ−1ˆη =Ex∗y2 ∗−EXˆηTˆΣ−1ˆη (F.10) =Ex∗h y∗−xT ∗Σ−1η2+... | https://arxiv.org/abs/2502.12999v2 |
i=0Aixi+1!2 = ∞X i̸=1AiEXxi+1+A1EXx2!2 EXx2µ(x)2=EXx2 ∞X i=0Aixi+1!2 =EX ∞X i=0,j=0AiAjxi+j+2! =EX ∞X i̸=1AiEXxi+1! ∞X j̸=1AjEXxj+1! + 2A1 ∞X j̸=1Ajxj+3! + 2A2 1x4! EXx3µ(x) =EX ∞X i=0AiEXxi+3! =∞X i̸=1AiEXxi+3+A1EXx4 EXx3µ(x)·EXx′µ(x′) = ∞X i̸=1AiEXxi+3+A1EXx4! ∞X i̸=1AiEXxi+1+A1EXx2! Then, EXn 2σ2 ϵ·OptRX ≍1 2σ2 ϵ 6... | https://arxiv.org/abs/2502.12999v2 |
nEXh Ex∗ y∗−xT ∗Σ−1 kη+xT ∗ Σ−1 k−Σ−1 η 2 2 xT ∗ Σ−1 k−Σ−1 k+Σ−1 x∗i +Op1 n3/2 (K.1) =2 nEXEx∗h y∗−xT ∗Σ−1 kη 2 2+ 2 xT ∗ Σ−1 k−Σ−1 η y∗−xT ∗Σ−1 kη + xT ∗ Σ−1 k−Σ−1 η 2 2 (K.2) · xT ∗Σ−1 kx∗ + xT ∗ Σ−1−Σ−1 k x∗ +Op1 n3/2 Then, we suppose that Σ=UΛVTis the singular value decomposition with o... | https://arxiv.org/abs/2502.12999v2 |
in (L.4): Ex∗h xT ∗ Σ−1 λη−ˆΣ−1 λˆηi2 =Ex∗ Σ−1 λη−ˆΣ−1 λˆηT x∗xT ∗ Σ−1 λη−ˆΣ−1 λˆη = Σ−1 λη−ˆΣ−1 λˆηT Σ Σ−1 λη−ˆΣ−1 λˆη (L.5) Unlike the cross-product term ηT−ηTΣ−1Σ = 0in (F.1) vanishes, we noticed that ηT− ηTΣ−1 λΣ̸= 0for any λ̸= 0. Ex∗ y∗−xT ∗Σ−1 ληT·xT ∗ Σ−1 λη−ˆΣ−1 λˆη =Ex∗ y∗xT ∗−ηTΣ−1 λx∗xT ∗ ... | https://arxiv.org/abs/2502.12999v2 |
y∗−xT ∗ˆβ 2 2=Ex∗ y∗−xT ∗(Σ+λI)−1η2 +1 nEx∗ Σ1/2Σ−1 λ x∗y∗− x∗xT ∗+λI Σ−1 λη 2 2 +2 n ηT−ηTΣ−1 λΣ Σ−1 λ Ex∗Σ−1 λ x∗xT ∗−Σ Σ−1 λ Σ−x∗xT ∗ Σ−1 λx∗y∗ +2 n ηT−ηTΣ−1 λΣ Σ−1 λh Ex∗Σ−1 λ x∗xT ∗−Σ x∗xT ∗+λI−1(x∗y∗−η)i +Op1 n3/2 by the magnitude in (L.13) . (L.14) Similarlyfortrainingerror,werecallthedefin... | https://arxiv.org/abs/2502.12999v2 |
2 2i +1 nE Σ−1/2 λ x∗y∗− x∗xT ∗+λI Σ−1 λη 2 2 +4 n ηT+ηTΣ−1 λ−ηTΣ−1 λΣΣ−1 λ +h Ex∗Σ−1 λ x∗xT ∗−Σn Σ−1 λ Σ−x∗xT ∗ Σ−1 λx∗y∗+ x∗xT ∗+λI−1(x∗y∗−η)oi −ηTΣ−1 λη+ηTΣ−1 λΣΣ−1 λη+Op1 n3/2 (L.22) But we have the following computation: Ex∗Σ−1 λ Σ−x∗xT ∗ Σ−1 λx∗y∗+ x∗xT ∗+λI−1(x∗y∗−η) =Ex∗ Σ−1 λΣΣ−1 λ−Σ−1... | https://arxiv.org/abs/2502.12999v2 |
arXiv:2502.13115v1 [cs.LG] 18 Feb 2025Near-Optimal Private Learning in Linear Contextual Bandit s Fan Chen fanchen@mit.eduJiachun Li jiach334@mit.eduAlexander Rakhlin rakhlin@mit.eduDavid Simchi-Levi dslevi@mit.edu February 19, 2025 Abstract We analyze the problem of private learning in generalized li near contextual b... | https://arxiv.org/abs/2502.13115v1 |
Local DPZheng et al. [2020] (dT)3/4/α ✓ ✓ Han et al. [2021]†/radicalbig dlog|A|·T/(λ⋆ minα) ✓ ✓ Li et al. [2024] |A|2logd(T)·√ T/α ✗ ✗ Chen and Rakhlin [2025]√ d3T/α ✓ ✓ Theorem 5.3√ d5T/α ✓ ✗ Lower bound [ Chen and Rakhlin ,2025]√ d2T/α / / Table 1: Summary of the existing results for private learnin g in (generalized... | https://arxiv.org/abs/2502.13115v1 |
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