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r= 100 and for T= 100. The curves represent the relative mean squared error with respect to the disjoint block maxima estimator, MSE( ·)/MSE(cRL(db) max). Top row: AR-models. Bottom row: ARMAX-models. 56 200 400 Number of blocks0.70.80.911.1RL(50,100) 200 400 Number of blocksRL(100,100) 200 400 Number of blocksRL(200,1... | https://arxiv.org/abs/2502.15036v1 |
series at extreme levels: new estimators for the limiting cluster size distribution”. In: Stochastic Process. Appl. 149 (2022), pp. 75–106. doi:10.1016/j.spa.2022. 03.004 . [BS14] B¨ ucher, A. and Segers, J. “Extreme value copula estimation based on block maxima of a multivariate stationary time series”. In: Extremes 1... | https://arxiv.org/abs/2502.15036v1 |
“Estimation of the generalized extreme-value distribution by the method of probability-weighted moments”. In: Technometrics 27.3 (1985), pp. 251–261. doi:10.2307/1269706 . [Lea83] Leadbetter, M. R. “Extremes and local dependence in stationary sequences”. In: Z. Wahrsch. Verw. Gebiete 65.2 (1983), pp. 291–306. doi:10.10... | https://arxiv.org/abs/2502.15036v1 |
arXiv:2502.15116v1 [math.PR] 21 Feb 2025UNIFORM MEAN ESTIMATION VIA GENERIC CHAINING DANIEL BARTL AND SHAHAR MENDELSON Abstract. We introduce an empirical functional Ψ that is an optimal uni form mean estimator: Let F⊂L2(µ) be a class of mean zero functions, uis a real valued function, andX1,...,X Nare independent, dis... | https://arxiv.org/abs/2502.15116v1 |
to obtain a uniform bound on the me ans{Eu(f(X)) :f∈F}; in Sections 4 and 5 we present examples of that flavour that hav e been studied extensively in recent years. Over the years, several attempts have been made to construct uniform mean estimators forarbitraryclasses offunctionsthat outperformtheempi rical mean(notabl... | https://arxiv.org/abs/2502.15116v1 |
in this case. Similar examples show that even for p= 2, ifFsatisfies a slightly weaker norm equiva- lence than (1.4), the empirical mean functional1 N/summationtextN i=1f2(Xi) does not satisfy (1.3). The reason behind both facts is that the empirical mean performs well only in ‘light-tailed’ sit- uations, and if the fun... | https://arxiv.org/abs/2502.15116v1 |
not significantly more restrictive than the bare-minimum neede d to ensure that Question 1.2 makes sense, namely that supf∈FE|u(f)|= supf∈FE|f|p<∞. Remark 1.6. Clearly, ifuis smooth and convex (as is the case in most interesting ap- plications), one can choose v=u′. The integrability assumption on vis there to balance b... | https://arxiv.org/abs/2502.15116v1 |
estimated from random data. This question ha s been completely resolved, and there are various procedures that achieve the optimal be haviour: Theorem 2.1. There are absolute constants c1andc2for which the following holds. For everyexp(−c1N)≤δ≤1/2there is a mapping ψδ:RN→Rwhich satisfies that for any random variable Xwi... | https://arxiv.org/abs/2502.15116v1 |
is that the expected supremumof gaussian processes is a purely metric object— determined by the γ2functional of the indexing set endowed with the right metric —, is nothing short of remarkable. There are only a handful of proc esses other than the gaussian that have similar metric characterizations, and those char acte... | https://arxiv.org/abs/2502.15116v1 |
with probability at least 1−exp(−2s+4), (3.3) |ψδs(h(X1),...,h(XN))−Eh| ≤c22s/2 √ N·/bardblh/bardblL2. Using (3.3) for every h∈Hs, combined with the fact that |Hs| ≤22s+2, it follows from the union bound that with probability at least 1 −exp(−2s+3), for every f∈F, |Ψs(f)−E(u(πs+1f)−u(πsf))| ≤c22s/2 √ N·/bardblu(πs+1f)−... | https://arxiv.org/abs/2502.15116v1 |
rather coarse and does not allow one to pin-point the sets Kpaccurately. Taking Question 4.1 as a starting point, it is natural to look formembership oracles of the setsKp. To be more accurate, let T⊂Sd−1be a centrally symmetric set, and consider the cone CT={λt:t∈T, λ≥0}. Everyt∈Tcorresponds to a direction in Rdand th... | https://arxiv.org/abs/2502.15116v1 |
(4.3) becomes (4.4) |Ψ1(X1,...,XN,z)−E|/a\}bracketle{tX,z/a\}bracketri}ht|p| ≤c5(p)εE|/a\}bracketle{tX,z/a\}bracketri}ht|p. Finally, set /hatwideKT,p={z∈ CT: Ψ1(X1,...,X N,z)≤1}, fix a realization for which (4.2) holds, and let z∈ CT. It is straightforward to verify that if z∈/hatwideKT,pthen E|/a\}bracketle{tX,z/a\}bra... | https://arxiv.org/abs/2502.15116v1 |
explore the case u(t) =t2andT=Sd−1—corresponding tocorrupted covariance estimation inRd. LetXbe an unknown, centred random vector in Rd. One receives as data a random sample consisting of Nindependent copies of Xwhere at most ηNof them have been corrupted by an adversary. The goal is to find a symmetric, pos itive semi-... | https://arxiv.org/abs/2502.15116v1 |
using a much simpler argument, though with the caveat of Assumption 1.3. Setu(t) =t2and thenv(t) =tis a valid choice. Let F={/a\}bracketle{t·,z/a\}bracketri}ht:z∈Sd−1}, and note that dF= sup z∈Sd−1/bardbl/a\}bracketle{tX,z/a\}bracketri}ht/bardblL2=/radicalbig λ1. By theL4−L2norm equivalence, R(F) = 2 sup z∈Sd−1/parenle... | https://arxiv.org/abs/2502.15116v1 |
and S. Mendelson. Risk minimization by median -of-means tournaments. Journal of the European Mathematical Society , 22(3):925–965, 2019. [19] G. Lugosi and S. Mendelson. Sub-gaussian estimators of the mean of a random vector. Annals of Statistics , 47(2):783–794, 2019. [20] G. Lugosi and S. Mendelson. Robust multivaria... | https://arxiv.org/abs/2502.15116v1 |
arXiv:2502.15118v1 [math.ST] 21 Feb 2025DO WE REALLY NEED THE RADEMACHER COMPLEXITIES? DANIEL BARTL AND SHAHAR MENDELSON Abstract. We study the fundamental problem of learning with respect to the squared loss in a convex class. The state-of-the-art sample complex ity estimates in this setting rely on Rademacher complex... | https://arxiv.org/abs/2502.15118v1 |
te-of-the-art, we must first describe the setting of the learning problems we focus o n. 1.1.The classical learning problem. In what follows we consider the classical learning scenario: learning with respect to the squared loss, and for an underlying class of functions that is convex and compact. The study of such learn... | https://arxiv.org/abs/2502.15118v1 |
example t he references mentioned previously and [6, 11, 16, 20, 22]). And what was believed to b e an (almost) complete solution in the setup studied here was established in [16] (a nd then extended to the non- convex case in [23]): it was shown that there is a learning pro cedure for which the tradeoff is governed by ... | https://arxiv.org/abs/2502.15118v1 |
infor- mation on the noise level of the problem and the resulting acc uracy/confidence tradeoff depends on σ. The first two components of Assumption 1.5 are that the class F∪{0}satisfiesL4−L2 norm equivalence with constant L, as does the class {f−Y:f∈F}; neither assumption is really restrictive. Example 1.6. There are many... | https://arxiv.org/abs/2502.15118v1 |
symmetric set, and therefore so is ( T−T)∩rD. As such, ( T−T)∩rDis the unit ball of a norm on Rd(or on DO WE REALLY NEED THE RADEMACHER COMPLEXITIES? 7 a subspace of Rd), and denote that norm by /bardbl·/bardblKr. Thus, the Rademacher complexities are determined by the behaviour of the functions (1.9) Φ N(r) =E/vextend... | https://arxiv.org/abs/2502.15118v1 |
true. Despite that, our main result, presented in the next section , is that essentially, max{rQ(κ),rM(κ)}is an upper bound on the optimal tradeoff even in heavy tailed p rob- lems, once we assume that the learner is given a little more in formation on the problem. 1.4.The main result. The ‘little more information’ that... | https://arxiv.org/abs/2502.15118v1 |
1.17. There are constants c1,c2,c3that depend only on Landηfor which the following holds. Given εandδ, letN0(ε)be the smallest integer for which max{r∗(c1),λ∗(c1)} ≤c2√ε. If N≥N0(ε)+c3/parenleftbiggσ2 ε+1/parenrightbigg log/parenleftbigg2 δ/parenrightbigg , (1.12) then with probability at least 1−δ, /bardbl/hatwidef−f∗... | https://arxiv.org/abs/2502.15118v1 |
given the data the learner has access to . Throughout this article, c,c0,c1denote strictly positive absolute constants that may change their value in each appearance. If a constant cdepends on a parameter α, that is denoted by c=c(α). We write α/lessorsimilarβif there is an absolute constant cfor whichα≤cβ andα∼βif bot... | https://arxiv.org/abs/2502.15118v1 |
obtain the isomo rphic estimator for the noise level stated in Lemma 1.14. Proof of Lemma 1.14. : Recall that σ2= inff∈FE(f−Y)2and that /hatwideσ2= inf f∈FΨ/BV(f). Note that in the high probability event in which (2.1) holds, /hatwideσ2≤2max{r2,σ2}, and if σ>rthen/hatwideσ2≥1 2σ2. Hence, ifσ>rthen1 2σ2≤/hatwideσ2≤2σ2, ... | https://arxiv.org/abs/2502.15118v1 |
d(f∗,h)< ηr. Indeed, if not andd(f∗,h)≥ηr, then Ψ L(f∗,h)≥0. At the same time, by (2.4), /bardblf∗−h/bardblL2≥r, and thanks to (2.5) and (2.6), ΨL(f∗,h)≤ L(f∗,h)+1 2/bardblf∗−h/bardbl2 L2 =−L(h,f∗)+1 2/bardblf∗−h/bardbl2 L2 ≤ −/bardblf∗−h/bardblL2+1 2/bardblf∗−h/bardblL2<0, which is impossible. Thus d(f∗,h)<ηr, and in ... | https://arxiv.org/abs/2502.15118v1 |
considering other processes—for example—an empirica l process in a heavy tailed situation. What is true, however, is that there are mean esti mation procedures that exhibit subgaussian tails even for general random variables. For the remainder of this article, denote by ψδ:RN→Ran optimal mean-estimation procedure, i.e.... | https://arxiv.org/abs/2502.15118v1 |
3.5. His a centrally symmetric class of mean zero functions that sa tisfies for everyf,h∈H∪{0}, /bardblf−h/bardblL4≤L/bardblf−h/bardblL2. Also, assumeas always that the learner has access to a symmet ricfunctional dthat satisfies η−1/bardblf−h/bardblL2≤d(f,h)≤η/bardblf−h/bardblL2 for everyf,h∈H∪{0}. Remark 3.6. Clearly t... | https://arxiv.org/abs/2502.15118v1 |
LetHbe a class of mean zero functions that satisfies L4−L2norm equivalence with constant L. Let2s0≤c1Nand setU={ξj: 1≤j≤2exp(−2s0−1)} ⊂L4 to be a collection of functions. Then there is a procedure Φ/C5for which, with probability 1−2exp(−2s0−1), for every ξ∈ U, sup h∈H|Φ/C5(h,ξ)−Eξh| ≤c(L)/bardblξ/bardblL4√ N/parenleftbi... | https://arxiv.org/abs/2502.15118v1 |
2:To complete the proof, it suffices to show that for every f∈H1andh∈H2, |Efh−Eπs1fπs1h| ≤c4(η)√ N/parenleftBigg RH1Esup h∈H2Gh+RH2Esup f∈H1Gf/parenrightBigg . (3.11) To that end, note that |Efh−Eπs1fπs1h| ≤ /bardblf−πs1f/bardblL2/bardblh/bardblL2+/bardblh−πs1h/bardblL2/bardblπs1f/bardblL2, and/bardblh/bardblL2≤RH2,/bard... | https://arxiv.org/abs/2502.15118v1 |
{u(w+2(v−Y)) :u∈U, w∈W, v∈V}. Theprocedureisgiventheidentitiesof u,w,vandthesecondhalfofthesample( Xi,Yi)2N i=N+1, and returns its best guess of the mean Eu(w+2(v−Y)). As will become clear immediately, that leads to a uniform mea n estimation procedure for all the functions of the form ( f−Y)2−(h−Y)2. Theorem 5.1. Ther... | https://arxiv.org/abs/2502.15118v1 |
of Ψ ∗is its behaviour on the scaled down versions, which we explore next. Proposition 5.3. In the setting of Theorem 5.1 and using its notation, we have tha t with probability at least 1−2exp(−c22s0), for every u,w∈(F−F)∩r0Dandv∈F, |ΨQ(u,w)−Euw| ≤c3θr2, |ΨQ(u,v−vj)−Eu(v−vj)| ≤c3θr2,and |ΨM(u,vj−Y)−u(vj−Y)| ≤c3θr2·r+/b... | https://arxiv.org/abs/2502.15118v1 |
Springer-Verlag, New York, 1996. [10] L.Devroye, M.Lerasle, G.Lugosi, andR.Oliveira. Sub- gausssianmeanestimators. Annals of Statistics , 44:2695–2725, 2016. DO WE REALLY NEED THE RADEMACHER COMPLEXITIES? 31 [11] L. Gy¨ orfi, M. Kohler, A. Krzy˙ zak, and H. Walk. A Distribution-Free Theory of Nonparametric Regres- sion.... | https://arxiv.org/abs/2502.15118v1 |
Lemma A.1. There is an absolute constant csuch that for every κ∈(0,1),λ∗(κ)≤ /tildewider∗(cκ). Moreover, if r∗(cκ)≤2dFsatisfies r∗(cκ)/radicalBigg log/parenleftbigg4dF r∗(cκ)/parenrightbigg ≤σ then λ∗(κ)≤r∗(cκ)/radicalBigg log/parenleftbigg4dF r∗(cκ)/parenrightbigg . Proof.For everys,t>0, set H(s,t) = sup f∈FN(F∩(f+sD),... | https://arxiv.org/abs/2502.15118v1 |
index 1 ≤s≤Ns.t.|xs|=k1/4 c0and/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay i=1εi/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2√ N/bracerightBigg . On that event, /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdispl... | https://arxiv.org/abs/2502.15118v1 |
Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling Yufan Li YUFAN LI@G.HARVARD .EDU Pragya Sur PRAGYA @FAS.HARVARD .EDU Department of Statistics, Harvard University 1 Oxford Street, Cambridge, MA Abstract We study the fundamental problem of calibrating a li... | https://arxiv.org/abs/2502.15131v1 |
modern neural networks and other interpolation learners under overparametrization [1, 28, 42, 43, 46, 54, 60]. For binary classification in this proportional regime, a substantial line of work [63, 64, 73] establishes that classical logistic regression yields seriously biased estimates; building upon these, [2] shows t... | https://arxiv.org/abs/2502.15131v1 |
x n]⊤∈ Rn×dandy= [y1, . . . , y n]⊤∈Rn. To quantify the degree of miscalibration, we define the calibration error at level pfor any pre- dictor ˆfas ∆cal p(ˆf) =p−Exnewh σ w⊤ ⋆xnew |ˆf(xnew) =pi , where Exnewdenotes the expectation over xnew∼N(0,Σ). A predictor is said to be well-calibrated if∆cal p(ˆf) = 0 for all p... | https://arxiv.org/abs/2502.15131v1 |
in (13) andEZdenotes expectation with respect to the Gaussian noise Z∼N(0,1). We will later refer to ˆfangas the angular predictor for simplicity. Theorem 2 below shows that the angular predictor is well-calibrated. We defer the proof to Appendix A Theorem 2 Assume the link function σis continuous. Then, the predictor ... | https://arxiv.org/abs/2502.15131v1 |
F⋆= arg min F∈FExnew[Dϕ(q⋆,ˆqF)]. Further suppose that the link function σis continuous. We then have that as n, d→ ∞ , we have ˆqang(ˆθ)−F∗(bw⊤xnew) 2 2→0 in probability. That is, the label prediction probability vector from angular calibration converges to the optimal label prediction probability vector given by F∗. ... | https://arxiv.org/abs/2502.15131v1 |
defer the proof of the Theorem below to Appendix C. Theorem 5 Consider the predictor ˆfnho platt(u)calibrated by the Platt scaling procedure, that is, ˆfnho platt(bw⊤xnew) =σ(ˆAnho·bw⊤xnew+ˆBnho), with ˆAnho,ˆBnho= argmin (A,B)∈Hˆℓnho(u7→σ(Au+B))(7) forˆℓnho(·)defined in (5). If the link function σsatisfies σ(x) = Φ( a... | https://arxiv.org/abs/2502.15131v1 |
13, as n, d→ ∞ , we have that |ˆθ−θ∗| →0in probability where nho=α·nfor a fixed constant α >0. 7. Simulations This section presents a simple simulation to demonstrate results in Section 6. We generate i.i.d. sam- plesxiiid∼N(0,Σ), i= 1, ..., n where Σ =1 d¯Σand¯Σkl= 0.5|k−l|,∀k, l∈ {1, ..., d}; we also generate labels ... | https://arxiv.org/abs/2502.15131v1 |
we have σ(u)≈Φ(p π/8(3u+ 1)) . We then obtain 9 LISUR 0.10 0.30 0.50 0.70 0.90 Predicted Probability0.100.300.500.700.90Empirical Probability45° line Uncalibrated Angular Platt scaling (nho=100) 0.10 0.30 0.50 0.70 0.90 Predicted Probability0.100.300.500.700.90Empirical Probability45° line Uncalibrated Angular Platt sc... | https://arxiv.org/abs/2502.15131v1 |
(3)is well-calibrated at all p∈[0,1]when . That is, for anyp∈[0,1]and any d, n∈N+ ∆cal p ˆfang(·;θ∗) =p−Exnewh σ w⊤ ⋆xnew |ˆfang bw⊤xnew;θ∗ =pi = 0. Proof [Proof of Theorem 8] Let us define the following event A:=n ˆfang w⊤ ⋆xnew;θ∗ =po . We have Exnewh σ w⊤ ⋆xnew | Ai (i)=Exnewh Exnewh σ w⊤ ⋆xnew |x⊤ newbw... | https://arxiv.org/abs/2502.15131v1 |
suffices to show that ˆqang(ˆθ)−ˆqang(θ∗) 2→0 in probability. This is an immediate consequence of the continuous mapping theorem under the assumption σis continuous. Appendix C. Proof of Theorem 5 Before proving Theorem 5, we first state two classic analysis results that we will later use. Proposition 11 (Theorem 5.7, ... | https://arxiv.org/abs/2502.15131v1 |
B1)) which implies that Exnewf(t(A1, B1) + (1 −t)(A1, B1))< tExnewf((A1, B1)) + (1 −t)Exnewf((A1, B1)). The claim that ℓ⋆(A, B)is strictly convex follows. It then follows from Theorem 11 that ˆAnho→A∗,ˆBnho→B∗in probability as nho→ ∞ . The uniform convergence ˆfnho platt(u)→ˆfang(u)follows immediately. 15 LISUR Appendi... | https://arxiv.org/abs/2502.15131v1 |
in linear models. The Annals of Statistics , 51(2):391–436, 2023. [10] Eugene Berta, Francis Bach, and Michael Jordan. Classifier calibration with roc-regularized isotonic regression. In International Conference on Artificial Intelligence and Statistics , pages 1972–1980. PMLR, 2024. [11] Bj ¨orn B ¨oken. On the approp... | https://arxiv.org/abs/2502.15131v1 |
of statistics , 50(2):949, 2022. [29] Hong Hu and Yue M Lu. Universality laws for high-dimensional learning with random fea- tures. IEEE Transactions on Information Theory , 69(3):1932–1964, 2022. [30] Adel Javanmard and Andrea Montanari. Debiasing the lasso: Optimal sample size for Gaus- sian designs. The Annals of St... | https://arxiv.org/abs/2502.15131v1 |
Basil N Saeed. Universality of empirical risk minimization. In Confer- ence on Learning Theory , pages 4310–4312. PMLR, 2022. [50] Andrea Montanari, Subhabrata Sen, et al. A friendly tutorial on mean-field spin glass tech- niques for non-physicists. Foundations and Trends® in Machine Learning , 17(1):1–173, 2024. [51] ... | https://arxiv.org/abs/2502.15131v1 |
Yeming Wen, Dustin Tran, and Jimmy Ba. Batchensemble: an alternative approach to efficient ensemble and lifelong learning. arXiv preprint arXiv:2002.06715 , 2020. [70] Bianca Zadrozny and Charles Elkan. Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers. In Icml, volume 1, pag... | https://arxiv.org/abs/2502.15131v1 |
TENSOR PRODUCT NEURAL NETWORKS FOR FUNCTIONAL ANOVA M ODEL Seokhun Park1Insung Kong2Yongchan Choi3Chanmoo Park1Yongdai Kim1 1Seoul National University2University of Twente3Toss bank {shrdid,chanmoo13,ydkim903}@snu.ac.kr insung.kong@utwente.nl pminer32@gmail.com ABSTRACT Interpretability for machine learning models is b... | https://arxiv.org/abs/2502.15215v3 |
for the functional ANOV A model such that each component is identifiable but they are learnable by standard stochastic gradient descent algorithms. Identifiability makes our proposed neuralarXiv:2502.15215v3 [stat.ML] 28 Feb 2025 networks be good at estimating the components and thus provide reliable interpretation. In... | https://arxiv.org/abs/2502.15215v3 |
functional ANOV A model, and [ 9,16] proposed methods for estimating the functional ANOV A model using basis expansion. In addition, the functional ANOV A model has been applied to various problems such as sensitivity analysis [ 17], survival analysis [18], diagnostics of high-dimensional functions [19] and machine lea... | https://arxiv.org/abs/2502.15215v3 |
=f∗ 1(x1) +f∗ 2(x2) +f∗ 1,2(x1, x2), 3 Table 1: Stability scores on real datasets. Lower stability score means more stable interpretation. The textbf faces highlight the best results among GAM and GA2M. GA1M GA2M DatasetANOV A T1PNNNA1M NB1MANOV A T2PNNNA2M NB2M CALHOUSING [23] 0.012 0.045 0.039 0.035 0.071 0.075 WINE[... | https://arxiv.org/abs/2502.15215v3 |
of ANOV A-SHAP is that it is significantly faster to compute compared to existing methods of calculating SHAP values such as Deep-SHAP and Kernel-SHAP proposed by [2], as well as Tree-SHAP proposed by [22]. 3.2 ANOV A-TPNN We first propose basis neural networks for the main effects and extend these basis neural network... | https://arxiv.org/abs/2502.15215v3 |
function ℓ,we learn the parameters by minimizing the empirical riskPn i=1ℓ(yi, f(xi))by a gradient descent algorithm. Overfitting can be avoided by selecting the number of epochs and learning rate carefully. Data preprocessing. The term cj(α, γ)could be too large when η(b, γ)is close to 0, which can happen when µj is t... | https://arxiv.org/abs/2502.15215v3 |
all z1, z2∈ Z, where ∥ · ∥ is a certain norm defined on Z. 6 Table 3: Prediction performance. We report the averages (standard deviations) of the prediction performance measure. In addition, we report the averages of ranks of prediction performance of each model on nine datasets. The optimal (or suboptimal) results are... | https://arxiv.org/abs/2502.15215v3 |
We generate synthetic datasets from Y=fk(x) +ϵ,where fkis the true prediction model defined in Appendix B.1 for k= 1,2,3. Then, we apply ANOV A-T2PNN, NA2M and NB2M to calculate the importance scores of the main effects and second order interactions and examine how well they predict whether a given component is signal.... | https://arxiv.org/abs/2502.15215v3 |
out unnecessary second order interactions a priori and include only selected second order interactions (and all the main effects) into the model. In the experiment, we use Neural Interaction Detection (NID) proposed by [ 33] for the interaction screening. The numbers of selected interactions are given in Appendix B.2. ... | https://arxiv.org/abs/2502.15215v3 |
0.009 WINERMSE↓(std) 0.725 (0.02) 0.720 (0.02) Stability score ↓ 0.011 0.017 4.7 ANOV A-TPNN with monotone constraints Monotone constraint. In practice, prior knowledge that some main effects are monotone functions is available and it is needed to reflect this prior knowledge in the training phase. A notable example is... | https://arxiv.org/abs/2502.15215v3 |
ANOV A-TPNN is comparable to its competitors. Additionally, we proposed NBM-TPNN that improves the scalability of ANOV A-TPNN using the idea from NBM [ 8]. One advantage of the basis function in NBM-TPNN is that the number of basis functions does not depend on the dimension of the input feature. Even though it is compu... | https://arxiv.org/abs/2502.15215v3 |
models. In International Conference on Artificial Intelligence and Statistics , pages 1–9. PMLR, 2024. [17] Gaëlle Chastaing, Fabrice Gamboa, and Clémentine Prieur. Generalized hoeffding-sobol decomposition for dependent variables-application to sensitivity analysis. 2012. [18] Jianhua Z Huang, Charles Kooperberg, Char... | https://arxiv.org/abs/2502.15215v3 |
Tang, Stephen Mussmann, Emma Pierson, Been Kim, and Percy Liang. Concept bottleneck models. In International conference on machine learning , pages 5338–5348. PMLR, 2020. [43] Hugh A Chipman, Edward I George, and Robert E McCulloch. Bart: Bayesian additive regression trees. 2010. [44] Adam Paszke, Sam Gross, Francisco ... | https://arxiv.org/abs/2502.15215v3 |
up, we have |g0,j(x)−fE,j(x)| ≤LKX k=1 EXj[ℓj,k(Xj)|x−Xj|] EXj[ℓj,k(Xj)] ℓj,k(x) =LX k∈{r−1,r,r+1} EXj[ℓj,k(Xj)|x−Xj|] EXj[ℓj,k(Xj)] ℓj,k(x) +X k/∈{r−1,r,r+1} EXj[ℓj,k(Xj)|x−Xj|] EXj[ℓj,k(Xj)] ℓj,k(x) ≤L3C′ K+1 1 + exp( K) . 14 Step 2. Decomposing the approximation function fE,jinto a sum of TPNNs. Now, for fE,j(x)... | https://arxiv.org/abs/2502.15215v3 |
loss of generality, we consider S={1, ..., d}. Similarly to the case of S={j}, we define an interval partition ofXjas{Ωj k}K k=1={[χj k−1, χj k]}K k=1such that µj(Ωj k) =1 Kand|χj k−1−χj k| ≤1 pLKforj= 1, ..., d . Additionally, letΩk1,....,k d= Ω1 k1× ··· × Ωd kdforkj∈[K], j= 1, ..., d . ForΩk1,....,k d, we define ℓk1,... | https://arxiv.org/abs/2502.15215v3 |
10+ϵ Table 9: Distributions of input features corresponding to each synthetic function. f(1)X1, X2, X3, X4, X5∼iidU(0,1) f(2)X1, X2, X3, X6, X7, X9∼iidU(0,1)andX4, X5, X8, X10∼iidU(0.6,1). f(3)X1, X2, X3, X4, X5, X6, X7, X8, X9, X10∼iidU(−1,1) The synthetic function f(1)is a slightly modified version of Friedman’s synt... | https://arxiv.org/abs/2502.15215v3 |
fixed the model’s hyper-parameters and used the 10 train-test dataset pairs obtained from the previous data splitting to train the model on the train datasets and evaluate its performance on the test datasets. For XGB, the range of hyper-parameters for the grid search is as below. • The number of tree : {50,100,200,300... | https://arxiv.org/abs/2502.15215v3 |
All models are trained via the Adam optimizer with the learning rate 1e-3 and the batch size for training 256. For ANOV A- T1PNN, KSis determined through grid search on [10,30,50]. For NA1M and NB1M, the neural network consists of 3 hidden layer with sizes (64,64,32) and (256,128,128), respectively. Due to the limitati... | https://arxiv.org/abs/2502.15215v3 |
2.122 (0.08) C.2 Impact of the initial parameter values to stability We investigate how the choice of initial parameter values affects the stability of the estimated components of ANOV A- TPNN by analyzing synthetic datasets generated from f(1). We fit ANOV A-T2PNN on 5 independently generated datasets, and the 5 estim... | https://arxiv.org/abs/2502.15215v3 |
The results are bit different from those of ANOV A-T1PNN. In particular, the interaction between ‘latitude’ and ‘longitude’ emerges as a new important feature while the main effects of ‘latitude’ and ‘longitude’ become less important. Figure 3: Plots of the functional relations of the main effects in ANOV A-T1PNN on CA... | https://arxiv.org/abs/2502.15215v3 |
2-2 of Figure 4, we observe that ANOV A-T1PNN with the monotone constraint assigns a negative score to ‘No Beard’ that increases the probability of being classified as female. However, ANOV A-T1PNN without the monotone condition assigns a positive score to ‘No Beard’ that increases the probability of being classified a... | https://arxiv.org/abs/2502.15215v3 |
datasets. We observe that the 5 estimates of each component estimated by ANOV A-TPNN are much more stable compared to those by NAM and NBM. Note that in Figure 9, we observe that some components are estimated as a constant function in NA2M, which would be partly because the main effects are absorbed into the second ord... | https://arxiv.org/abs/2502.15215v3 |
(0.03)3.474 (0.03)3.511 (0.03) f(2)RMSE↓0.076 (0.001)0.088 (0.005)0.075 (0.001) f(3)RMSE↓0.161 (0.003)0.183 (0.016)0.137 (0.003) 36 I Comparison of ANOV A-SHAP and Kernel-SHAP In this section, we conduct an experiment to investigate the similarity between ANOV A-SHAP and Kernel-SHAP [ 2]. We calculate ANOV A-SHAP from ... | https://arxiv.org/abs/2502.15215v3 |
10 6.6 sec 3.0 sec 1.6 sec 1.5 sec CALHOUSING 21K 8 14.1 sec 4.1 sec 3.8 sec 3.5 sec ONLINE 40K 58 68 sec 15.6 sec 65 sec 9.8 sec We conduct experiments to assess the scalability of NBM-TPNN. We consider NA1M, which has 3 hidden layers with 16, 16, and 8 nodes; 10 basis DNNs for NB1M, which have 3 hidden layers with 32... | https://arxiv.org/abs/2502.15215v3 |
relations of the main effects on CALHOUSING and WINE dataset in GAM-T2PNN. We observe that GAM-T2PNN estimates the components more unstable compared to ANOV A-T2PNN. 42 N On the post-processing for the sum-to-zero condition Table 26: Stability scores for ‘Latitude’ and ‘Longitude’ of C ALHOUSING dataset after post-proc... | https://arxiv.org/abs/2502.15215v3 |
the spline model can be adjusted by tuning λS>0. Inaccurate prediction beyond the boundary. Spline-GAM is a model that sets knots based on the training data and interpolates between the knots using cubic spline basis functions. If the test data contains input outliers, the prediction performance of Spline-GAM may deter... | https://arxiv.org/abs/2502.15215v3 |
SEMIPARAMETRIC BERNSTEIN -VON MISES PHENOMENON VIA ISOTONIZED POSTERIOR IN WICKSELL ’S PROBLEM Francesco Gili∗Geurt Jongbloed†Aad van der Vaart‡ Delft University of Technology, Mekelweg 4, Delft 2628CD, The Netherlands. February 24, 2025 ABSTRACT In this paper, we propose a novel Bayesian approach for nonparametric est... | https://arxiv.org/abs/2502.15352v1 |
more likely to be sampled. Thus the distribution of the squared radii of the spheres actually cut is not F∗ 0but a biased version of it; we write for the squared radius of the sphere actually cut: X∼F0. Since we assume that the samples are gathered uniformly, by the Pythagorean Theorem, the observable squared circle ra... | https://arxiv.org/abs/2502.15352v1 |
speed : The Dirichlet posterior (1.3) is explicit and its projection can be effectively computed by the Pool-Adjacent-Violators Algorithm (PA V A). (ii)Efficiency and adaptation : The IIP achieves the asymptotic variance of the efficient estima- tors introduced in [13] and [16] and it is adaptive to level of local smoo... | https://arxiv.org/abs/2502.15352v1 |
esti- mator is not efficient. However, it behaves in practice very closely to informed estimators that make use of the fact that the cdf is constant on that interval. Although the Bayesian methodology for addressing inverse problems is largely adopted in scientific and statistical circles, theoretical guarantees that e... | https://arxiv.org/abs/2502.15352v1 |
start by proving a BvM result for the Naive Bayes Posterior (NBP), analogous to the result ob- tained for the naive estimator (c.f. Theorem 2 in [17]). For our first result, we need our version of g0 to be well defined and bounded, i.e.: ∥g0∥∞= sup z≥0 Z∞ zdV0(s) π√s−z <∞. (2.1) In equation (2.2) and (2.9) below we use... | https://arxiv.org/abs/2502.15352v1 |
|Z1, . . . , Z n⇝N 0,g0(x) 2γ . (2.9) Proof. For sequences δn=√logn/√nandδ∗ n=δ1/γ nwhere γ > 1/2satisfies condition (2.8), define ZG0n(t) :=δ−1 n(δ∗ n)−1{Un(x+δ∗ nt)−Un(x)−U0(x+δ∗ nt) +U0(x)} (2.10) = 2δ−1 n(δ∗ n)−1Znp (z−x)+−p (z−x−δ∗nt)+o d(Gn−G0) (z). (2.11) By Lemma 1 given in Appendix A, for any bounded 1-Lipsc... | https://arxiv.org/abs/2502.15352v1 |
have Wni=nεiPn j=1εj=εi/¯εn,with ¯εn:=1 nPn j=1εj. In fact, by Proposition G.2 in [12] (ε1 n¯εn, . . . ,εn n¯εn)∼Dir(n; 1, . . . , 1). Because |¯εn−1|=Op(1/√n), for almost every sequence Z1, . . . , Z nconclude that, uniformly in tin compacta: Z∗ n(t) =√n√logn(Bn−Gn)fn t=1√nlognnX i=1(Wni−1)fn t(Zi) =Z∗ n(t) +op(1), (2... | https://arxiv.org/abs/2502.15352v1 |
continuous sample paths and a.s. unique maxima (see Lemma 2.6 in [23]) and that ˆtG n:= inf t≥ −(δ∗ n)−1x:˜ZG n(t)−atis max is an argmax process conditionally on Z1, . . . , Z n, satisfying the second condition on Lemma 2. It is thus enough to show that there exists a conditionally stochastically bounded sequence (˜Mn... | https://arxiv.org/abs/2502.15352v1 |
frequentist approach of [13] would entail estimating both g0(x)andγ, the latter being particularly challenging. The current Bayesian approach requires neither of the two, making this method preferable. In Figure 3, we apply the methodology to star position data from globular cluster Messier 62. Let Z= (Zp1, Zp2, Zp3)re... | https://arxiv.org/abs/2502.15352v1 |
highlights its potential for Bayesian nonparametrics in inverse problems. 13 Semiparametric Bernstein-von Mises Phenomenon via Isotonized Posterior in Wicksell’s problem Regarding potential future work, several interesting directions emerge. First, it would be important to verify whether the bootstrap procedure of [33]... | https://arxiv.org/abs/2502.15352v1 |
Phenomenon via Isotonized Posterior in Wicksell’s problem [13] G ILI, F., J ONGBLOED , G., and VAN DER VAART , A. (2024a). Adaptive and Efficient Isotonic Estimation in Wicksell’s Problem. Journal of Nonparametric Statistics , pages 1–41. https: //doi.org/10.1080/10485252.2024.2397680 . [14] G ILI, F., J ONGBLOED , G.,... | https://arxiv.org/abs/2502.15352v1 |
A. (2020). Semiparametric Bayesian Causal Inference. The Annals of Statistics , 48:2999–3020. https://doi.org/10.1214/19-AOS1919 . [32] R IVOIRARD , V. and R OUSSEAU , J. (2012). Bernstein-von Mises Theorem for Linear Function- als of the density. The Annals of Statistics , 40(3):1489–1523. http://www.jstor.org/stable/... | https://arxiv.org/abs/2502.15352v1 |
n→ ∞ : √n√logn ˆVn(x)−V0(x) =ZG0n(1) + oG0(1). where ZG0n(1)is given in (2.10) . Proof. Fixx≥0such that the conditions stated above are satisfied. Define, for Ungiven in (2.5) and a >0, the process Tnas follows: Tn(a) = inf {t≥0 :Un(t)−atis maximal }. The relation between TnandˆVnis given by the switch relation (Lemm... | https://arxiv.org/abs/2502.15352v1 |
made arbitrarily small and lim supn→∞P(ˆtn∈Kc|Z1, . . . , Z n)converges to 0. Lemma 3. Ifg0(x) =R∞ x(4s(s−x))−1/2dF0(s)<∞, then as t↓0: E1 Z−x1{Z≥x+t} ∼log1 t g0(x), (A.8) where ∼means that the ratio of the left-hand side and of the right-hand side goes to 1 as t↓0. Moreover let Z⊥ ⊥Λ∼Exp(1) , then it is also true:... | https://arxiv.org/abs/2502.15352v1 |
Conclude again by applying the Dominated Convergence Theorem. Lemma 4. Letξn1, . . . , ξ nnbe a triangular array of independent random variables that satisfy,Pn i=1E|ξni|=O(1),and∀ε > 0,Pn i=1E|ξni|1|ξni|>ε→0.asn→ ∞ . Then:Pn i=1(ξni− Eξni)P→0. Proof. Fixε >0 Var nX i=1ξni1|ξni|≤ε! =nX i=1Var ξni1|ξni|≤ε ≲nX i=1εE|ξn... | https://arxiv.org/abs/2502.15352v1 |
n. Therefore we need to show: nX i=1P |εi−1|fn 1(Zi)≥p δ nlogn Z1, . . . , Z nG0→0,∀δ >0, (A.16) 1√nlognnX i=1E (εi−1)fn 1(Zi)1{|εi−1|fn 1(Zi)≥√δ nlogn} Z1, . . . , Z nG0→0, (A.17) 24 Semiparametric Bernstein-von Mises Phenomenon via Isotonized Posterior in Wicksell’s problem and: 1 nlognnX i=1Var (εi−1)fn 1(Zi)1{... | https://arxiv.org/abs/2502.15352v1 |
n(t),Z∗ n(s)|Z1, . . . , Z n) =1 nlognnX i=1fn t(Zi)fn s(Zi) E (εi−1)2 |{z} =1. (A.23) Now we apply Lemma 4 with ξni:=fn t(Zi)fn s(Zi) nlognto show: 1 nlogn(nX i=1fn t(Zi)fn s(Zi)−nEfn t(Z)fn s(Z)) G0→0. First, by using (A7) from [13] and that δ∗ n= (log n/n)1/2γ: 1 lognEfn t(Zi)fn s(Zi) =−stg0(x)logδ∗ n logn+O1 logn... | https://arxiv.org/abs/2502.15352v1 |
where ˜ZG nis given in (2.12)-(2.13). Thus note that we can rewrite it as: ˜Tn= (δ∗ n)−1arg maxt Mn(t) , where: Mn(t) := 2Zp (z−x)+−p (z−x−t)+−1 2V0(x)t d(G−Gn+G0)(z)−atδn(A.28) −t δ∗n2Zp (z−x)+−p (z−x−δ∗n)+ d(Gn−G0)(z) (A.29) + 2Zp (z−x)+−p (z−x−t)+ d(Gn−G0)(z). (A.30) Moreover define the deterministic function... | https://arxiv.org/abs/2502.15352v1 |
PROBABILISTIC MORPHISMS AND BAYESIAN SUPERVISED LEARNING HˆONG V ˆAN L ˆE Abstract. In this paper, we develop category theory of Markov kernels to study categorical aspects of Bayesian inversions. As a result, we present a unified model for Bayesian supervised learning, encompassing Bayesian density estimation. We illu... | https://arxiv.org/abs/2502.15408v2 |
XtoY, identifying f(x)∈ Y with the Dirac measure δf(x)∈ P(Y) [Vapnik98]. The i.i.d. assumption is not always satisfied in life. In Bayesian statistics, one replaces the i.i.d. assumption by the weaker condition of condition- ally i.i.d. assumption [Schervish1997]. In this paper, using the category of probabilistic morp... | https://arxiv.org/abs/2502.15408v2 |
× Y we denote by Π Xthe canonical projection to the factor X. •We denote by Meas the category of measurable spaces, whose objects are measurable spaces and morphisms are measurable mappings. Important examples of probabilistic morphisms are measurable mappings κ:X → Y , since the Dirac map δ:X → P (X), x7→δx,is measura... | https://arxiv.org/abs/2502.15408v2 |
morphisms. We denote by Probm the category whose objects are measurable spaces and morphisms are prob- abilistic morphisms T:X;Y. The identity morphism is Id X. For κ∈Meas (X,Y) we also use the shorthand notation (3.4) κ:=δ◦κ. For any T∈Probm (X,Y),µ∈ S(X) and B∈ΣYwe set (3.5) S(T)∗µ(B) =Z XT(B|x)dµ(x). Chentsov proved... | https://arxiv.org/abs/2502.15408v2 |
denote the space of simple functions on Y. Since the LHS and RHS are linear in g, the equality (3.8) is also valid for any g∈ Fs(Y). It is known thatFs(Y) is dense in Fb(Y) in the sup-norm, see, for instance, [Chentsov72, §5], taking into account that T∗ Y|Xis a bounded linear map of norm 1, we conclude Assertion (3). ... | https://arxiv.org/abs/2502.15408v2 |
the first assertion of Theorem 3.9 we compute the value of (Γ T)∗µXforA×Bwhere A∈ΣXandB∈ΣY (3.13) (Γ T)∗µX(A×B) =Z XΓT(x)(A×B)dµX(x) =Z AT(B|x)dµX. Comparing with Equation (3.1), we conclude that (Γ T)∗µX=µif and only ifT:X → P (Y) is a product regular conditional probability measure for µ. (2) Now assume that T,T′:X →... | https://arxiv.org/abs/2502.15408v2 |
parameter θ∈Θ. One interprets the value of a Bayesian inversion q(n)(Sn)∈ P(Θ) of pnrelative to a prior distribution µΘ∈ P(Θ) as the updated value of our certainty about parameter θafter seeing Sn∈ Xn[Schervish1997], [GV2007]. Let (Θ , µΘ,p,X) be a Bayesian statistical model. By Lemma 3.10, µX= (p)∗µΘ. Equivalently, fo... | https://arxiv.org/abs/2502.15408v2 |
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