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Storey, J. D., Taylor, J. E. and Siegmund, D. (2004). Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach, J. R. Stat. Soc. Ser. B Stat. Methodol. 66(1): 187–205. Strimmer, K. (2008). A unified approach to false discovery rate estimation, ... | https://arxiv.org/abs/2502.16005v1 |
arXiv:2502.16019v1 [math.ST] 22 Feb 2025A Generalisation of Ville’s Inequality to Monotonic Lower Bounds and Thresholds Wouter M. Koolena,b, Muriel F. P´ erez-Ortizc, Tyron Lardya,d aCentrum Wiskunde & Informatica, Amstedam, The Netherlands bUniversity of Twente, Enschede, The Netherlands cTechnical University Eindhove... | https://arxiv.org/abs/2502.16019v1 |
−f(n) for some increasing function f, and fix some increasing target threshold g(n). What can we say aboutP{∃n≥0 :Mn≥g(n)}? And what applications could this tackle? Ville & LIL. We focus on the paradigmatic application of Ville’s inequality for deriving anytime-valid concentration inequalities of iterated-log arithm typ... | https://arxiv.org/abs/2502.16019v1 |
any supermartingale (Mn)n≥0bounded below by Mn≥ −f(n)for all n≥0and with initial expectation E[M0]∈[−f(0),g(0)], we have P/braceleftbig ∃n≥0 :Mn≥g(n)/bracerightbig ≤1−g(0)−E[M0] g(0)+f(0)∞/productdisplay n=1g(n)+f(n−1) g(n)+f(n).(4) (b)For anym∈[−f(n),g(n)]there is a martingale (Mn)n≥0bounded below byM≥ −fand with E[M0... | https://arxiv.org/abs/2502.16019v1 |
jump is succesful, then Mn+1=g(n+1), whichwouldimplythat Ln+1≥1. However, theprobability of success (5) can be made arbitrarily close to one by taking f(n+1)→ ∞, which gives E[Ln+1|Fn] = 1≤Ln. It follows that Ln≥1 for all n, so the implication Mn≥g(n)⇒Ln≥1 is powerless. The only way to avoid this is forKnto account for... | https://arxiv.org/abs/2502.16019v1 |
becomes apparent in the proof of the upper bound in Section 2.2, which operates by applying V ille’s inequality to an auxiliary non-negative supermartingale. Indeed if on e is able to encode an event of interest directly in the form of that auxilia ry supermartingale, classic Ville suffices. Is our extension more user fr... | https://arxiv.org/abs/2502.16019v1 |
A. Ramdas. Sequential nonparametric testin g with the law of the iterated logarithm. In Proceedings of the Thirty-Second Conference on Uncertainty in Artificial Intelligence , 2016. D. A. Darling and H. Robbins. Iterated logarithm inequalities. Proceedings of the National Academy of Sciences , 57(5):1188–1192, 1967. V. ... | https://arxiv.org/abs/2502.16019v1 |
τ+2√ 2with τ(n) =ln(n+1)/√ 2π+e δ′ln2/parenleftBig ln(n+1)√ 2π+e/parenrightBig . Furthermore, the remainder term R=R(n)is harmless: R≤1/√ 2, the function n/ma√sto→R(n)is increasing, and R(n)∼1√ 2/parenleftBig 1−1√ 2lnlnn/parenrightBig asn→ ∞. If one considers a version Mκ nofMnwith “prior”e−η2κ2/2 |η|dηfor some κ >0, t... | https://arxiv.org/abs/2502.16019v1 |
d(e+x)ln(e+x)2,and letg(x) :=u(f(x))−f(x). Then, the following two hold. 1.If0< d≤1, theng≥f, and the function gis increasing. 2./integraltext∞ 0f′(x) f(x)+g(x)dx=d We can now proceed to prove Lemma Appendix B.3. Proof of Lemma Appendix B.3. By Lemma Appendix B.2, the martingale Mncan be rewritten as Mn=I(Sn/√n+1)−ln(1... | https://arxiv.org/abs/2502.16019v1 |
u= (a+b)/√ 2 andv= (b−a)/√ 2. The integral becomes I=/radicalbigg 2 π/integraldisplayx′ 0/integraldisplay2x′−a aeabdbda, (B.5) wherex′=x/√ 2. Use now the change of variables u= ln(/radicalbig b/a) and v=√ ab. Then, I=/radicalbigg 2 π/integraldisplay∞ 0/integraldisplayx′√ 1−tanh2(u) 0ev22vdvdu (B.6) =/radicalbigg 2 π/in... | https://arxiv.org/abs/2502.16019v1 |
PLS- BASED APPROACH FOR FAIR REPRESENTATION LEARNING UNDER REVIEW Elena M. De-Diego Institute of Data Science and Artificial Intelligence (DATAI), Universidad de Navarra, Ismael Sánchez Bella Building, Campus Universitario 31009 Pamplona, Spain e.dediego.m@gmail.es Adrián Perez Suay Image Processing Laboratory (IPL) Un... | https://arxiv.org/abs/2502.16263v1 |
dimension; (ii) preserves information about the input space; (iii) is useful for predicting the target; (iv) is approximately independent of the sensitive variable. Our formulation has the same complexity as standard Partial Least Squares, or Kernel Partial Least Squares, and have applications on different domains and ... | https://arxiv.org/abs/2502.16263v1 |
measure DP requires that sensitive attributes should not influence the algorithm’s outcome, that is ˆY⊥ ⊥S; while for EO such independence is conditional to the ground-truth, that is ˆY⊥ ⊥S|Y. In the particular setting of binary classification, Y={0,1}, a classifier c:Rd→ {0,1} is said to be DP-fair, with respect to th... | https://arxiv.org/abs/2502.16263v1 |
fairness of any model trained on top of this new representation. FRL has been initially considered by Zemel et al. (2013) where they propose to learn a representation that is a probability distribution over clusters where learning the cluster of a sample does not give any information about the sensitive attribute S. Si... | https://arxiv.org/abs/2502.16263v1 |
we suppose thatX∈Rn×dis the original centred dvariables of nobservations and Y∈Rn×mbe the centred target. In this section, we begin by reviewing the Partial Least Squares (PLS) technique and then we introduce the first theoretical formulation of Fair Partial Least Squares. Our approach involves incorporating a regulari... | https://arxiv.org/abs/2502.16263v1 |
a consequence, the problem of finding the vectors wandcsuch that the components tanduare the ones with maximal covariance among all components in XandYspace respectively, is equivalent to the problem of computing the singular vectors of the singular value decomposition (SVD) of the matrix A=XTY. This is, the weight vec... | https://arxiv.org/abs/2502.16263v1 |
the empirical cross covariance matrix between AandBandWh={w∈Rd|wTw= 1,wTXTXwl= 0 ∀l∈[h−1]}. This problem is solved in an efficient manner with the Gradient Descent algorithm (Shalev-Shwartz and Ben-David, 2014). Then, at each iteration, we take a step in the direction of the negative of the gradient at the current poin... | https://arxiv.org/abs/2502.16263v1 |
score; while for η= 1.0they are: event, age, juv other count, juv misd count and priors count. Hence, the sensitive variable does not impact the Fair PLS components. 4 Extensions 4.1 Kernelizing Fair PLS Let us now extend our Fair PLS approach (Section 3.2) to the non-linear version of PLS by means of reproducing kerne... | https://arxiv.org/abs/2502.16263v1 |
w=ΦTα (Rosipal and Trejo, 2002). Hence, the Kernel Fair PLS (KFPLS) is: ∀h∈[k]wh= arg max w∈WKernel hC2 ϕ(X)w,Y−η∥Cϕ(X)w,ψ(S)∥2 HS≡ ∀h∈[k]αh= arg max α∈ℵh1 n2Tr(αTeKXYYTeKXα)−η1 n2Tr(αTeKXeKSeKXα) ,(7) 6 PLS-based approach for fair representation learning UNDER REVIEW where η >0is the regularization parameter and WKe... | https://arxiv.org/abs/2502.16263v1 |
understand the behavior of the language model. This framework has been recently presented in Jourdan et al. (2023a) and bias analysis in this context is discussed in Jourdan et al. (2023b) for instance. Hence, for SVD , the matrix Ais decomposed into A=U0Σ0V⊤ 0, where U0∈Rn×nandV0∈Rd×dare orthonormal matrices, and Σ0∈R... | https://arxiv.org/abs/2502.16263v1 |
the analyses carried out consist, based on our Fair PLS formulation, of evaluating the behaviour of the covariance between the projections and the target, and between the projections and the sensitive attribute, as the parameter ηincreases for six different datasets. Evidence is shown Table 1: The table summarizes the ... | https://arxiv.org/abs/2502.16263v1 |
As the ignored subspace is the orthogonal complement of the principal subspace, then the reconstruction error can be seen as the average squared distance between the original data points and their respective projections onto the principal subspace. For our purposes, an optimal representation is one for which the recons... | https://arxiv.org/abs/2502.16263v1 |
model trained on top of the Fair PLS representation and better predictive performance for the same level of fairness when is compared to existing methods for FRL as Fair PCA. References Katharina Morik. Medicine: Applications of Machine Learning , pages 654–661. Springer US, Boston, MA, 2010. ISBN 978-0-387-30164-8. do... | https://arxiv.org/abs/2502.16263v1 |
on Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems , volume 30. Curran Associates, Inc., 2017. URL https://proceedings.neurips.cc/ paper_files/paper/2017/file/a486cd07e4ac3d270571622f4f316ec5-Paper.pdf . Lucas De Lara, Alberto Gonzál... | https://arxiv.org/abs/2502.16263v1 |
volume 30. Cur- ran Associates, Inc., 2017. URL https://proceedings.neurips.cc/paper_files/paper/2017/file/ 8cb22bdd0b7ba1ab13d742e22eed8da2-Paper.pdf . 11 PLS-based approach for fair representation learning UNDER REVIEW Jian Liao, Chang Huang, Peter Kairouz, and Lalitha Sankar. Learning generative adversarial represen... | https://arxiv.org/abs/2502.16263v1 |
International Conference on Artificial Intelligence and Statistics , volume 108 of Proceedings of Machine Learning Research , pages 155–166. PMLR, 26–28 Aug 2020. URL https://proceedings.mlr.press/v108/tan20a.html . Kenji Fukumizu, Arthur Gretton, Xiaohai Sun, and Bernhard Schölkopf. Kernel measures of conditional depe... | https://arxiv.org/abs/2502.16263v1 |
Duchesnay. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research , 12:2825–2830, 2011. 7 Appendix to Sections 3 and 4.1 We provide here the algorithms for NIPALS. 13 PLS-based approach for fair representation learning UNDER REVIEW Algorithm 2: Nonlinear Iterative Partial Least Squares (NIPALS):... | https://arxiv.org/abs/2502.16263v1 |
described as values of 7 features: 2 categorical and 7 numerical. The objective of this dataset is to accurately predict if an applicant will have a high FYA, taking into account factors such as students entrance exam scores (LSAT), their grade-point average (GPA) collected prior to law school, and their first year ave... | https://arxiv.org/abs/2502.16263v1 |
group from two multivariate normal distributions on R7; with means (9,8,10,10,0,0,0) and(10,10,10,10,0,0,0), respectively, and different covariance matrix. For females, binary variables 4 and 5 strongly impact variable 0, while variable 6 influences variable 1. In contrast, for males, binary variables have little to no... | https://arxiv.org/abs/2502.16263v1 |
5.0 7.5 10.0 η0.000.020.04Objective functionsLaw School Dataset (d = 3, k = 2) 0.0 2.5 5.0 7.5 10.0 η0.000.010.020.030.04Objective functionsDiabetes Dataset (d = 27, k = 5) 0.0 2.5 5.0 7.5 10.0 η0.00.20.40.6Objective functionsCOMPAS Dataset (d = 6, k = 5) 0.0 2.5 5.0 7.5 10.0 η0.00.20.40.6Objective functionsCrimes Data... | https://arxiv.org/abs/2502.16263v1 |
.0012±0.0011 DT 0.9792±0.0431 0 .7613±0.1223 0 .0012±0.0014 0 .0012±0.0011 XGB 0.9651±0.0844 0 .814±0.1163 0 .0043±0.0071 0 .0012±0.0011 DiabetesLR 1.0±0.0 0 .601±0.0 0 .0±0.0 0 .0013±0.0011 DT 0.9924±0.0309 0 .5161±0.011 0 .0006±0.0005 0 .0013±0.0011 XGB 0.9212±0.0581 0 .5748±0.0057 0 .0036±0.0028 0 .0013±0.0011 COMPA... | https://arxiv.org/abs/2502.16263v1 |
C2 r(X),Yincreases but the C2 r(X),Salso increases, therefore fairness goal is not achieved. In contrast, this is not the case with Fair PLS as shown in the right column of Figure 5 Figure 5: Comparison of the covariance with respect to the target Yand the sensitive attribute Sbetween the new representation r(X)obtaine... | https://arxiv.org/abs/2502.16263v1 |
PCA unsupervised dimensionality reduction technique is based on the covariance matrix XTX. Nevertheless, in many applications it is important to weight the covariance matrix, this is, to replace XTXwithXTVX, being Va positive definite matrix. PLS algorithm choose as Vthe representative matrix for the "size" of the data... | https://arxiv.org/abs/2502.16263v1 |
Rectifying Conformity Scores for Better Conditional Coverage Vincent Plassier* 1Alexander Fishkov* 2 3Victor Dheur* 4Mohsen Guizani2Souhaib Ben Taieb2 4 Maxim Panov2Eric Moulines2 5 Abstract We present a new method for generating confi- dence sets within the split conformal prediction framework. Our method performs a t... | https://arxiv.org/abs/2502.16336v1 |
Another branch of work parti- tions the covariate space Xinto multiple regions and applies classical conformal prediction within each region (LeRoy & Zhao, 2021; Alaa et al., 2023; Kiyani et al., 2024). However, such partitioning based on the calibration set often leads to overly large prediction regions (Bian & Barber... | https://arxiv.org/abs/2502.16336v1 |
denote by P X,YoverX × Y the joint distribution of (X, Y).Construction of prediction regions for regression problems is often based on distributional regression that focuses on fully characterizing the conditional distribution of a re- sponse variable given a covariate (Klein, 2024). This ap- proach improves uncertaint... | https://arxiv.org/abs/2502.16336v1 |
specially transformed (rec- tified) scores to enhance conditional coverage. To construct the rectified scores, it builds on the key observation that marginal and conditional coverage coincide precisely when the conditional (1−α)-quantile of the conformity score is independent of the covariates. RCP then applies the SCP... | https://arxiv.org/abs/2502.16336v1 |
=f−1 τ⋆(x) V(x, y) : C∗ α(x) = y∈ Y:˜V⋆(x, y)≤φ . (9) In Appendix B.4, we establish that the rectified score sat- isfies (5), more precisely, setting ˜V⋆=˜V⋆(X, Y), for all x∈ X, φ=Q1−α P˜V⋆|X=x =Q1−α P˜V⋆ . (10) With the rectified score, conditional and unconditional cover- age coincide. However, while the orac... | https://arxiv.org/abs/2502.16336v1 |
this setting, are given, e.g., in (Chen & Wei, 2005; Koenker, 2005). Non-parametric methods have also been explored, mainly using kernel-based approaches; see (Christmann & Stein- wart, 2008). In these cases, gis a smoothness penalty (e.g a norm of an appropriately defined RKHS). More recently, Shen et al. (2024) intro... | https://arxiv.org/abs/2502.16336v1 |
score mea- sures the prediction error associated with the predictor µ(Nouretdinov et al., 2001; V ovk et al., 2005; 2009). Setting ft(v) = tvandφ= 1 , the rectified confor- mity scores are given by ˜V(x, y) =V(x, y)/bτ(x)where bτ(x)≈Q1−α PV∞|X=x , withV∞=V∞(X, Y). Thus, RCP is similar to the approach proposed in (Lei... | https://arxiv.org/abs/2502.16336v1 |
validity of split-conformal method. Theorem 3. Assume H1-H2 hold and suppose the rectified conformity scores {˜Vk}n+1 k=1are almost surely distinct. Then, 5 Rectifying Conformity Scores for Better Conditional Coverage for any α∈(0,1), it follows 1−α≤P(Yn+1∈ Cα(Xn+1))<1−α+1 n+ 1. The proof is postponed to Appendix B.3. ... | https://arxiv.org/abs/2502.16336v1 |
goal is to investigate the influ- ence of the quality of the (1−α)-quantile estimate bτon performance. We set α= 0.1and consider the conformity score V(x, y) =|y−µ(x)|. In this case, the (1−α)-quantile of 6 Rectifying Conformity Scores for Better Conditional Coverage 1 01X 2 02 Y11 012 Y2 Oracle 1 01X 2 02 Y11 012 Y2Re... | https://arxiv.org/abs/2502.16336v1 |
whereas PCP andDCP are compatible with the mixture predictor . Finally, SLCP , like RCP, is compatible with any conformity score and base predictor. For RCP, we compuate an estimate bτ(x)(see Section 4) using quantile regression with a fully connected neural network composed of 3 layers with 100 units. Experimental set... | https://arxiv.org/abs/2502.16336v1 |
discuss the choice of adjustment function. For certain adjustment functions, the domain of the scores v=V(x, y)must be restricted to a subset of Rto satisfy H2. Notably, ft(v) =tvrequires v >0,ft(v) = exp( tv) requires require v >1. Appendix A.5 presents an additional study comparing RCP with CP and CQR methods, that w... | https://arxiv.org/abs/2502.16336v1 |
2023.Dheur, V . and Taieb, S. B. Probabilistic calibration by design for neural network regression. In International Conference on Artificial Intelligence and Statistics , pp. 3133–3141. PMLR, 2024. Dheur, V ., Bosser, T., Izbicki, R., and Ben Taieb, S. Distribution-free conformal joint prediction regions for neural ma... | https://arxiv.org/abs/2502.16336v1 |
, 2021. Marx, C., Zhao, S., Neiswanger, W., and Ermon, S. Modular conformal calibration. In International Conference on Machine Learning , pp. 15180–15195. PMLR, 2022. Messoudi, S., Destercke, S., and Rousseau, S. Copula-based conformal prediction for multi-target regression. Pattern Recognition , 120:108101, 2021. Nal... | https://arxiv.org/abs/2502.16336v1 |
V ovk, V ., Gammerman, A., and Shafer, G. Algorithmic learning in a random world , volume 29. Springer, 2005. V ovk, V ., Nouretdinov, I., and Gammerman, A. On-line predictive linear regression. The Annals of Statistics , pp. 1566–1590, 2009. Wang, Z., Gao, R., Yin, M., Zhou, M., and Blei, D. Prob- abilistic conformal ... | https://arxiv.org/abs/2502.16336v1 |
blog datataxi10−1100Conditional coverage error RCP (−)-DCP RCP (∗)-DCP DCP scm20drf1 rf2scm1dmeps 21 meps 19 meps 20housebio blog datataxi10−1Conditional coverage error RCP (−)-PCP RCP (∗)-PCPRCP (−)-ResCP RCP (∗)-ResCP Figure 7: Marginal coverage and conditional coverage error obtained for two types of adjustments. 13... | https://arxiv.org/abs/2502.16336v1 |
the response). These graphs provide some important insights: •Methods based on classic conformal prediction (Const-CP, MLP-CP) often struggle to maintain conditional coverage. •RCP improves conditional coverage: Const-RCP outperforms Const-CP in conditional coverage and set size. •RCP in combination with any of these c... | https://arxiv.org/abs/2502.16336v1 |
assumption α≥ {n+ 1}−1, we have kα∈ {1, . . . , n }. Additionally, remark that ˜Vkhas the same distribution that F−1 ˜V(Uk). Therefore, by independence of the data, we can write P ˜Vn+1≤Q(1−α)(1+n−1)(1 nPn k=1δ˜Vk) =P F−1 ˜V(Un+1)≤F−1 ˜V(U(kα)) , where U(1), . . . , U (n)denotes the order statistics. Additionally, ... | https://arxiv.org/abs/2502.16336v1 |
We can rewrite the conditional coverage as follows P(Yn+1∈ Cα(Xn+1)|Xn+1=x) =Z P f−1 τ(x)(V(x, Y))≤t|X=x PQ(dt) =Z P ˜f−1 t(V(x, Y))≤τ(x)|X=x PQ(dt) =P ˜f−1 φ(V(x, Y))≤τ(x)|X=x +Z ∆t(x)PQ(dt) = 1−α+ϵτ(x) +Z ∆t(x)PQ(dt). Moreover, consider a set of ni.i.d. uniform random variables {Uk}1≤k≤n, and let U(1)≤. . .≤U(n... | https://arxiv.org/abs/2502.16336v1 |
+i =Zβ 0n!(β−u) (k−1)!(n−k)!uk−1(1−u)n−kdu. (36) Furthermore, we have the following derivations: Zβ 0(β−u)uk−1(1−u)n−kdu =βZβ 0uk−1(1−u)n−kdu−Zβ 0(β−u)uk(1−u)n−kdu =βI(k−1, n−k)−I(k, n−k). (37) 20 Rectifying Conformity Scores for Better Conditional Coverage Lastly, combining (36) with (37) yields the next result Eh β−... | https://arxiv.org/abs/2502.16336v1 |
hX+ supt∈R+{MtK1(t)}s 2 logm mE[ ˜wk(x)]2+2DhX(x) E˜wk(x)! ≤2 + 4E[ ˜wk(x)]−1Var[ ˜wk(x)] m, where DhX(x)is defined in (47). 22 Rectifying Conformity Scores for Better Conditional Coverage Proof. Letx∈ X andv∈Rbe fixed. First, recall that F˜Vφ|X(v|x) =P(V(X, Y)≤v|X=x). We will now control ˆF˜Vφ|X(v|x)−F˜Vφ|X(v|x)as be... | https://arxiv.org/abs/2502.16336v1 |
previous inequality implies that P sup v∈R{G(v)} ≥∆ ≤exp −∆2 m∥KhX∥∞ . (43) For any γ∈(0,1), setting ∆ =p m∥KhX∥∞log(1/γ)gives P sup v∈R{G(v)}<p m∥KhX∥∞log(1/γ) ≥1−γ. (44) The following statement controls P(supv∈R{G(v)} ≥ϵ). Its proof is similar to the extension of the Dvoretzky–Kiefer– Wolfowitz inequality provi... | https://arxiv.org/abs/2502.16336v1 |
H3 holds and let γ∈(0,1). With probability at least 1−γ, it holds sup v∈R mX k=1˜wk(x)n F˜Vφ|X(v|Xk)−F˜Vφ|X(v|x)o ≤mDhX(x) + sup t∈R+{MtK1(t)}r mlog(1/γ) 2. Proof. First of all, using H3 implies that sup v∈R mX k=1˜wk(x)n F˜Vφ|X(v|Xk)−F˜Vφ|X(v|x)o ≤mX k=1˜wk(x) min{1,M∥x−Xk∥}. For every k∈[m], let’s consider Zk= ˜wk(x)... | https://arxiv.org/abs/2502.16336v1 |
2−1ρ <|ϵτ(X)|;X∈An∩Bm,hX∩Em,hX ≤r. Equation (53) implies P(|P(Y∈ Cα(X)|X)−1 +α|> ρ)≤P(X /∈An∩Bm,hX∩Em,hX) +P(X /∈G) +r. Lastly, (54) combined with the previous inequality shows lim sup hX→0lim sup n→∞P(|P(Y∈ Cα(X)|X)−1 +α|> ρ)≤E 1fX(X)<r +r. Asris arbitrary fixed, from the dominated convergence theorem we can conclu... | https://arxiv.org/abs/2502.16336v1 |
arXiv:2502.16355v1 [math.ST] 22 Feb 2025Monotonicity Testing of High-Dimensional Distributions w ith Subcube Conditioning∗ Deeparnab Chakrabarty Dartmouth College deeparnab@dartmouth.eduXi Chen Columbia University xichen@cs.columbia.edu Simeon Ristic University of Pennsylvania sristic@seas.upenn.eduC. Seshadhri Univers... | https://arxiv.org/abs/2502.16355v1 |
. . . . . 9 3 Lower Bound for Testing Monotonicity 11 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The one-dimensional mean distributions . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 The distributions DyesandDno. . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.16355v1 |
•There is a growing body of work seeking to make high-dimensio nal distribution testing practical (particularly in the context of testing software that produces samples), where one can often implement more powerful query oracles, and in part icular, the subcube conditional sampling oracle (e.g., see [ MPC20 ,PM22 ,BCP+... | https://arxiv.org/abs/2502.16355v1 |
where the function value decreases). Directed isoperimetric inequalities relate the “non-monotone edge boundary” (the measure of surface ar ea) to the distance to monotonicity (the measure of volume). These inequalities have played a ma jor role in testing monotonicity of functions. In this work, we use a real-valued v... | https://arxiv.org/abs/2502.16355v1 |
is larger than that of 1, it can safely output “reject” (mean ing non-monotone). The challenge is understanding, when pisε-far from monotone, how likely it is that an “edge” {x,x(i)}is biased toward −1, and is this bias detectable from few samples. The connection to directed isoperimetry then becomes clear : considerin... | https://arxiv.org/abs/2502.16355v1 |
subcube conditioning for test ing and learning k-juntas [ CJLW21 ]. Uniformity of Monotone Distributions. The lower bound on testing uniformity of monotone distributions also proceeds by Yao’s minimax principle. We note that, the fact that testing uni- formity of product distributions required Ω(√n/ε2) samples was know... | https://arxiv.org/abs/2502.16355v1 |
real-valued function is ε-far from being monotone in the ℓ0-sense (which is usual in property testing), then the expect ed square-root of the negative influence is Ω(ε) thereby generalizing [ KMS18 ]. The authors use this result to give an O(r√ d/ε2)-query non- adaptive tester for real-valued monotone functions, where r... | https://arxiv.org/abs/2502.16355v1 |
on {−1,1}n. Furthermore, we receive the accuracy parameter ε∈(0,1). We let c0denote a sufficiently small constant. 1. For all integers w≥0 such that 2w=˜O(n/ε2), repeat the following t=O(2wlog(n/ε)) times: •Samplex∼pandi∼[n], and consider the restriction ρ∈ {− 1,1,∗}ngiven by ρj=xjifj/n⌉}ationslash=i, andρi=∗. •Letη=c2 0... | https://arxiv.org/abs/2502.16355v1 |
Prx∼p i∼[n] /parenleftBigg (p(x(i→−1))−p(x(i→1)))+ p(x(i→−1)) +p(x(i→1))/parenrightBigg2 ≥η ≥1 r·2γ+ℓ. forη=c2 0ε2·22γ+ℓ/(16h·n·logn). When the algorithm iterates over all w≥0 such that 2w= ˜O(n/ε2), it will eventually consider w= 2γ+ℓ. This implies that except with probability 0 .01, one of thetsamplesx∼pandi∼[n] ... | https://arxiv.org/abs/2502.16355v1 |
area by the distance to monotonicity. 8 Theorem 6 ([KMS18 ,PRW22 ]).There exists a universal constant C > 0such that for every f:{−1,1}n→ {0,1},Ex/ba∇⌈bl∇−f(x)/ba∇⌈bl2≥C·dist0(f). Theorem 3gives a real-valued generalization of the above theorem, wi th a√lognloss in the bound. The proof appears in Subsection 2.2, but we... | https://arxiv.org/abs/2502.16355v1 |
BRY14 ]).For any function f:{−1,1}n→[0,1],dist1(f) =/integraltext1 0dist0(ft)dt= Et[dist0(ft)]. The main work is in relating the (directed) gradients of fto the corresponding gradients of ft. This is where we suffer a√lognloss. Lemma 2.7. For allx∈ {− 1,1}n,/ba∇⌈bl∇−f(x)/ba∇⌈bl2= Ω(1/√logn)Et/ba∇⌈bl∇−ft(x)/ba∇⌈bl2. Proo... | https://arxiv.org/abs/2502.16355v1 |
distribution supported on {−1,1}n, andAlgbe a deterministic q-query algorithm with subcube conditioning access. There ex ists a func- tion Alg′:{−1,1}nq→ {“accept”,“reject”}which exactly simulates the algorithm on independent samples, i.e., Pr[Alg(p) =“accept” ] =Pr x1,...,xq∼p/bracketleftbig Alg′(x1,...,xq) =“accept”/... | https://arxiv.org/abs/2502.16355v1 |
det(A) = det(V) =/productdisplay i,j∈[ℓ] i<j(αj−αi)/n⌉}ationslash= 0, as long as α1,...,α ℓare distinct, and this means that Ais invertible. We now compute zusing Cramer’s rule: z0= 1 and for each i∈[ℓ],ziis given by zi=det(Ai) det(A)= (±1)·det(V(−1;α−i)) det(V). The numerator det( Ai) denotes the determinant of the ma... | https://arxiv.org/abs/2502.16355v1 |
at least 0.99. Proof: Letp∼ Dnoand letµ= (µ1,...,µn) be the corresponding mean vector, where µi∼ Dn for alli∈[n]. LetN⊂[n] denote the coordinates i∈[n] whereµi=−ε/√n. From the third item of Lemma 3.2and standard concentration inequalities, |N|def=mhas size at least c0n/2 with probability 1 −o(1). We assume this is the ... | https://arxiv.org/abs/2502.16355v1 |
x1,...,xq fromDyesorDnoby sampling r∼ RyorRnand then generating the samples x1,...,xqconditioned on the vector of counts r. Thus, we will derive ( 5) by showing that dTV/parenleftbig Ry,Rn/parenrightbig =o(1),whenq≤n ε2·polylog(n). Toward this end, we define G⊂ /CIn ≥0(in Lemma 3.6below) and show that •Lemma 3.6:r∈Gwith... | https://arxiv.org/abs/2502.16355v1 |
from Dno. 4.1 The distribution Dno A draw of p∼ Dnois generated as follows: •First, we let Ddenote the distribution over vectors µwhere we independently set µito be ε/n1/4with probability 1 /√nand 0 otherwise. •Then, we let pbe the monotone product distribution on {−1,1}nwhose mean vector is µ. The fact that p∼ Dnois f... | https://arxiv.org/abs/2502.16355v1 |
q≤√n/(cε2log4n) for some sufficiently large constant cand show below that the algorithm cannot distinguish betwee n the two cases. We set up some notation for the proof. We use i∈[q] as an index over the set of queries, while j∈[n] indexes the dimension/coordinate. We write fy,fn: ( /CAn)q→ /CA≥0to denote the probability... | https://arxiv.org/abs/2502.16355v1 |
≥exp/parenleftbigg −qε2 2√n/parenrightbigg ·/parenleftbigg n·exp/parenleftbiggqε2 2√n/parenrightbigg −√n log0.25n/parenrightbigg > n−√n log0.25n where in the last inequality we used exp( −qε2/2√n)<1. Substituting in (10), we get ln/parenleftbiggfn(x1,...,xq) fy(x1,...,xq)/parenrightbigg >−1 log0.25n−1 log2n(11) Hence, ... | https://arxiv.org/abs/2502.16355v1 |
coordinate condit ional sampling. In Proceedings of the 36th Annual Conference on Learning Theory (COLT ’2023) , 2023. [BDKR05] Tugkan Batu, Sanjoy Dasgupta, Ravi Kumar, and Roni tt Rubinfeld. The complexity of approximating the entropy. SIAM Journal on Computing , 35(1):132–150, 2005. [BFR+00] Tugkan Batu, Lance Fortn... | https://arxiv.org/abs/2502.16355v1 |
with subcube conditioning. In Proceedings of the 35th ACM-SIAM Symposium on Discrete Algori thms (SODA ’2024) , 2024. 24 [CR14] Cl´ ement L. Canonne and Ronitt Rubinfeld. Testing pr obability distributions underlying aggregated data. In Proceedings of the 41st International Colloquium on Automata , Languages and Progra... | https://arxiv.org/abs/2502.16355v1 |
Symposium on Discrete Algori thms (SODA ’2021) , 2021. [OS18] Krzysztof Onak and Xiaorui Sun. Probability–reveal ing samples. In Proceedings of the 21st International Conference on Artificial Intelligence and S tatistics (AISTATS ’2018) , 2018. [Pin23] Renato Ferreira Pinto Jr. Directed poincar´ e inequ alities and l1mo... | https://arxiv.org/abs/2502.16355v1 |
arXiv:2502.16462v1 [cs.LG] 23 Feb 2025Improved Margin Generalization Bounds for Voting Classifiers Mikael Møller Høgsgaard Aarhus University hogsgaard@cs.au.dkKasper Green Larsen Aarhus University larsen@cs.au.dk Abstract In this paper we establish a new margin-based generalizatio n bound for voting classifiers, refininge... | https://arxiv.org/abs/2502.16462v1 |
belong to a finite hypothesis class H⊆{−1,1}X. If we useLγ S(f)to denote the fraction of samples (x,y)inSfor which fhas margin no more than γ, then [Schapire et al., 1998 ] showed that for any data distribution DoverX×{−1,1}, any 0<δ<1 and 0<γ≤1, it holds with probability at least 1 −δover a training sequence S∼Dnthat e... | https://arxiv.org/abs/2502.16462v1 |
unlike in the case of finit eH(see discussion afterEq. (4)), there does not seem to be a way of tweaking the proof of the pre vious bound in Eq. (5)to improve the factors ln (m/slash.leftd)and lnmto ln(γ2m/slash.leftd). New Boosting Results. One of the prime motivations for studying generalization bounds for large margi... | https://arxiv.org/abs/2502.16462v1 |
the first 4 such algorithm. This algorithm uses the sub-sampling idea of [ Hanneke, 2016 ] from op- timal realizable PAC learning and runs AdaBoost on mlg43many sub-samples Si⊂Sof the training data. It combines the produced voting classifiers by ta king a majority vote among their predictions, i.e. a majority-of-majoriti... | https://arxiv.org/abs/2502.16462v1 |
D(f)= P(x,y)∼D[yf(x)≤γ],Lγ S(f)is the fraction of training examples (x,y)∈Swithyf(x)≤γ. Forγ=0 we will useLD(f)=L0 D(f)=P(x,y)∼D[yf(x)≤γ]=P(x,y)∼D[sign(f(x))/slash.left=y]. Partitioning into Intervals. To establish Eq. (7), we first simplify the task by parti- tioning the range of γandLγ S(f)into small intervals [γi 0,γ... | https://arxiv.org/abs/2502.16462v1 |
fixed - however we still want to be able to show that an imbalance between Lγ0/slash.left2 S(f′)andLγ0/slash.left2 S′(f′)for some f′∈Nis highly unlikely. As in previous works, we employ the following way of viewing the sampling of SandS′. First, we draw X∼D2m, consisting of 2 mi.i.d. training examples from D, and then l... | https://arxiv.org/abs/2502.16462v1 |
−γorγ. Wewill show that ∆ (H)⌈γ1⌉hasa small γ0/slash.left2-cover Nof cardinality just N∞(X,∆(H)⌈γ1⌉,γ0/slash.left2)=exp/parenleft.alt4O/parenleft.alt4d γ2 0Γ/parenleft.alt4mγ2 0 d/parenright.alt4/parenright.alt4/parenright.alt4, (12) where the notation N∞(⋅,⋅,⋅)for a point set X⊆Xfunction classF⊆RXand precision paramet... | https://arxiv.org/abs/2502.16462v1 |
Eq. (12) and completes our bound on the covering number and. All that remains is thus to argue that dcγ0ε/slash.left2=O(d(γ0ε)−2). Bounding Fat Shattering Dimension. To bound dcγ0ε/slash.left2, we use anargument similar to the proof of [ Larsen and Ritzert, 2022 ] [Lemma 9]. Assume ∆ (H)⌈γ1⌉cγ0ε/slash.left2-shatters a ... | https://arxiv.org/abs/2502.16462v1 |
the probability that a fixed set of kof the voting classifiers all err on xis precisely pkx. A union bound over all /parenleft.alt12k−1 k/parenright.alt1≤22kchoices of kvoting classifiers implies that P S[Maj(fS1,⋯,fS2k−1)(x)≠t(x)]≤22kpk x. Byswappingtheorderofexpectation, wecanboundtheexpected errorofMaj(fS1,⋯,fS2k−1) as... | https://arxiv.org/abs/2502.16462v1 |
the Rademacher complexity for a function class with VC-dimension dofO(/radical.alt1 d/slash.leftm)due to [Dudley, 1978 ] [See e.g. Theorem 7.2 [ Hajek and Raginsky, 2021 ]], instead of the weaker O(/radical.alt1 dln(m/slash.leftd)/slash.leftm)used in [ Schapire and Freund, 2012 ]. This bound has the right be- haviour f... | https://arxiv.org/abs/2502.16462v1 |
Eq. (21), whereby we have shown Eq. (19). We now bound the probability of the expression on the second line of Eq. (19) by sup X∈X2mN∞(X,∆(H)⌈2γ1⌉,γ0 6)δ e, which would conclude the proof. Now consider any γsuch that γ0≤γ≤γ1.We now recall that for a function f, we definedf⌈2γ1⌉as follows: f⌈2γ1⌉(x)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2γ1iff(x)>... | https://arxiv.org/abs/2502.16462v1 |
2+/roottop /rootmod/rootmod/rootbotLγ0 2 S(f)ln/parenleft.alt1e δ/parenright.alt1 m<Lγ0 2 S′(f) 2−/roottop /rootmod/rootmod/rootbotLγ0 2 S′(f)ln/parenleft.alt1e δ/parenright.alt1 m. (27) This implies that the event Lγ0 2 S′(f)≥τ1+32βandLγ0 2 S(f)≤τ1is in the complement of the eventLγ0 2 S(f)≥µ−/radical.alt2 µ2ln/parenl... | https://arxiv.org/abs/2502.16462v1 |
We now move on to prove Lemma 5 . To prove Lemma 5 , we need the following two lemmas. The first lemma bounds the infinity cover of ∆ (H)⌈γ⌉in terms of the fat shattering dimension of ∆ (H)⌈γ⌉. Lemma 7. There exists universal constants ˇC≥1andˇc>0such that: For X= {x1,...,x m}⊆Xa point set of size m,H⊆{−1,1}Xa hypothesis... | https://arxiv.org/abs/2502.16462v1 |
that f⌈γ⌉(xi)≤ri−βifbi=−1 (36) f⌈γ⌉(xi)≥ri+βifbi=1. (37) We now recall that f⌈γ⌉∈F⌈γ⌉is generated by a function f∈∆(H)in the following way f⌈γ⌉(x)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩γiff(x)>γ −γiff(x)<−γ f(x)else, i.e.falways being below f⌈γ⌉iff⌈γ⌉is less strictly less than γandfalways being above f⌈γ⌉iff⌈γ⌉is strictly larger than −γ. This im... | https://arxiv.org/abs/2502.16462v1 |
LD(f)≤Lγ S(f)+64/parenleft.alt2/roottop /rootmod/rootmod/rootbotLγ S(f)⋅2/parenleft.alt2ln/parenleft.alt11640e δ/parenright.alt1+9600˜c2min(c,1)d 32γ2Γ/parenleft.alt232mγ2 min(c,1)d/parenright.alt2/parenright.alt2 m(41) +5/parenleft.alt2ln/parenleft.alt11640e δ/parenright.alt1+9600˜c2min(c,1)d 32γ2Γ/parenleft.alt232mγ2... | https://arxiv.org/abs/2502.16462v1 |
the end of employing Corollary 6 withγ=min(2γi 1,1)andα=γi 0 4γi 1, we notice that forγ=min(2γi 1,1)andα=γi 0 4γi 1we have that N∞(X,∆(H)/uni2308.alt12γi 1/uni2309.alt1,γi 0 2)≤N∞(X,∆(H)⌈γ⌉,αγ) (44) where the inequality follows from in the case that γ=min(2γi 1,1)=1,then since the functions in ∆(H)only attains values i... | https://arxiv.org/abs/2502.16462v1 |
allj∈Ji, for allγ∈[γi 0,γi 1], for allf∈∆(H), that either Lγ S(f)/slash.left∈[τi,j 0,τi,j 1] or LD(f)<τi,j 1+βi,j=τi,j 1+64⎛ ⎜⎜ ⎝/roottop /rootmod/rootmod/rootbotτi,j 1⋅2ln/parenleft.alt1e δi/parenright.alt1 m+2ln/parenleft.alt1e δi/parenright.alt1 m⎞ ⎟⎟ ⎠, where the last equality uses that βi,j=64/parenleft.alt2/radic... | https://arxiv.org/abs/2502.16462v1 |
byP(x,y)∼Dt[(x,y)∈A]=Px∼D[(x,t(x))∈A]for anyA⊆X×{−1,1}. Furthermore, for R⊂X, such that Px∼D[x∈R]/slash.left=0 , we defineDt/divides.alt0Ras P (x,y)∼Dt/divides.alt0R[(x,y)∈A]=P x∼D[(x,t(x))∈A/divides.alt0x∈R]=Px∼D[(x,t(x))∈A,x∈R] Px∼D[x∈R]. We define a learning algorithm Las a mapping from (X×{−1,1})∗ ttoRX, forS∈ (X×{−1... | https://arxiv.org/abs/2502.16462v1 |
/summation.disp i=02−2i8C(i+1)22id γ2m=96Cd γ2m, where the second inequality follows from ∑∞ i=02−i(i+1)2=12 and this gives the claim ofLemma 12 . We thus proceed to show that for each i∈{0,1,2,...}, we have that PX∼D[Ri]≤8C(i+1)22id γ2m. To this end let i∈{0,1,2...}.IfPx∼D[Ri]=0 then we are done, thus we consider the ... | https://arxiv.org/abs/2502.16462v1 |
γ2i+cln/parenleft.alt1e δ/parenright.alt1 i≤max⎛ ⎜ ⎝2cdLn2/parenleft.alt2γ2i d/parenright.alt2 γ2i,2cln/parenleft.alt1e δ/parenright.alt1 i⎞ ⎟ ⎠(57) where we have upper bounded Γ by Ln2, which holds since for x≤eewe have that Γ(x)= Ln2(Ln(x))Ln(x)=Ln(x)≤Ln2(x)and forx>eeΓ(x)=ln2(ln(x))ln(x)≤ln2(x), and used that a+b≤2m... | https://arxiv.org/abs/2502.16462v1 |
Larsen, K. G. (20 20). Margins are insufficient for explaining gradient boosting. In NeurIPS. [Grønlund et al., 2019] Grønlund, A., Kamma, L., Larsen, K. G., Mathias en, A., and Nelson, J. (2019). Margin-based generalization lower bounds for b oosted classifiers. In NeurIPS, pages 11940–11949. [Hajek and Raginsky, 2021] H... | https://arxiv.org/abs/2502.16462v1 |
arXiv:2502.16574v1 [stat.ME] 23 Feb 2025Regularized zero-inflated Bernoulli regression model Mouhamed Ndoye(1)∗andAba Diop(1)† (1)Equipe de Recherche en Statistique et Mod` eles Al´ eatoires D´ epartement de Math´ ematiques Universit´ e Alioune Diop, Bambey, S´ en´ egal Abstract Logistic regression model is widely used ... | https://arxiv.org/abs/2502.16574v1 |
zero-inflated and overdispersed count data framework, and recent 2 research has focused on penalized ZIP models ((Czado et al., 2007), (Buu et al., 2011) and (Preisser et al., 2012)). Penalized regression methods are a useful theoretical approach for bothdeveloping predictive modelsandselecting key indicatorswithina nof... | https://arxiv.org/abs/2502.16574v1 |
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