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Nauki ,40(3), 111–118. 16. McKendrick, A.G. (1925). Applications of Mathematics to Medical Problems. Proceedings of the Edinburgh Mathematical Society 44, 98–130. 17. Olay ´e, J., Bouzidi, H., Aristov, A., Guti ´errez Ramos, S., Baroud, C., Bansaye, V . (2024). Estimation of the lifetime distribution from fluctuations ... | https://arxiv.org/abs/2504.06516v1 |
arXiv:2504.06569v1 [math.ST] 9 Apr 2025An Unbiased Variance Estimator with Denominator N Dai Akitaa aGraduate School of Information Science and Technology, The Univ ersity of Tokyo, Japan ARTICLE HISTORY Compiled April 10, 2025 ABSTRACT Standard practice obtains an unbiased variance estimator b y dividing by N−1 rather... | https://arxiv.org/abs/2504.06569v1 |
biased version. Moreover, [4] propo ses an alternative measure of dispersion that does not rely on explicitly estimating th e mean, thereby moving beyond the conventional NversusN−1 debate. Such discussions highlight the multi- faceted considerations in choosing avariance estimator, m otivating furtherexploration of ap... | https://arxiv.org/abs/2504.06569v1 |
numerous w ays to construct an average-adjusted unbiased variance estimator, their prac tical appeal remains limited. In general, among unbiased quadratic-form estimators of th e variance, the usual un- biased variance exhibits the smallest variance [13]. Becau se AAUVs also boil down to quadratic forms in the sample, ... | https://arxiv.org/abs/2504.06569v1 |
Equation (16), then choosing λ=/radicalbig (N−M)/Mgives ˜Xλ=1 MM/summationdisplay n=1Xn (31) and s2 λ=1 N−1+(N−M)/MN/summationdisplay n=1(Xn−˜Xλ)2, (32) where the mean estimator uses only Mdata points. As with AAUV, symmetrization recovers the standard unbiased variance. Theorem 4.1. Leti1,...,iNbe a permutation of the... | https://arxiv.org/abs/2504.06569v1 |
by adjusting the mean estimator rather than the more familia r approach of correcting 11 thedenominator. Thehalf-sample approach illustrated tha t usingonly part of thedata to estimate the mean could still yield an unbiased variance e stimator when dividing byN. Generalizing this idea led us to introduce average-adjust... | https://arxiv.org/abs/2504.06569v1 |
Weak Signals and Heavy Tails: Machine-learning meets Extreme Value Theory Stephan Cl´ emen¸ conaand Anne Sabourinb aLTCI, T´ el´ ecom Paris, Institut Polytechnique de Paris, Palaiseau, France bUniversit´ e Paris Cit´ e, CNRS, MAP5, F-75006 Paris, France April 10, 2025 Abstract The masses of data now available have open... | https://arxiv.org/abs/2504.06984v1 |
Covariate Tails . . . . . . . . . . . . . . . . . . . . . . 21 5 High Dimensional Extreme Covariates - XLASSO 23 5.1 Framework and Preliminaries . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Asymptotic linear Model on Extreme Covariates . . . . . . . . . . . 25 5.3 XLASSO: Statistical Guarantees . . . . . . . . . ... | https://arxiv.org/abs/2504.06984v1 |
into a bias term arising from the finite-distance nature of the data, and a variance term capturing deviations from the mean, conditional to an excess. Often, the bias term is excluded from the analysis, although regular variation assumptions ensure that it vanishes above sufficiently large thresholds. Through this rev... | https://arxiv.org/abs/2504.06984v1 |
in distribution of random elements Zn, n≥1 to a non degenerate limit Z∞(i.e.weak convergence) is denoted by Znw→Z∞. 2.2 Multivariate Extremes and Regular Variation Most of the material presented in this paper focuses on learning problems in multivariate (and possibly high dimensional) spaces, typically Rdwhen d > 1. We... | https://arxiv.org/abs/2504.06984v1 |
r(x), θ(x)) where r(x) =∥x∥andθ(x) =r(x)−1xis a point on the positive orthant S+of the sphere, which we call the angle ofx. Then the homogeneity property of µimplies that µ◦Polar−1is a product measure on R∗ +×S+, namely d( µ◦Polar−1)(r, θ) =dr r2⊗dΦ(θ). The angular component Φ, usually called the angular measure has fi... | https://arxiv.org/abs/2504.06984v1 |
n≥1. A bound of order O(log(n)/n) is thus obtained for maximum deviations in expectation. Upper confidence bounds are established in a similar way, using the bounded differences concentration inequality. Many simple classes ( e.g. half-spaces, hyperrectangles, ellipso¨ ıds, unions and intersections of such classes) hav... | https://arxiv.org/abs/2504.06984v1 |
functions, see 2.6 in van der Vaart and Wellner (1996)), the generalization capacity of classifiers learned using ERM can be assessed, the stochastic term being then of order OP(1/√n) up to a logarithmic factor. One may refer to Devroye et al. (2013) or Vapnik (2000) for a detailed presentation of statistical learning ... | https://arxiv.org/abs/2504.06984v1 |
sup A∈A|Pn(A)−P(A)| ≤O r p[log(1 /δ) + log( SA(np))] n! . The following normalized VC-inequality (Vapnik and Chervonenkis (2015); Anthony and Shawe-Taylor (1993), see Boucheron et al. (2005), Section 5 for further discussions) comes close to this goal: with probability 1 −δ, sup A∈AP(A)−Pn(A)p P(A)≤2s logSA(2n) + log4 ... | https://arxiv.org/abs/2504.06984v1 |
significantly the analysis of the error induced by marginal standardization, compared with the rectangular regions involved in the analysis of the standard tail dependence function. Indeed, the marginal errors bFj(x)−Fj(x) may not be analyzed separately from the devations of the pseudo- empirical process involving the ... | https://arxiv.org/abs/2504.06984v1 |
evtas regions close to the axes constitute a recurrent issue that have motivated various censoring approaches (Ledford and Tawn, 1996). The following bound is then valid with probability at least 1 −δ(see Theorem 3.1 in Cl´ emen¸ con et al. (2023)) sup A∈A|bΦ(A)−Φ(A)| ≤C1(δ, d,VΓ, k)√ k+C2(δ, d,VΓ, k) k+ Bias( k, n), (... | https://arxiv.org/abs/2504.06984v1 |
The material presented here is taken from Thomas et al. (2017); Cl´ emen¸ con et al. (2023). Minimum volume sets ( mv-sets in short), extending univariate quantiles, are the smallest sample space subsets containing at least αprobability mass, at some level α(Einmahl and Mason, 1992). This approach shares similarity wit... | https://arxiv.org/abs/2504.06984v1 |
(2006), together with algorithmic approaches to compute such solutions. Denoting by λthe Lebesgue measure on S+, the optimization problem solved in Thomas et al. (2017) to produce an empirical angular mv-setbΩαon the positive orhtant S+of the sphere is min Ω∈Aλ(Ω) subject to bΦ(Ω)≥α−ψ(δ). (14) where ψ(δ) is a tolerance... | https://arxiv.org/abs/2504.06984v1 |
learning problems. It is also one of a most natural framework in which uniform concentration bounds such as those introduced as background in Section 2 reveal themselves fruitful for proving generalization guarantees of classifiers obtained viaerm. Consider a classification problem where a random pair ( X, Y) is observ... | https://arxiv.org/abs/2504.06984v1 |
tail errors compared to bulk errors. To address data scarcity in the tails, it is assumed that the class distributions P(X∈ · |Y=σ),σ∈ {− 1,+1}, are regularly varying. Additionally, the ratio P(Y= +1| ∥X∥> t)/P(Y=−1| ∥X∥> t)must converge to a finite, non-zero limit to ensure the problem is neither trivial nor insoluble... | https://arxiv.org/abs/2504.06984v1 |
empirical estimation of the angular measure to the classification problem is that evaluating the empirical risk of a classifier gon extreme covariates involves counting the positive (resp. negative) instances ( θ(bVi), Yi= +1resp. −1) such that ∥bVi∥ ≥ ∥bV(k)∥, observed in positively (resp. negatively) assigned regions... | https://arxiv.org/abs/2504.06984v1 |
target distribution psatisfying the regularity condition (3) with b(t) =t, and (ii)the classification loss of a multilayer perceptron trained on the code Zbe small. The study also introduces a novel data augmentation mechanism ( GENELIEX ), generating synthetic sequences that maintain the original labels, building upon... | https://arxiv.org/abs/2504.06984v1 |
regression settings (Sections 4.4, 5 below). A broader class of problems where cross-validation comes as a natural approach include unseupervised contexts, e.g. for goodness-of-fit evaluation or model selction in parametric modelic of tail dependence (Einmahl et al., 2012, 2018, 2016; Kiriliouk et al., 2019) and model ... | https://arxiv.org/abs/2504.06984v1 |
z), g∈ G} associated with the predictor class Gis a VC subgraph class (see e.g. van der Vaart and Wellner, 1996, Section 2.6) and (ii) the cost function is bounded. It should be noted that the boundedness assumption precludes application to extreme quantile regression, leaving the extension to unbounded losses with app... | https://arxiv.org/abs/2504.06984v1 |
to negligible terms, for some context-dependent factors B1, B2. 20 a bounded loss may be seen as restrictive, although it aligns well with the ’learning on extreme covariates’ setting. This setting encompasses prediction problems in multivariate regularly varying random vectors, as seen in Example 4.1, or in the predic... | https://arxiv.org/abs/2504.06984v1 |
B ) =t−αP∞(A, B). Importantly, denoting by (X∞, Y∞) a random pair distributed according to P∞, the homogeneity of P∞implies that∥X∞∥ ⊥ ⊥ (θ(X∞), Y∞). A major consequence is that the regression function fP∞for the limit pair ( X∞, Y∞), defined by fP∞(X∞) =E[Y∞|X∞]almost surely, does not depend on the radial component r(... | https://arxiv.org/abs/2504.06984v1 |
p(x, y) =∥x∥π(x,∥x∥y), with limit density p∞(x, y) = ∥x∥π∞(x,∥x∥y), and same scaling function as that of the pair (X, Z), namely sup ∥x∥≥1,y∈R b(t)tdp(tx, y)−p∞(x, y) − − − → t→∞0. (22) Finally, under the additional condition that the limit marginal density πx,∞(x) =R Rπ∞(x, y) dyis lower bounded on SinRd,i.e. infSπx,∞... | https://arxiv.org/abs/2504.06984v1 |
which is the solution to Equation (23). Straightforward extensions to the constrained Lasso can be readily derived using similar, albeit simpler, arguments. For instance, in Aghbalou et al. (2024a), a constrained logistic-Lasso algorithm is examined as a primary example of a model selection problem involving extreme co... | https://arxiv.org/abs/2504.06984v1 |
where εis a bounded noise, |ε| ≤Mϵalmost surely, and b:Rd→Ris a bounded function that vanishes at infinity, sup x:r(x)>t|b(x)| − − − → t→∞0. Notice that the above assumption of a linear relationship between Yandθ(X) only holds asymptotically as ∥X∥ → ∞ , and the inclusion of a bias term b(X) must be somehow acknowledge... | https://arxiv.org/abs/2504.06984v1 |
where tn,κdenotes the 1−κ/nquantile of the random variable ∥X∥, ˜k(δ) =k 1 +r 3 log(1 /δ) k+3 log(1 /δ) k , and¯b(t) = sup∥x∥>tb(x), see Assumption 5.1. Our main result derives immediately from Lemma 5.1 and Proposition 5.1. Theorem 5.1 (XLASSO: prediction guarantees) .With the notations of Proposition 5.1, letλsatis... | https://arxiv.org/abs/2504.06984v1 |
n= 13577). In this work we take the Trans variable (transportation sector) as a target, to be predicted given that the other variables are large. In this example the covariate vector Xhas dimension d= 48. Based on the experiments in previous works mentioned above bringing evidence of multivariate regular variation, we ... | https://arxiv.org/abs/2504.06984v1 |
to the proof of Proposition 4.1. The result is unsuprising and similar ones are likely to be found in other works focused on graphical structures for extremes, however we find it simpler to give a short, sef-contained proof. Notice that the required conditions on the norms on RdandRd+1in the statement hold true for any... | https://arxiv.org/abs/2504.06984v1 |
The latter display is precisely Assumption 4.2 with P∞=˜P∞◦φ−1. Finally the relationship between Π ∞andP∞derives immediately from the relationship between Π∞and ˜P∞stated in Lemma A.1. Observe also the following identity: let Π ∞be the limit distribution for ( X, Z) defined by L(t−1(X, Z)| ∥(X, Z)∥> t)− − − → t→∞Π∞, an... | https://arxiv.org/abs/2504.06984v1 |
random variables Ti,j=Zi,jε(i)are independent. Also |Ti,j| ≤Mεalmost surely and by independence, E[Ti,j|(Zℓ,j, ℓ≤k)]=Zi,jE[ε(i)] = 0. A direct application of McDiarmid’s inequality (conditionally on the Zi,j’s) yields that for t >0, for fixed j≤d, almost surely, P k−1X iTi,j ≥t (Zi,j, i≤k) ≤2 exp−2kt2 M2ε . Integra... | https://arxiv.org/abs/2504.06984v1 |
and how well does it do it? Journal of the American Statistical Association , pages 1–12. 34 Blanchard, G., Lee, G., and Scott, C. (2010). Semi-supervised novelty detection. The Journal of Machine Learning Research , 11:2973–3009. Boucheron, S., Bousquet, O., and Lugosi, G. (2005). Theory of Classification: A Survey of... | https://arxiv.org/abs/2504.06984v1 |
, 51(1):43–60. Daouia, A., Gardes, L., and Girard, S. (2013). On kernel smoothing for extremal quantile regression. Bernoulli , 19(5B):2557–2589. De Haan, L. and Resnick, S. (1987). On regular variation of probability densities. Stochastic processes and their applications , 25:83–93. Devroye, L., Gy¨ orfi, L., and Lugo... | https://arxiv.org/abs/2504.06984v1 |
measures on metric spaces. Publications de l’Institut Mathematique , 80(94):121–140. Jalalzai, H., Cl´ emen¸ con, S., and Sabourin, A. (2018). On binary classification in extreme regions. In Advances in Neural Information Processing Systems , pages 3092–3100. 37 Jalalzai, H., Colombo, P., Clavel, C., Gaussier, E., Varn... | https://arxiv.org/abs/2504.06984v1 |
en, H. and Tajvidi, N. (2006). Multivariate generalized pareto distributions. Bernoulli , 12(5):917–930. Rudelson, M. and Zhou, S. (2012). Reconstruction from anisotropic random measurements. In Conference on Learning Theory , pages 10–1. JMLR Workshop and Conference Proceedings. Sabourin, A. (2021). Extreme Value Theo... | https://arxiv.org/abs/2504.06984v1 |
Assessing dominance in survival functions: A test for right-censored data F´ elix Belzunce, Carolina Mart´ ınez-Riquelme and Jaime Valenciano Dpto. Estad´ ıstica e Investigaci´ on Operativa Universidad de Murcia Facultad de Matem´ aticas, Campus de Espinardo 30100 Espinardo (Murcia), SPAIN belzunce@um.es, carolina.mart... | https://arxiv.org/abs/2504.07012v1 |
the Kaplan-Meier estimator is a widely used non-parametric method for estimating the sur- vival function. Additionally, several statistical tests are available to assess whether survival functions differ. The log-rank test is the most commonly used statistical test for comparing survival curves. It compares the observe... | https://arxiv.org/abs/2504.07012v1 |
and let Ci,i= 1, . . . , n , be independent and identically dis- tributed censoring times with common survival function HC(t). It is assumed that failure and censoring times are independent. The dataset consists of bivariate random vectors ( Xi, δX i), where Xi=Ti∧Ci, where ∧denotes the minimum, with common distributio... | https://arxiv.org/abs/2504.07012v1 |
Di,i= 1, . . . , m , be inde- pendent and identically distributed censoring times with common survival function HD(t). Again, it is assumed that failure and censoring times are independent. The second dataset consists of bivariate random vectors ( Yi, δY i), where Yi=Ui∧DiandδY i=I(Ui≤Di) indi- cates whether Yiis censo... | https://arxiv.org/abs/2504.07012v1 |
provide the upper bound (2.1) for the p-value we need to compute the probability for the supremum of a Gaussian process. A common computational method is to approximate the probability via Monte Carlo simulation. The idea is to generate Nsamples of the Gaussian process over a discretized grid, then compute the supremum... | https://arxiv.org/abs/2504.07012v1 |
page 6. 6 Test Test statistic p-value Log-Rank 10.3267 0.0013 Gehan 12.4721 0.0004 Tarone–Ware 12.4555 0.0004 Peto–Peto 12.7078 0.0003 Modified Peto–Peto 12.7091 0.0003 Table 1: Test statistics and p-values of some classical tests for the lung dataset. In this case we compare times to infection for Group 1 and Group 2.... | https://arxiv.org/abs/2504.07012v1 |
different shape parameter, λ. In each case, we have selected two different λ’s to get approximately a rate of 20% and a 50% of censored observations. To get a 20% and a 50% rate of censoring we have considered λequal to −log(0.9)/q20and log(2) /q50, respectively, where q20andq50 are the 20% and 50% quantile of the corr... | https://arxiv.org/abs/2504.07012v1 |
to crossings is a key advantage over traditional survival comparison tests (like log-rank), which often fail to detect any difference when survival curves intersect. Such improved detection 9 can lead to better insights in studies where survival functions may cross, ensuring that meaningful survival differences are not... | https://arxiv.org/abs/2504.07012v1 |
dominance evolves over time. Overall, our test represents a novel and powerful alternative for comparing survival func- tions, particularly when the interest lies in detecting dominance or structural changes in survival relationships. By incorporating supremum-based differences in Kaplan-Meier esti- mators, our approac... | https://arxiv.org/abs/2504.07012v1 |
Can SGD Select Good Fishermen? Local Convergence under Self-Selection Biases and Beyond Alkis Kalavasis Anay Mehrotra Felix Zhou Yale University Yale University Yale University alkis.kalavasis@yale.edu anaymehrotra1@gmail.com felix.zhou@yale.edu Abstract We revisit the problem of estimating klinear regressors with self... | https://arxiv.org/abs/2504.07133v1 |
. . . . . . . . . . . . . . . . 12 1.5 Towards Global Convergence Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6 Takeaways for Self-Selection and Open Questions . . . . . . . . . . . . . . . . . . . . 14 1.7 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... | https://arxiv.org/abs/2504.07133v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.2 Local Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.3 Projection Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6.4 Stochastic Gradient Oracle and Second Moment . .... | https://arxiv.org/abs/2504.07133v1 |
as functions of features xencoding, e.g., location, size, and amenities. For a given x, only the trans- action price (x, min{yS(x),yD(x)})is observed instead of the complete sample (x,yS(x),yD(x)), leaving it unclear whether the market imbalance stems from excess supply or demand. Another classic model due to Roy [Roy5... | https://arxiv.org/abs/2504.07133v1 |
Fair and Jaffee [FJ72]. Remarkably, [CDIZ23] presented an algorithm to estimate regression parameters w⋆ 1, . . . , w⋆ k to within poly (1/k)-error in poly (d)·epoly(k)time under Definition 1. This result is surprising since even the identifiability of the parameters under Definition 1 is highly non-trivial. Subsequent... | https://arxiv.org/abs/2504.07133v1 |
fields, including Economics, Engineering, Medical and Biological Sciences, and all areas of the Physical Sciences ( e.g., [HR90; HR91; Hei93; LEA97; GMBA+20]). One of the simplest forms of coarsening is rounding, where data values are mapped to the nearest point on a specified lattice. The problem of estimation from co... | https://arxiv.org/abs/2504.07133v1 |
revisit the notion of information preservation of a partition P, introduced by [FKKT21], and provide a more general formulation. Definition 2 (Information Preservation ).Given constants α∈[0, 1], true parameters (µ⋆,Σ⋆)of a d-dimensional Gaussian distribution N(µ⋆,Σ⋆), and a partition Pof the domain Rd,Pis said to be α... | https://arxiv.org/abs/2504.07133v1 |
the opti- mum given a sufficiently good warm start. A natural algorithm of this kind is stochastic gradient descent (SGD) performed on (1). However, proving theoretical guarantees for SGD is highly non- trivial because it relies on showing that the negative log-likelihood objective of (1) is convex in a sufficiently la... | https://arxiv.org/abs/2504.07133v1 |
and gives us Informal Theorem 1. Next, we show how to implement each one of these steps. Step 1 (From Self-Selection to Coarse-Inference). Recall that in self-selection, given the feature vector x∈Rd, we observe the maximum of kvariables y1, . . . , ykdetermined by linear regression models yi=x⊤w⋆ i+ξi. We view the obs... | https://arxiv.org/abs/2504.07133v1 |
norm, it holds that ∇2L(W)⪰0 , whereL(·)is the population negative log-likelihood objective of the self-selection problem evaluated at W . This result shows that the radius where the landscape is convex is of order poly (1/k)which has no dependence on d. As for information preservation, we postpone the technical detail... | https://arxiv.org/abs/2504.07133v1 |
preservation). Challenge II: No Notion of Distance to Pymax.A natural approach to lower bounding αis to use Definition 2. This requires showing that if a parameter Wisε-far from W⋆, then the distributions M(W)andM(W⋆)of the observed maximum ymaxwith parameters Wand W⋆respectively are alsoε-far from each other (in total... | https://arxiv.org/abs/2504.07133v1 |
linear dependence. Our Approach to Show Information Preservation. To obtain the poly (1/k)-information preserva- tion, we draw inspiration from the events considered in [GM24] to define certain, slightly differ- ent, ”good” events Eifori∈[k]over the randomness in xthat happens with constant probability (Definition 6). ... | https://arxiv.org/abs/2504.07133v1 |
Local Convexity. Instead, we follow a more algebraic route. First, thanks to information preservation, we know that L(W)is strongly convex at the true optimal param- eters W⋆; also we know that Lhas quadratic growth (Lemma 1.1). We hence directly bound the change in Hessian value, i.e., upper bound the difference ∇2L(W... | https://arxiv.org/abs/2504.07133v1 |
. . ,ξkare independent N(0, 1)random variables, and 3. y max,2 is the second-highest bid ,i.e., ymax,2=max j̸=ix⊤w⋆ j+ξj. The unknown parameters w⋆ 1, . . . , w⋆ ksatisfy the same conditions as in Definition 1. Figure 6 illustrates the coarsening arising from this model. This illustration shows that the par- tition ari... | https://arxiv.org/abs/2504.07133v1 |
is dropped, then the problem becomes computationally hard in general. They prove their hardness result using a reduction from MAXCUT . Given a MAXCUT instance, they design partitions of Rdconsisting of intersections of two halfspaces, ellipsoids, and their complements and (roughly speaking) show that if there were an e... | https://arxiv.org/abs/2504.07133v1 |
to implement this step (see Appendix C). 1.6 Takeaways for Self-Selection and Open Questions The tractability for convex partitions and intractability for some non-convex partitions positions the problem of self-selection in between the algorithm in Informal Theorem 6 and the hardness result of [FKKT21]: on the one han... | https://arxiv.org/abs/2504.07133v1 |
of Ghosh, Pananjady, Guntuboyina, and Ramchandran [GPGR22]. We mention that our techniques could be used to obtain local convergence guarantees for max-affine regression; however, such a result already exists by Kim and Lee [KL24] (handling sub-Gaussian noise). Both [CDIZ23] and [GM24]’s algorithms for the model in Def... | https://arxiv.org/abs/2504.07133v1 |
the support of the distribution, called survival set , are not observed. Truncation arises in a variety of fields from Econometrics [Mad83], to Astronomy and other phys- ical sciences [Woo85], to Causal Inference [IR15; HR23]. Another recent line of work tackles the problem of testing whether a given source of data is ... | https://arxiv.org/abs/2504.07133v1 |
machine autopsy – which have received significant attention [Mei81; Ger88; Now90; ADH93]; while many of these works focus on the nonparametric setting, they have natural parametric counterparts where our results are applicable. 2 Preliminaries and Notation In this section, we introduce preliminaries and (standard) nota... | https://arxiv.org/abs/2504.07133v1 |
a family of distributions {P(θ):θ∈Θ}over it. Let f:X→Ybe a (deterministic) distortion mechanism that transforms each element x∈Xto some element of Y. This mapping induces, for any distribution P(θ), a new distribution Pf(θ)overY, where Pf(θ;y) =R x:f(x)=yP(θ;x)dxfor any y∈Y. The statistical task of interest is the foll... | https://arxiv.org/abs/2504.07133v1 |
(x,y1, . . . , yk), one observes (x,f(y1, . . . , yk))for some known function f:Rk→R, which prevents us from using learning each regressor w⋆ iseparately. We focus on the max-self-selection bias, where f(·) =max(·)is the maximum function. This set- ting was studied by Cherapanamjeri, Daskalakis, Ilyas, and Zampetakis [... | https://arxiv.org/abs/2504.07133v1 |
strongly convex in a poly (1/k)-sized neighborhood of W⋆ (Theorem 3.5). Note that globally the negative log-likelihood is highly non-convex: it has at least k! distinct stationary points (corresponding to the permutations of W⋆). This local strong convex- ity is sufficient to deduce Theorem 3.3 as a point within this p... | https://arxiv.org/abs/2504.07133v1 |
coarse partition created by this model. We study this problem under the same assumptions as we considered for the max-self-selection problem. y1 y2 y3 Figure 6: This figure illustrates one possible observation from the second-price auction model (Definition 3). The figure illustrates the set Pcorresponding to the obser... | https://arxiv.org/abs/2504.07133v1 |
eµsatisfying ∥eµ−µ⋆∥2≤ε 24 with probability 1−δ. Moreover, the algorithm requires m=eOdD2log(1/δ) α4+dlog(1/δ) α4ε2 i.i.d. samples from NP(µ⋆,I)andpoly(m,Ts)time, where T sis the time complexity of sampling from a Gaussian distribution truncated to a set P ∈P. Moreover, if the facet-complexity5of every observed set P... | https://arxiv.org/abs/2504.07133v1 |
and C (21)Bounds on Terms A, B, and CLower Bound on Eq. (20) Bound on Term CBound on Term BBound on Term A Lemma 5.4 Figure 7: Outline of Proof of Information Preservation for Self-Selection. Toward proving Equation (7), fix any parameters V= [v1, . . . , vk]and W= [w1, . . . , wk]close to each other and W⋆in the follo... | https://arxiv.org/abs/2504.07133v1 |
two reasons which are outlined in the left sub-branch of the right branch of Figure 7. Property 1 ( Eoccurs with poly(1/k)probability). First,Eoccurs with a constant probability for any fixed γand R. Lemma 5.2. For any γ∈(0,1/2], R≥2, and the corresponding event E=Ei,γ,R(Definition 6), Pr[E]=γ28+6R2. Eventually, we wil... | https://arxiv.org/abs/2504.07133v1 |
much, we can just focus on small changes in ρi,Vandρi,W(for the index iin Equation (10)). • So far, we mentioned that due to the first two properties, to lower bound the total variation distance in Equation (14) it suffices to focus on the i-th coordinate. The last condition shows thatρi,Vandρi,Ware not too close to ea... | https://arxiv.org/abs/2504.07133v1 |
the corresponding vectors in the translated space. Elementary Set Witnesses Information Perservation Conditional on E.Next, we move to the right sub-branch of the right branch of Figure 7. Since d TV(M(V|E),M(W|E))is defined as a supremum over all sets S⊆R dTV(M(V|E),M(W|E))=max S⊆R|Pr[ymax, V∈S|E]−Pr[ymax, W∈S|E]|, (1... | https://arxiv.org/abs/2504.07133v1 |
e20· vj−wj 2·c k. (28) Step 1.2 (Bound on Term B).Substituting the values of Randγ(Equation (24)) into the first part of Lemma 5.4 implies that B/∈1±1 100C·min( 1 ,√ 72C c·r 20+3 logk c+2 log C·∥vi−wi∥2) . (29) Observe that Equation (9) implies that ∥vi−wi∥2≤∥V−W∥F≤c√ 72C2·1 10 log k/c(k≥2,c≤1,C≥1) ≤c√ 72C·1p 20+3 log ... | https://arxiv.org/abs/2504.07133v1 |
5.8 Properties of “Norm-Bound” Event E Tail Bounds Bounds on Conditional Expectations (Lemma 5.13)Bounds on Unconditional Expectations (Lemma 5.14)High Probabil- ity Guarantee (Lemma 5.10)Change of Measure II (Corollary 5.12) Change of Measure I (Lemma 5.11) Figure 8: Outline of Proof of Local Convexity for Self-Select... | https://arxiv.org/abs/2504.07133v1 |
is given in Section 5.2.4. Given these two lemmas, we are ready to prove the strong convexity property near W⋆. Proof of Theorem 5.6. FixVwith∥V∥F=1. We will prove the desired statement for ρ∈(0, 1)but only need the smaller range to establish Theorem 5.6. Define t=ρ·eO1 C2B4k17/2 and ζ=ρ·1 k11/2C2. (42) 37 By constru... | https://arxiv.org/abs/2504.07133v1 |
in Section 5.2.6. This change-of-measure allows us to multiplicatively relate expectations of Wtand W0. While, in general, these expectations are not easily related but we show that they are similar conditioned on the good event E. Lemma 5.11 (Change of Measure I) .For function f :Rk→Rand t∈[0, 1], e−O(tB4k3)≤Et[f(z)|E... | https://arxiv.org/abs/2504.07133v1 |
Ex,ymax γ(x,ymax)·Ik⊗xx⊤ . We use the following fact, which we proved at the end of this section. Fact 5.15 (Expectations of Powers of L∞-norms of Gaussian Vectors) .For1≤ℓ≤6andµ∈Rk, E v∼N(µ,I)∥v∥ℓ ∞≤12+384·(10+logk)ℓ/2+ℓ∥µ∥ℓ ∞. Step 1 (Upper bound Eymaxγ(x,ymax)):Fix any x∈Rd. By linearity of expectation, E ymaxγ(x,... | https://arxiv.org/abs/2504.07133v1 |
Observe that, E[Mℓ] =Z∞ 0Prh Mℓ>wi dw=Z∞ 0ℓrℓ−1Pr[M>r]dr. We divide the integral into two parts: Z ℓrℓ−1Pr[M>r]dr=Z2√ 10+logk 0ℓrℓ−1Pr[M>r]dr+Z∞ 2√ 10+logkℓrℓ−1Pr[M>r]dr. (56) Since Pr [M>r]≤1, the first term satisfies the following upper bound Z2√ 10+logk 0ℓrℓ−1Pr[M>r]dr≤ℓ·2ℓ·(10+logk)ℓ/2. Next observe that, for any r... | https://arxiv.org/abs/2504.07133v1 |
|(44)|≤ζ·r Pr[E]·E0h ⟨v,z⟩4i and |(46)|≤ζ·Pr[E]·E0h ⟨v,z⟩2i . Since Pr [E]≤1, the upper bounds further simplify to |(44)|≤ζ·r E0h ⟨v,z⟩4i and |(46)|≤ζ· E0h ⟨v,z⟩2i2 . Part B (Bounding terms Equations (45),(47) and(48)):First, note that |(45)|≤Pr[¬E]· Et[⟨v,z⟩2| ¬E] +E0[⟨v,z⟩2| ¬E] . Applying the Cauchy–Schwarz ineq... | https://arxiv.org/abs/2504.07133v1 |
The gradient of the negative log-likelihood function (Equation (88)) at W∈Kis given by ∇L(W) =E xE ymax E z∼N(W⊤x,I)|z∈P(ymax)h xz⊤i −xx⊤W! . Lemma 5.17. Consider an instance of the max-self-selection problem and suppose Assumptions 1 and 2 hold. Fix W in the projection set K (Equation (65)). Suppose (x,ymax)is drawn f... | https://arxiv.org/abs/2504.07133v1 |
Suppose we wish to bound the joint TV distance of every coordinate by ξwith probability 1 −δover the observed covariate. This leads to a total running time of k·polylog (k/δ,k/ξ)for all sampling procedures. In the δprobability event of failure, we can simply output an arbitrary vector in P(ymax), and the TV distance is... | https://arxiv.org/abs/2504.07133v1 |
Projected Stochastic Gradient Descent We are now ready to prove Theorem 3.4 by applying the iterative PSGD algorithm (Theorem 8.3). Our complete algorithm (Theorem 3.3) follows by combining this result with an appropriate warm start. We restate the theorem below for convenience. Theorem 3.4 (Polynomial Time Local Conve... | https://arxiv.org/abs/2504.07133v1 |
. . . , vk]and W= [w1, . . . , wk]close to each other and W⋆in the following sense ∥V−W⋆∥F,∥W−W⋆∥F≤c 400Clog k/c. (71) Since the only difference between the two problems is in the observations, we can reproduce the proof in Section 5.1 until Equation (20). At this point, we get that it is sufficient to prove that, for ... | https://arxiv.org/abs/2504.07133v1 |
lemmas can be proved by adapting the proofs of lemmas in Section 5.2. Negative Log-Likelihood and Its Hessian. To state the negative log-likelihood, define the fol- lowing set: given a value of the second max sand an index of the winner i, define P(s,i):=n z∈Rk:zmax,2=sand zimax=zmaxo . Now, the conditional negative lo... | https://arxiv.org/abs/2504.07133v1 |
tx,1)⟨v,z⟩2|z∈P(ymax) and E z∼N(W⊤ tx,1)h ⟨v,z⟩4|z∈P(ymax)i . Its proof divides the moment into k-parts one corresponding to each coordinate of zand, further, breaks each conditioning z∈P(ymax)into k-subparts corresponding to the k-parts of Pymax; see 58 Figure 9. One can repeat the same proof for P(ymax,2 ,imax)sinc... | https://arxiv.org/abs/2504.07133v1 |
them corresponds to P(ymax,2 ,imax). Compare this figure with Figure 6 which illustrates P(ymax,2 ,imax). In this section, we bound the second moment of the gradient, as promised in the proof of Lemma 6.5. Fix any Vwith∥V∥=1. For each t∈R, define Wt=W⋆+tV. Our goal is to prove the following upper bound (for 0 ≤t≤1) E x... | https://arxiv.org/abs/2504.07133v1 |
,imaxE zh ∥z−W⊤ tx∥2 2|z∈P(ymax,2 ,imax)i ·∥x∥2 2 ≤O(k4C2logk)·E x>2kh O(log2k) +∥x>2k∥2 2·O(logk)i . Finally, using Ex>2k[∥x>2k∥2 2] = ( d−2k)Eu∼N(0,1)[u2] =O(d−2k), we get that E x E ymax,2 ,imaxE zh ∥z−W⊤ tx∥2 2|z∈P(ymax,2 ,imax)i ·∥x∥2 2 ≤(d−2k)·O(k4C2log3k)≤d·O(k4C2log3k). 6.5 Projected Stochastic Gradient Desc... | https://arxiv.org/abs/2504.07133v1 |
C.1 for the formal definition and more details. 63 7.2 Convexity and Local Growth of Log-Likelihood Under Convex Partitions It can be verified (Appendix A.3) that the coarse negative log-likelihood of the mean under canon- ical parameterization is given by the following LP(µ) =∑ P∈PN(µ⋆,I;P)LP(µ), where LP(µ):=−log(N(µ... | https://arxiv.org/abs/2504.07133v1 |
can then bound the first term using a deterministic bound and the second term using the second moment of N(µ⋆,I). E P∼NP(µ⋆,I)E x∼N(µ,I,P)h ∥x∥2 2·1B∞(0,R)i +E P∼NP(µ⋆,I)E x∼N(µ⋆,I,P)h ∥x∥2 2·1B∞(0,R)ci ≤ dR2+E x∼N(µ⋆,I)h ∥x∥2 2i = dR2+d+∥µ⋆∥2 2 = O(dR2+D2). 7.5 Projected Stochastic Gradient Descent We run the iterativ... | https://arxiv.org/abs/2504.07133v1 |
(η-Local Growth Condition) .We say that F :K→Rsatisfies an (η,ρ)-local growth condition if ∥w−w⋆∥2≤(F(w)−F(w⋆))1 2 η for every w ∈Sρ. We remark that a function satisfying a (η0,ρ0)-local growth condition also satisfies (η,ρ)-local growth for every η∈(0,η0],ρ∈(0,ρ0]. We also note that by Lemma 1.1, our log-likelihood fu... | https://arxiv.org/abs/2504.07133v1 |
bound over τ stages. The base case for ℓ=0 holds by assumption. We consider some ℓ≥1. Remark that Dℓ=εℓ−1 η√εand γℓ=εℓ 100G2τ. By Proposition 8.2, we have (w(ℓ−1))† ε−w(ℓ−1) 2≤1 η√ε(F(w(ℓ−1))−F(w(ℓ−1))† ε)≤εℓ−1 η√ε≤Dℓ. Then Theorem 8.1 on the ℓ-th stage of PSGD yields E[F(w(ℓ))−F((w(ℓ−1))† ε)]≤γℓG2 2+D2 ℓ 2ηℓT=εℓ 200τ+... | https://arxiv.org/abs/2504.07133v1 |
Avrim Blum, Yishay Mansour, and Jamie Morgenstern. “Learning Valuation Distri- butions from Partial Observation”. In: Proceedings of the Twenty-Ninth AAAI Confer- ence on Artificial Intelligence . AAAI’15. Austin, Texas: AAAI Press, 2015, pp. 798–804 (cit. on p. 17). [Bor87] George J. Borjas. Self-Selection and the Ear... | https://arxiv.org/abs/2504.07133v1 |
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