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0.9977 0.9970 0.5501 0.9969 0.9974 1 Ionosphere 33 0.9311 0.9287 0.9190 OT 0.9376 0.5723 0.5227 0.4699 0.8865 0.9327 0 WPBC 33 0.5304 0.5326 0.4433 OT 0.5950 0.9384 0.8379 0.5290 0.5011 0.5370 0 letter 32 0.9221 0.9093 0.8523 OT 0.7813 0.7609 0.8387 0.6419 0.9207 0.9286 1 WDBC 30 0.9958 0.9944 0.9845 OT 1.0000 0.9915 0... | https://arxiv.org/abs/2504.07522v1 |
mnist 100 0.9771 0.9779 0.8867 OT 0.8349 0.8685 0.9489 0.4804 0.9618 0.9780 1 optdigits 64 0.9979 0.9981 0.9975 OT 0.5289 0.9077 0.9994 0.7617 0.9844 0.9973 0 SpamBase 57 0.3579 0.3584 0.3758 OT 0.7329 0.4295 0.3511 0.4495 0.3057 0.3601 0 landsat 36 0.7850 0.7842 0.7815 OT 0.7824 0.7624 0.8215 0.5057 0.7715 0.7857 1 sa... | https://arxiv.org/abs/2504.07522v1 |
0.8216 0.9393 ERR 0.5920 0.8118 0.8216 1 MNIST-C 512 0.6473 0.6470 0.5242 OT 0.6288 0.7073 ERR 0.4760 0.6364 0.6490 1 MVTec-AD 512 0.9399 0.9403 0.9331 OT 0.9234 0.8127 ERR 0.5296 0.8566 0.9415 0 SVHN 512 0.5609 0.5608 0.4566 OT 0.6007 0.5634 ERR 0.3949 0.5232 0.5653 1 speech 400 0.5451 0.5421 0.4884 OT 0.5325 0.5814 E... | https://arxiv.org/abs/2504.07522v1 |
1 Agnews 768 0.5235 0.5235 0.4441 OT 0.4661 0.4985 ERR 0.5110 0.4818 0.5242 0 Amazon 768 0.5715 0.5715 0.5608 OT 0.4758 0.4984 ERR 0.5220 0.5719 0.5731 0 Imdb 768 0.5119 0.5118 0.5178 OT 0.4716 0.5488 ERR 0.5374 0.5029 0.5121 0 Yelp 768 0.6186 0.6186 0.5638 OT 0.4704 0.5913 ERR 0.5219 0.5945 0.6190 0 CIFAR10 512 0.7126... | https://arxiv.org/abs/2504.07522v1 |
0.4749 0.5196 0.5302 0.5282 ERR ERR 0.5057 0.5191 1 30 Table 11: Full table of raw AUCs from Section 5.3 using CBLOF. We denoted time-outs by OT, and ODM errors by ERR. Dataset Features Feature Bagging Full Space CAE CLIQUE HiCS GMD PCA UMAP ELM V-GAN Myopicity InternetAds 1555 0.7347 0.7211 0.4141 OT ERR ERR 0.1804 0.... | https://arxiv.org/abs/2504.07522v1 |
0.9462 0.6591 0 WBC 9 0.5151 0.5254 0.5772 0.8281 0.8756 0.9587 0.8979 0.7153 0.7213 0.4733 0 Pima 8 0.7218 0.7151 0.4886 0.5906 0.5766 0.5696 0.5702 0.5443 0.5388 0.7403 0 yeast 8 0.5942 0.6083 0.5064 0.5590 0.5670 0.5786 0.5502 0.5429 0.6042 1 thyroid 6 0.4465 0.4454 0.6131 0.6687 0.6731 0.5980 0.6129 0.6238 0.6109 0... | https://arxiv.org/abs/2504.07522v1 |
Natural Language We will try to extract lens operators from the token space of Natural Language data directly. We could do this by considering E=NsandΘ(X) =Diags×s({0,1}), wheresis the predifined sentence length for each dataset. We also utilize a different architecture than before, featuring 3 sequential layers, each ... | https://arxiv.org/abs/2504.07522v1 |
arXiv:2504.07704v1 [math.ST] 10 Apr 2025Measures of non-simplifyingness for conditional copulas and vines Alexis Derumigny Department of Applied Mathematics, Delft University of Tec hnology, Mekelweg 4, 2628 CD, Delft, Netherlands. E-mail:a.f.f.derumigny@tudelft.nl ; Abstract In copula modeling, the simplifying assumpt... | https://arxiv.org/abs/2504.07704v1 |
will tend to 1 under any fixed alternative. This paper starts by introducing a more general concept of “meas ure of non- constantness” (Section 2). These are operators that measures how non-constant a function is. In a similar way, we present the new concept of “measur e of non- simplifyingness” for conditional copulas ... | https://arxiv.org/abs/2504.07704v1 |
distinction. We now give several more applicable examples of ways on how to const ruct (pseudo-)measures of non-constantness in the case where the vector space Eis equipped with a pseudo-norm } ¨ }E. 3 Example 4. First, we can define the Kolmogorov-Smirnov pseudo-measure of no n- constantness by ψKSpfq:“sup x,yPZ}fpxq ´... | https://arxiv.org/abs/2504.07704v1 |
non-simplifyingness for conditional copulas 3.1 Framework Remember that XandZare two random vectors, of respective dimensions dandp. We can define the conditional copula of XgivenZ“zby the conditional version of Sklar’s theorem: FX|Zpx|zq “CX|Z´ FX1|Zpx1|zq,...,F Xn|Zpxn|zqˇˇˇZ“z¯ , forevery zPRp,whereFX|Zisthecondition... | https://arxiv.org/abs/2504.07704v1 |
in general because we have not assumed that the random vectors are exchangeable. Remark 12. A similar comment can be made on the structure of the set of all measures of (pseudo)-non-simplifyingness as was done in Remark 8. Indeed, we can see that the set of ( PZ-)measures of non-simplifyingness is a convex cone, and th... | https://arxiv.org/abs/2504.07704v1 |
probability measure µ. Indeed, the mapping ψ“ }pu,z,z1q ÞÑφpCX|Z“zpuqq ´φpCX|Z“z1puqq}. is a measure of non-simplifyingness. For example, ψ“ sup pu,z,z1qPr0,1sdˆRpˆRp}CX|Z“zpuqq ´CX|Z“z1puq}, or ψ“ż pu,z,z1qPr0,1sdˆRpˆRp}CX|Z“zpuqq ´CX|Z“z1puq}. Note that these measures are more expensive to compute since th ey require... | https://arxiv.org/abs/2504.07704v1 |
this means Hequality 0:@z,z1PZ2,CpFX|Z“zq “CpFX|Z“z1q. This may too strict to be useful. Indeed, if for some z, the conditional marginal distributionsof X|Z“zarecontinuousandforsomeother z1,theconditionalmarginal distributions X|Z“z1are discrete, then Hequality 0will fail to hold. Such phenomenon is contrary to the com... | https://arxiv.org/abs/2504.07704v1 |
tree of the vine decomposition is alwaysmade up of unconditional copulas; therefor e there is no condi- tioning at these levels. More generally, we can define a measure of no n-simplifyingness of the copula cXfor the vine structure Vby ψpcX,Vq:“›››` ψpcae,be|Deq˘ k“2,...,d ´1,ePEk›››, for any norm } ¨ }onRřd´1 k“2Card p... | https://arxiv.org/abs/2504.07704v1 |
Following [ 14], the conditional Kendall’s tau τ1,2|Z“zbetweenX1andX2can be estimated by pτ1,2|Z“z:“řn i,j“1wi,npzqwj,npzqsign` pXi,1´Xj,1qpXi,2´Xj,2q˘ 1´řn i“1w2 i,npzq where sign pxq:“1txą0u´1txă0u, andwi,npzq:“KhpZi´zq{řn j“1KhpZj´zq, with Khp¨q:“h´pKp¨{hqfor some kernel KonRp, andh“hpnqdenotes a bandwidth sequence ... | https://arxiv.org/abs/2504.07704v1 |
arXiv:2504.07921v1 [math.ST] 10 Apr 2025Note on the identification of total e ffect in Cluster-DAGs with cycles Cl´ement Yvernes March 2025 Abstract In this note, we discuss the identifiability of a total e ffect in cluster-DAGs, al- lowing for cycles within the cluster-DAG (while still assuming the associated un- derl... | https://arxiv.org/abs/2504.07921v1 |
not be available, but we assume that we have the causal knowledge about the clusters. Definition 1 (Cluster-DAG) .A Cluster-DAG (or C-DAG) GC= VC,(EC D,EC B) is an ADMG whose nodes represent a (non empty) cluster of variables. The cardinal of each cluster is displayed in the upper left corner of each node as rep- res... | https://arxiv.org/abs/2504.07921v1 |
C- DAGs directly, without going through exhaustive enumeration? Q2 : What properties of C-DAGs are shared by all the compatible causal diagrams? We formalize this in Problem 1. Problem 1. LetGCbe an admissible cluster-DAG. We aim to find out an e fficient way to characterise over GCthe following property: let XC,YCandZ... | https://arxiv.org/abs/2504.07921v1 |
a subgraph σ⊆G that satisfies all conditions of Definition 3 and connects XandYunderZ. •2⇒1: By definition, σhas a single connected component. Thus XandYare connected by σ. Letπbe a path fromXandYinσ. Let us first prove that without loss of generality, we can assume that all colliders on πare ancestors of Z. If it’s no... | https://arxiv.org/abs/2504.07921v1 |
follows directly from Definition 2. □ Proposition 4 (Self-loops are not a big deal) .LetGCbe an admissible cluster-DAG. The following properties hold: •For anyGm∈C GC , the micro graph obtained from Gmby removing all intra- cluster edges belongs to C(GC′), whereGC′denotes the cluster-DAG obtained fromGCby removing al... | https://arxiv.org/abs/2504.07921v1 |
π=⟨V1,..., Vn⟩be a walk from XtoYinG. If ˜πisZ-active, thenπ=⟨U1,..., Um⟩, where U1=V1andUk+1=Vmax{i|Vi=Uk}, is a isZ-active path from XtoY. Proposition 8. LetGCbe an admissible cluster-DAG, Gmbe a compatible graph and VCbe a cluster. Let XC,YCandZCbe pairwise disjoint subsets of nodes. Let πmbe aZm-active path from Xm... | https://arxiv.org/abs/2504.07921v1 |
from Vjis now pointing away from Vi. This creates a new fork on Viand a new path that by- passes Vj←···→ Vi.πmis active, thus Vi<Zm. Thus the new fork is not blocking and the new path is indeed a Zm-active path fromXm toYm. Moreover, this path does not pass through Vjanymore. In this case, we reduce the number of chain... | https://arxiv.org/abs/2504.07921v1 |
result, we obtain a compatible graph in which at most 4 elements of Vmhave arrows. Consequently, we can remove all other elements, yielding a graph compatible withGCin which the cardinality of Vmhas been reduced to 4. Repeating this process for all other clusters Vm, we obtain a compatible graph with Gc ≤4that still co... | https://arxiv.org/abs/2504.07921v1 |
set: Eto choose BUi→Vj UC→VC⊆GC,and, Ui→Vjdoes not create a cycle in Gm min ThenEuBEmin∪E ”to choose” Remark 1. In general, the complexity of building Gm,C uis polynomial with respect to the size ofGCand the size of each cluster. It can be intractable if some clusters are very large. Nonetheless, i... | https://arxiv.org/abs/2504.07921v1 |
min∪σmis acyclic. Proof. Let us prove the two implications: •1⇒2: IfXC/⊥ ⊥GCYC|ZC, then there exists a compatible graph Gmin which there exists a structure of interest σmwhich connectsXmandYmunderZm. By Lemma 3,Gmis a subgraph ofGm,C u. Thusσmis also a structure of interest which connectsXmandYmunderZminGm,C u. By Lemm... | https://arxiv.org/abs/2504.07921v1 |
A Piecewise Lyapunov Analysis of Sub-quadratic SGD: Applications to Robust and Quantile Regression YIXUAN ZHANG, University of Wisconsin-Madison, USA DONGYAN (LUCY) HUO, Cornell University, USA YUDONG CHEN, University of Wisconsin-Madison, USA QIAOMIN XIE, University of Wisconsin-Madison, USA Motivated by robust and qu... | https://arxiv.org/abs/2504.08178v3 |
New York, USA, dh622@cornell.edu; Yudong Chen, University of Wisconsin-Madison, Department of Computer Sciences, Madison, Wisconsin, USA, yudong.chen@wisc.edu; Qiaomin Xie, University of Wisconsin- Madison, Department of Industrial and Systems Engineering, Madison, Wisconsin, USA, qiaomin.xie@wisc.edu.arXiv:2504.08178v... | https://arxiv.org/abs/2504.08178v3 |
the study of constant stepsize sub–quadratic SGD via a Markov chain perspective remains scarce. In recent developments, [ 58] introduced a novel technique to analyze the weak convergence of Markov chains, validating their approach with specific examples of sub–quadratic SGD involving solely additive (i.e., 𝑤(𝜃)is ind... | https://arxiv.org/abs/2504.08178v3 |
for SGD applied to sub–quadratic functions. Having established weak convergence, we further prove a central limit theorem for the Markov chain {𝜃𝑛}𝑛≥0, which is crucial for statistical inference [ 46]. Additionally, we 4 Yixuan Zhang, Dongyan (Lucy) Huo, Yudong Chen, and Qiaomin Xie characterize the asymptotic bias ... | https://arxiv.org/abs/2504.08178v3 |
Previous work focusing on Markov chains in general state spaces has proposed various techniques to verify convergence and provide convergence rates. Most convergence results are established by verifying drift and minorization (D&M) conditions [ 1,4,73]. However, verifying D&M conditions often relies on assuming that th... | https://arxiv.org/abs/2504.08178v3 |
the sub–quadratic objective function [ 24], we relax the assumption that the objective function is twice differentiable by only requiring the first-order differentiability. Although we need an additional assumption that ⟨𝜃−𝜃∗,∇𝑓(𝜃)⟩≥𝑏∥𝜃−𝜃∗∥𝑘, we argue that this is verifiable in both online robust regression and... | https://arxiv.org/abs/2504.08178v3 |
𝜃∈R𝑑,L(𝜃)represents its distribution. The Wasserstein 1-distance between two distributions 𝜇 and𝜈inP1(R𝑑)is defined as 𝑊1(𝜇,𝜈)=inf 𝜉∈Π(𝜇,𝜈)∫ R𝑑∥𝑢−𝑣∥𝑑𝜉(𝑢,𝑣)=inf{E[∥𝜃−𝜃′∥]:L(𝜃)=𝜇,L(𝜃′)=𝜈}, (3) where Π(𝜇,𝜈)denotes the set of all joint distributions in P(R𝑑×R𝑑)with marginals 𝜇and𝜈. For real-v... | https://arxiv.org/abs/2504.08178v3 |
function to bound the moments of the error |𝜃𝑛− 1A sequence{𝜃𝑛}converges to 𝐿with Q-convergence of order 1 if 0<lim𝑛→∞|𝜃𝑛+1−𝐿| |𝜃𝑛−𝐿|<1. 8 Yixuan Zhang, Dongyan (Lucy) Huo, Yudong Chen, and Qiaomin Xie 𝜃∗|[13,15,33,65]. The key challenge here is the construction of a proper Lyapunov function. Specifically,... | https://arxiv.org/abs/2504.08178v3 |
to Robust and Quantile Regression 9 3.2 A New Piecewise Lyapunov Function The discussions in Section 3.1 provide insights into constructing an appropriate piecewise Lyapunov function for a general sub–quadratic SGD (2). Building on the discussion and under Assumption 2, the suitable piecewise Lyapunov function 𝑉should... | https://arxiv.org/abs/2504.08178v3 |
Proposition 1 depends on Lemma 1 and a precise discussion of the value of noise 𝑤(𝜃), and is provided in Appendix B. Although Proposition 2 does not offer an exact one-step contraction, it establishes a recursive relationship among 𝑉𝑘,𝑝(𝜃𝑛+1),𝑉𝑘,𝑝(𝜃𝑛), and𝑉𝑘,𝑝−2(𝜃𝑛), which would allow us to upper bound... | https://arxiv.org/abs/2504.08178v3 |
convergence rate of E[∥𝜃𝑛−𝜃∗∥2]is influenced by 𝜉and𝜄in the stepsize 𝛼𝑛=𝜄 (𝑛+𝜅)𝜉. Specifically, by setting 𝛼𝑛=𝜄 𝑛+𝜅with𝜄>1/𝜇𝑘,0, we obtain the optimal convergence rate of O(1/𝑛). In contrast, when 𝜉∈(0,1), the convergence rate becomes sub-optimal atO(1/𝑛𝜉), but this rate comes with greater robust... | https://arxiv.org/abs/2504.08178v3 |
distance between their subsequent iterates shrinks compared to the Euclidean distance between the initial points. We remark that if Assumption 2 holds and E[∥𝑤(𝜃)−𝑤(𝜃′)∥2]∈O(∥𝜃−𝜃′∥2), then Assumptions 4 and 5 are readily satisfied. We need Assumption 5 since we employ the drift and contraction (D&C) condition tec... | https://arxiv.org/abs/2504.08178v3 |
objective function is differentiable to a higher order, we can further characterize the asymptotic biasE[𝜃(𝛼) ∞]−𝜃∗, as presented in the following corollary. Corollary 1 (Bias Characterizaion). Under the same setting as Theorem 3 and further assuming that the objective function 𝑓is three times differentiable, then ... | https://arxiv.org/abs/2504.08178v3 |
corrupted. In contrast, the online oblivious response corruption model [55] assumes that 𝑠follows a specific distribution and is independent of (𝑥,𝜖). Furthermore, Assumption 6–(3) ensures that the population-level loss 𝑓reg(𝜃)is well-defined for all𝜃∈R𝑑, particularly for loss functions with at least linear grow... | https://arxiv.org/abs/2504.08178v3 |
𝑛+𝜅. (3) When𝛼𝑛=𝜄 (𝑛+𝜅)𝜉with𝜉∈(0,1), for all𝑛≥0and𝜃0∈R𝑑, E[∥𝜃𝑛−𝜃∗ reg∥2]≤2(Δ𝑙−Δ𝜖,𝑠)2𝑉1,0(𝜃0) 𝑒𝜎2𝑥ln2(8E[∥𝑥∥4])exp −𝜇reg,0𝜄 1−𝜉 (𝑛+𝜅)1−𝜉−𝜅1−𝜉 +4𝜄(Δ𝑙−Δ𝜖,𝑠)2𝑐′ 1,0 𝜇reg,0𝑒𝜎2𝑥ln2(8E[∥𝑥∥4])·1 (𝑛+𝜅)𝜉. 16 Yixuan Zhang, Dongyan (Lucy) Huo, Yudong Chen, and Qiaomin Xie (4) When𝛼�... | https://arxiv.org/abs/2504.08178v3 |
on 𝑥. We denote𝐹𝑥(·)to be the cumulative distribution function (CDF) of 𝜖given x. Consequently, we have𝐹𝑥(0)=𝜏for all𝑥∈Ω𝑥. We focus on the setting where the covariate 𝑥and the conditional CDF𝐹𝑥(·)satisfies the following assumption. Assumption 9. The covariate 𝑥and the conditional cumulative distribution fu... | https://arxiv.org/abs/2504.08178v3 |
J. Since the update (13)is not smooth with respect to 𝜃, it is unclear whether constant stepsize online quantile regression exhibits weak convergence; we discuss why this may not hold in Appendix K. Nonetheless, building on Theorem 6, we extend the main results from Section 4.1 to online quantile regression (13), as s... | https://arxiv.org/abs/2504.08178v3 |
in the corresponding settings), as shown in Figures 1a and 1c. We observe that for all diminishing stepsizes, both algorithms converge, and converge more rapidly after 108iterations when using a larger𝜉. Additionally, we smooth the last iterates using a sliding window median, approximate it with a linear function, and... | https://arxiv.org/abs/2504.08178v3 |
Figures 3a and 3b show that using constant stepsizes enables faster convergence for TA iterates compared to diminishing stepsizes, with larger constant stepsizes converging faster. Notably, diminishing stepsizes with a large 𝜉=0.9result in slow initial convergence, as observed in the first1010iterations of Figures 3a ... | https://arxiv.org/abs/2504.08178v3 |
stepsizes and averaged iterates achieves a convergence rate of O 𝑑 𝑛(1−˜𝜂)2 . We also provide a comprehensive analysis for online robust regression with constant stepsize. For online quantile regression, we remove the previous assumption that the conditional density of the noise is continuous everywhere and provid... | https://arxiv.org/abs/2504.08178v3 |
estimation of a sparse linear model using ℓ1-penalized Huber’s𝑀-estimator. Advances in neural information processing systems 32 (2019). [18] Aaron Defazio, Francis Bach, and Simon Lacoste-Julien. 2014. SAGA: A fast incremental gradient method with support for non-strongly convex composite objectives. Advances in neura... | https://arxiv.org/abs/2504.08178v3 |
Springer Science & Business Media. [40] Stuart A Klugman, Harry H Panjer, and Gordon E Willmot. 2012. Loss models: from data to decisions . Vol. 715. John Wiley & Sons. [41] Roger Koenker. 2005. Quantile regression. Cambridge Univ Pr (2005). [42] Krzysztof Kurdyka. 1998. On gradients of functions definable in o-minimal... | https://arxiv.org/abs/2504.08178v3 |
[61] Jeffrey S Rosenthal. 1995. Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Amer. Statist. Assoc. 90, 430 (1995), 558–566. [62] David Ruppert. 1988. Efficient estimations from a slowly convergent Robbins-Monro process . Technical Report. Cornell University Operations Research and Indu... | https://arxiv.org/abs/2504.08178v3 |
differentiable everywhere for all 𝑘∈[1,2)and 𝑉′′ 𝑘,0(𝜃)= (1−Δ2−𝑘 ∥𝜃−𝜃∗∥2−𝑘)·𝑘2exp(𝑘∥𝜃−𝜃∗∥2−𝑘 (2−𝑘)Δ2−𝑘) Δ4−2𝑘∥𝜃−𝜃∗∥2𝑘·(𝜃−𝜃∗)(𝜃−𝜃∗)𝑇+𝑘exp(𝑘∥𝜃−𝜃∗∥2−𝑘 (2−𝑘)Δ2−𝑘) Δ2−𝑘∥𝜃−𝜃∗∥𝑘·𝐼𝑑 if|𝜃−𝜃∗|>Δ, 𝑘exp(𝑘/(2−𝑘)) Δ2·𝐼𝑑 if∥𝜃−𝜃∗∥≤Δ. (15) To verify that 𝑉𝑘,𝑝(𝜃)is twice differen... | https://arxiv.org/abs/2504.08178v3 |
2∥𝜃−𝜃∗∥,∀𝛼≤1 2𝑎Δ𝑘−2. Therefore, we finish proving Lemma 3. A Piecewise Lyapunov Analysis of Sub-quadratic SGD: Applications to Robust and Quantile Regression 29 C PROOF OF PROPOSITION 2 Based on Lemma 2, when ∥𝜃−𝜃∗∥≥Δand𝑝≥2, we have E[max 𝑦∈[0,1]∥𝑉′′ 𝑘,𝑝(𝜃−𝑦𝛼(∇𝑓(𝜃)+𝑤(𝜃)))∥(∥∇𝑓(𝜃)∥2+∥𝑤(𝜃)∥2)] (i) ... | https://arxiv.org/abs/2504.08178v3 |
Xie Verifying Condition A3. We aim to verify that log(1+𝛼𝐿+𝛼𝑐𝑤)log(1+2𝛼2𝑑𝑘)<log(1 1−𝛼𝑟)log 𝑘exp(𝑘/(2−𝑘)) 2+1 𝛼𝜇𝑘𝑘exp(𝑘/(2−𝑘)) 2+2𝛼2𝑑𝑘+1. For the LHS, by the fact that log(𝑥)≤𝑥−1,∀𝑥>0, we have LHS≤2𝛼3(𝐿+𝑐𝑤)𝑑𝑘 For the RHS, by the fact that log(𝑥)≥1−1/𝑥,∀𝑥>0, we have RHS≥𝛼𝑟·log 𝑘exp(�... | https://arxiv.org/abs/2504.08178v3 |
bounded, there exist 𝑐′ 𝑙≥0such that∥𝑙′(𝑡′)−𝑙′(𝑡′′)∥ ≤ 9𝑐𝑙 64E[∥𝑥∥4]E[∥𝑥∥]for all𝑡′,𝑡′′∈ {𝑡:𝑡≥𝑐′ 𝑙}and all𝑡′,𝑡′′∈ {𝑡:𝑡≤−𝑐′ 𝑙}. Therefore, for term 𝑇2, we have 𝑇2=E[(𝑙′(𝑥𝑇𝜃∗ reg−𝑥𝑇𝜃+𝜖+𝑠)−𝑙′(𝑥𝑇𝜃∗ reg−𝑥𝑇𝜃))(𝑥𝑇𝜃∗ reg−𝑥𝑇𝜃)1{|𝑥𝑇𝜃∗reg−𝑥𝑇𝜃|≥2𝑐′ 𝑙}] +E[(𝑙′(𝑥𝑇𝜃∗ reg−𝑥𝑇�... | https://arxiv.org/abs/2504.08178v3 |
and (ii) holds by following Theorem 5. Therefore, we verify the Assumption 𝐻𝑆in [24] for online robust regression. J PROOF OF THEOREM 6 J.1 A Few Preliminary Facts In this subsection, we present some useful preliminary lemmas. The proofs of Lemmas 9 and 10 are similar to those of Lemmas 6 and 7; hence, we omit them. ... | https://arxiv.org/abs/2504.08178v3 |
Deep Distributional Learning with Non-crossing Quantile Network Guohao Shen* Department of Applied Mathematics, The Hong Kong Polytechnic University Runpeng Dai Department of Biostatistics, University of North Carolina at Chapel Hill Guojun Wu ByteDance, Beijing, China Shikai Luo ByteDance, Beijing, China Chengchun Shi... | https://arxiv.org/abs/2504.08215v1 |
for deep quantile regression (Zhou et al., 2020; Brando et al., 2022; Padilla et al., 2022; Wu et al., 2023; Yan et al., 2023; Shen et al., 2024). We discuss these methods in Section 2 in detail. While the aforementioned methods have been applied to a variety of applications, much less is known regarding the theoretica... | https://arxiv.org/abs/2504.08215v1 |
constraint is not strictly imposed in Shen et al. (2024). Learning guarantees for non-crossing quantile estimators are not derived in Zhou et al. (2020); Padilla et al. (2022); Yan et al. (2023) and the optimality is unclear. closely related to our proposal. Section 3 introduces the proposed framework for distributiona... | https://arxiv.org/abs/2504.08215v1 |
et al. (2024) employed the rectified quadratic units (ReQU) activated neural networks to estimate the quantile process, and proposed to penalize the estimator’s partial derivative to encourage the monotonicity or the non-crossing of the quantile process. The resulting estimator is shown to attain a nonparametric rate o... | https://arxiv.org/abs/2504.08215v1 |
K)denote a set of Kpre-specified non-decreasing quantile levels, typically chosen as τk=k/(K+ 1) fork= 1, . . . , K . Our proposed NQ networks f(x;θ)output the vector f(x;θ) = (f1(x;θ), . . . , f K(x;θ)), where f(·;·)is a certain neural network model, xis the input data, θis the set of parameters in the network f, and ... | https://arxiv.org/abs/2504.08215v1 |
(g1, . . . , g K). Then the quantiles f1, . . . , f Kcan be calculated according to (2). Additionally, we denote the number of hidden layers (depth), the maximum number of neurons in the hidden layers (width), the number of neurons, and the total number of parameters (size) including weights and biases in the NQ networ... | https://arxiv.org/abs/2504.08215v1 |
QYbelongs to the hypothesis class FN. Under such a realizability assumption, thebiasinff∈FNR(f)is zero and the estimator’s converges at a “parametric” rate. Thevariance measures the estimation error induced by the randomness of the samples. It is closely associated with the loss function, the sample size, and the prope... | https://arxiv.org/abs/2504.08215v1 |
. . . , α d)⊤∈Nd 0and∥α∥1=Pd i=1αi. The classes of Lipschitz continuous and continuously differentiable functions are H ¨older function classes. To analyze the bias, we make the following assumptions. Assumption 1 (i) The target quantile curves QY= (Qτ1 Y, . . . , QτK Y)areβ-H¨older smooth with constant B. (ii) The dom... | https://arxiv.org/abs/2504.08215v1 |
width W= 38( K+ 1)(⌊β⌋+ 13 1)2d1d⌊β⌋+1 0Ulog2(8U)and depth D= 21(⌊β⌋+ 1)2d⌊β⌋+1 0Mlog2(8M), then we have inf f∈FN[R(f)]≤C(K+ 2)2h B2(⌊β⌋+ 1)4d2⌊β⌋+(β∨1) 0 (UM)−4β/d0+ exp( −2B)i , where C >0is a universal constant and d0is the input dimension of NQ neural networks in FN. Based on the error decomposition in Lemma 1, and... | https://arxiv.org/abs/2504.08215v1 |
sample size to achieve desired theoretical accuracy, often proving impractical in real-world settings. In modern statistics and machine learning, many high-dimensional data tend to lie in the vicinity of a low-dimensional manifold (Pope et al., 2021). This fact provides a way to mitigate the curse of dimensionality in ... | https://arxiv.org/abs/2504.08215v1 |
|s)maps each state s∈ S to a distribution over the action space A. For a fixed policy πand a given initial state-action pair (s, a), its return Zπ(s, a) =∞X t=0γtR(St, At), is a random variable representing the sum of discounted rewards observed along a trajectory {(St, At)}t≥0following π, conditional on that S0=sandA0... | https://arxiv.org/abs/2504.08215v1 |
networks FNdefined in (3) for the value distribution estimation as follows: F(RL) N={f:S × A → R:f(·, a)∈ F Nfor any a∈ A} . (8) By this definition, F(RL) N is a class of functions (may depend on the sample size N) that take state 18 Algorithm 1 Distributional RL with fitted NQ Iterations Require: Initial quantile esti... | https://arxiv.org/abs/2504.08215v1 |
2020; Li et al., 2021; Uehara et al., 2021), here we do not impose the i.i.d. assumption, as it is often violated in MDPs due to the temporal dependence between the observations (Hao et al., 2021). Nor do we require certain stationarity, ergodicity, or mixing conditions. These conditions are again frequently imposed (s... | https://arxiv.org/abs/2504.08215v1 |
Let πMdenote the greedy policy of Z(M). Then the expected cumulative reward following πMsatisfies J(π∗)−J(πM)≤2cMγ (1−γ)2|A|(logN)4N−2β/(4β+d0)+8γM+1 (1−γ)2Cp,R+C1×Cp,R (1−γ)K(p−1)/p,(9) where C1>0is a universal constant and cM>0is the concentration coefficient (Antos et al., 2007; Chen and Jiang, 2019; Fan et al., 202... | https://arxiv.org/abs/2504.08215v1 |
(IV) Non-crossing Quantile Regression Deep-Q-Network (Zhou et al., 2020), denoted by NC- QR-DQN , which is proposed for Distributional RL by using non-crossing quantile regression based on the QR-DQN architecture. NC-QR-DQN employs a feedforward network to extract features, then maps them to two scalars (a slope and an... | https://arxiv.org/abs/2504.08215v1 |
(CDF) of the Student’s t-distribution with 2 degrees of freedom. (ii)Wave Model :Qτ Y(x) = 2 xsin(4πx)+exp(4 x−2)Φ−1(τ),withΦ(·)representing the CDF of the standard normal distribution. (iii) Angle Model :Qτ Y(x) = 4(1 −|x−0.5|)+|sin(πx)|Φ−1(τ). These models allow us to evaluate the quantile estimation across various f... | https://arxiv.org/abs/2504.08215v1 |
conditional quantile curves at levels τ=0.05 (blue), 0.25 (orange), 0.5 (green), 0.75 (red), and 0.95 (purple) are depicted as solid curves. Table 3: Summary statistics for the “Wave” model with training sample size N= 512 and replications R= 100 . The averaged L1andL2 2test errors with the corresponding standard devia... | https://arxiv.org/abs/2504.08215v1 |
27 image-embedding network architecture and downstream networks with similar scales. Specifically, we employ ReLU activation for the “Gaps net” in our model, and denote the network by NQ-Net* . As in most game settings, the differences between quantiles are near zero, ReLU activation in NQ-Net* may facilitate the optim... | https://arxiv.org/abs/2504.08215v1 |
G. and Wakin, M. B. (2009). Random projections of smooth manifolds. Found. Comput. Math. , 9(1):51–77. Bartlett, P. L., Harvey, N., Liaw, C., and Mehrabian, A. (2019). Nearly-tight vc-dimension and pseudodimension bounds for piecewise linear neural networks. Journal of Machine Learning Research , 20(63):1–17. Bassett J... | https://arxiv.org/abs/2504.08215v1 |
(2020). Error bounds for approximations with deep relu neural networks in w s, p norms. Analysis and Applications , 18(05):803–859. Gy¨orfi, L., Kohler, M., Krzyzak, A., and Walk, H. (2006). A distribution-free theory of nonpara- metric regression . Springer Science & Business Media. Hao, B., Ji, X., Duan, Y ., Lu, H.,... | https://arxiv.org/abs/2504.08215v1 |
Journal of Machine Learning Research , 9(27):815–857. Padilla, O. H. M., Tansey, W., and Chen, Y . (2022). Quantile regression with relu networks: Estimators and minimax rates. Journal of Machine Learning Research , 23(247):1–42. Pope, P., Zhu, C., Abdelkader, A., Goldblum, M., and Goldstein, T. (2021). The intrinsic d... | https://arxiv.org/abs/2504.08215v1 |
treatment effects. Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining , 5215–5224. Xu, Y ., Shi, C., Luo, S., Wang, L., and Song, R. (2022). Quantile off-policy evaluation via deep conditional generative learning. arXiv preprint arXiv:2212.14466 . Yan, X., Su, Y ., and Ma, W. (2023). E... | https://arxiv.org/abs/2504.08215v1 |
and scaled cumulative sums of the softmax values model the gaps between adjacent quantiles. •Non-crossing quantile network in this paper, denoted by NQ-Net . Given a set of quantiles, NQ-Net estimates the mean of the conditional quantile curves and estimates the gaps between quantile curves by a non-negative activation... | https://arxiv.org/abs/2504.08215v1 |
of replications R= 100 . The averaged L1andL2 2test errors with the corresponding standard deviation (in parentheses) are reported for the estimators trained by different methods. τL1 L2 2 NQ-Net DQR DQR* NC-QR-DQN DQRP NQ-Net DQR DQR* NC-QR-DQN DQRP 0.05 0.135(0.037) 0.151(0.041) 0.319(0.091) 0.558(0.695) 0.265(0.168)... | https://arxiv.org/abs/2504.08215v1 |
0.057(0.057) 2.556(3.825) 0.149(0.122) 42 Table 7: Summary statistics for the “Angle” model with training sample size N= 2048 and the number of replications R= 100 . The averaged L1andL2 2test errors with the corresponding standard deviation (in parentheses) are reported for the estimators trained by different methods.... | https://arxiv.org/abs/2504.08215v1 |
1.157(0.634) 1.828(1.476) 0.9 0.443(0.073) 0.417(0.078) 0.612(0.106) 0.993(0.554) 0.987(0.199) 0.320(0.111) 0.277(0.105) 0.608(0.202) 1.536(1.369) 2.139(1.888) 0.95 0.553(0.130) 0.572(0.124) 0.744(0.172) 1.549(0.900) 1.277(0.200) 0.476(0.205) 0.508(0.213) 0.855(0.350) 3.639(3.294) 2.898(2.207) 44 Table 9: Summary stati... | https://arxiv.org/abs/2504.08215v1 |
0.226(0.049) 0.8 0.430(0.038) 0.442(0.043) 0.383(0.035) 0.483(0.068) 0.404(0.032) 0.287(0.050) 0.300(0.055) 0.230(0.041) 0.341(0.076) 0.277(0.058) 0.85 0.474(0.041) 0.493(0.046) 0.431(0.036) 0.514(0.121) 0.456(0.033) 0.344(0.058) 0.369(0.065) 0.285(0.048) 0.388(0.153) 0.345(0.067) 0.9 0.528(0.043) 0.553(0.048) 0.490(0.... | https://arxiv.org/abs/2504.08215v1 |
0.070(0.010) 0.7 0.220(0.016) 0.209(0.012) 0.195(0.013) 0.271(0.044) 0.232(0.015) 0.075(0.011) 0.066(0.009) 0.057(0.008) 0.110(0.027) 0.089(0.012) 0.75 0.251(0.014) 0.246(0.012) 0.231(0.013) 0.318(0.069) 0.268(0.015) 0.094(0.011) 0.090(0.010) 0.078(0.009) 0.149(0.053) 0.115(0.015) 0.8 0.287(0.014) 0.286(0.012) 0.270(0.... | https://arxiv.org/abs/2504.08215v1 |
foundation for deriving learning guarantees of many existing non-crossing deep quantile estimators (Bondell et al., 2010; Padilla et al., 2022; Yan et al., 2023). D Proofs of Theorems and Lemmas D.1 Proof of Lemma 1. For the empirical risk minimizer ˆfNand any f∈ F N, we have E[R(ˆfN)]≤E[R(ˆfN) + 2(RN(f)− R N(ˆfN))] =E... | https://arxiv.org/abs/2504.08215v1 |
NNX i=1[gk(f, Z i)]> t! . It then suffices to prove the uniform upper bound for the tail probability. By symmetrization, conditioning, and covering techniques, we can prove the following result for k= 1, . . . , K P ∃f∈ F N:E gk(f, Z) −1 NNX i=1 gk(f, Z i) > ϵ α+β+E g(f, Z) ! ≤14NNϵβ 7,F(k) N,∥ · ∥∞ exp −3ϵ2(1... | https://arxiv.org/abs/2504.08215v1 |
Consider a function ˆfN∈ F Ndepending upon Ssuch that En gˆfN(Z)|So −1 NNX i=1gˆfN(Zi)≥ϵ(α+β) +ϵEn gˆfN(Z)|So if such a function exists in FN, otherwise choose an arbitrary function in FN. Based on the definition of Qτky, it is the minimizer of E[ρτk(Y−f(X))|X, f]over fixed measurable functions f. Thus, gf(Z)≥0for any ... | https://arxiv.org/abs/2504.08215v1 |
similarly with the third inequality. Using this and the inequality Egf(Z)≥ 1 BEg2 f(Z) = 21 2BEg2 f(Z)we can bound the first probability on the right-hand side of (17) by 56 P( ∃f∈ F N:1 NNX i=1gf(Z′ i)−1 NNX i=1gf(Zi)≥ϵ(α+β)/2 +ϵ 2 1−ϵ 2B(1 +ϵ)1 NNX i=1g2 f(Zi)−ϵ(α+β) 2B(1 +ϵ)+1−ϵ 2B(1 +ϵ)1 NNX i=1g2 f(Z′ i)−ϵ(α+β) 2B... | https://arxiv.org/abs/2504.08215v1 |
g∈Gϵβ 7P( 1 NNX i=1Uig(zi) ≥ϵα 4−ϵ2α 4B(1 +ϵ)+ϵ(1−ϵ) 4B(1 +ϵ)1 NNX i=1g2(zi)) . STEP 5. Application of Bernstein’s inequality. In this step, we use Bernstein’s inequality to bound 59 P( 1 NNX i=1Uig(zi) ≥ϵα 4−ϵ2α 4B(1 +ϵ)+ϵ(1−ϵ) 4B(1 +ϵ)1 NNX i=1g2(zi)) , where z1, . . . , z Nare fixed and gsatisfies −B ≤ g(z)≤ B. Firs... | https://arxiv.org/abs/2504.08215v1 |
40(1 + ϵ)B2 . ForN≤64B ϵ2(α+β), we have exp −3ϵ2(1−ϵ)αN 40(1 + ϵ)B2 ≥exp −12 5 ≥1 14 and hence the assertion follows trivially. D.3 Proof of Lemma 2. Recall that the target quantile curves Qτ1 Y, . . . , QτK Yare H ¨older functions in Hβ([0,1]d0, B). On one hand, it is easy to show that the average of target quant... | https://arxiv.org/abs/2504.08215v1 |
the risk R, it is easy to show inf f∈FNR(f)≤18(K+ 2)B(⌊β⌋+ 1)2d⌊β⌋+(β∨1)/2 0 (UM)−2β/d0+ (K+ 2) exp( −B), which completes the proof. D.4 Proof of Lemma 3. The Lemma 3 can be proved by following the proof of Lemma 2. With the same construction of neural network f= (f1, . . . , f K)in the proof of Lemma 2, and we have E|... | https://arxiv.org/abs/2504.08215v1 |
.And it is easy to check A(Mρ)⊆A([0,1]d0)⊆E:= [−p d0/d∗ 0,p d0/d∗ 0]d∗ 0. Since Anearly preserves the distance on M, it is easy to see that for any z∈A(M), there exists a unique x∈ M such that Ax=z. For any z∈A(M), we define xz=SL({x∈ M : Ax=z})whereSL(·)returns a unique element of a set. And we can see that SL:A(M)→ M... | https://arxiv.org/abs/2504.08215v1 |
target to the maximum estimation errors for each iteration. Lemma S2 (Error Propagation) Let{Z(m)}M m=0be the iterates in Algorithm 1. Let πMbe the greedy policy with respect to Z(M), and let ZπMbe the action-value distribution corresponding to πM. Then for µ∈ P(S × A )being a distribution on S × A , we have ∥EZπM−EZ∗∥... | https://arxiv.org/abs/2504.08215v1 |
networks. Then by the calibration condition in Assumption 6, in the rest of the proof it is sufficient to give upper bounds for the excess risk R(m)where R(m)(f) :=E(S,A)∼σm1 KKX k=1[ρτk(TZ(m−1)(S, A)−fk(S, A))−ρτk(TZ(m−1)(S, A)−TZ(m−1) k (S, A))], forf∈ F(RL) N andZ(m)= (Z(m) 1, . . . , Z(m) K)is the estimation in the... | https://arxiv.org/abs/2504.08215v1 |
2B2 . (26) Note that for any random variable W, we have E[W]≤E[W×I(W > 0)]≤Z∞ t=0P(W > t ). Then taking ϵ= 1/Nand following the proof of Lemma 1, we can show that Esup f∈F(RL) N|R(m)(f)− R(m) N(f)| ≤8B2q log 2NN(1/N,F(RL) N,∥ · ∥∞) √ N form= 1, . . . , M . Note that the covering number in above bound NN(1/N,F(RL) N,∥ ... | https://arxiv.org/abs/2504.08215v1 |
EZ∗−EZ(m+1)=EZ∗− EZm+1−Eϱm+1 =EZ∗−EZm+1+Eϱm+1=EZ∗− TEZ(m)+Eϱm+1 =EZ∗− Tπ∗EZ(m)+ ETπ∗Z(m)−ETZ(m) +Eϱm+1 ≤EZ∗−ETπ∗Z(m)+Eϱm+1 where π∗is the greedy policy with respect to EZ∗, and the inequality follows from the fact that ETπ∗Z(m)≤ETZ(m). 72 Next, we establish a lower bound for EZ∗−EZ(m+1)based on EZ∗−EZ(m). By the de... | https://arxiv.org/abs/2504.08215v1 |
on S × A . Step (3): In the third step, we can conclude the proof by establishing an upper bound for ∥EZ∗−EZπM∥1,µ. Here µ∈ P(S × A )is a fixed probability distribution. To simplify the notation, for any measurable function f:S × A → R, we denote µ(f)to be the expectation of f 75 under µ, i.e., µ(f) =R S×Af(s, a)dµ(s, ... | https://arxiv.org/abs/2504.08215v1 |
exist f1, . . . , f N∈ F| xandF|x⊂SN i=1{f:∥f−fi∥ ≤δ}, i.e.,F|xis fully covered by Nspherical balls centered at fiwith radius δunder the norm ∥ · ∥. Moreover, the uniform covering number Np(δ,F,∥ · ∥)is defined to be the maximum covering number over all xinXp, i.e., Np(δ,F,∥ · ∥) := max {N(δ,F|x,∥ · ∥) :x∈ Xp}. Definit... | https://arxiv.org/abs/2504.08215v1 |
have E[X]−E[˜X] ≤E X−˜X =E|X·I(|X|> T)| =Z∞ 0P(|X| ·I(|X|> T)> u)du =ZT 0P(|X|> T))du+Z∞ TP(|X|> u)du ≤2(E|X|p)1/p (K+ 1)(p−1)/p. Meanwhile, ˜Xis random variable bounded by T= (E|X|p)1/p·(K+ 1)1/p. By the previous argument, we also have 1 KKX k=1qk−E[˜X] ≤C (K+ 1)(p−1)/p, where C >0is a constant not depending on K. Com... | https://arxiv.org/abs/2504.08215v1 |
arXiv:2504.08301v1 [stat.ME] 11 Apr 2025Enhanced Marginal Sensitivity Model and Bounds Yi Zhang∗†, Wenfu Xu∗‡, Zhiqiang Tan† May 17, 2025 Abstract. Sensitivity analysis is important to assess the impact of unmeasured confounding in causal inference from observational studies. The marginal sensitivity model (MSM) provid... | https://arxiv.org/abs/2504.08301v1 |
and Tan (2024). Various MSM-related sensitivity models have also been proposed to address different needs. For example, Jesson et al. (2022) extended MSM to continuous treatments, while Bonvini et al. (2022), Frauen et al. (2023) and Tan (2025) considered time-varying treatments in longitudinal settings. Another collec... | https://arxiv.org/abs/2504.08301v1 |
as follows. In Section 2, we review the marginal sensitivity model. In Section 3, we develop the proposed sensitivity model and its sharp population bounds, and then discuss specification and interpretation of sensitivity parameters. In Section 4, we use calibrated estimation to obtain both doubly robust point estimati... | https://arxiv.org/abs/2504.08301v1 |
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