Journal of Complexity最新文献

{"title":"Marcinkiewicz–Zygmund inequalities in quasi-Banach function spaces","authors":"Yurii Kolomoitsev ,&nbsp;Sergey Tikhonov","doi":"10.1016/j.jco.2025.101999","DOIUrl":"10.1016/j.jco.2025.101999","url":null,"abstract":"<div><div>We obtain Marcinkiewicz–Zygmund (MZ) inequalities in various Banach and quasi-Banach spaces under minimal structural assumptions. Our main results show that the Bernstein inequality in a general quasi-Banach function lattice <em>X</em> implies Marcinkiewicz–Zygmund type estimates in <em>X</em>. We present a unified approach to deriving MZ inequalities not only for polynomials, but also for other function classes, including entire functions of exponential type, splines, exponential sums, and more. As applications, we derive error estimates for sampling operators, Nikolskii-type inequalities, as well as inequalities for best approximations and moduli of smoothness.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"93 ","pages":"Article 101999"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145571799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Special Issue of the Journal of Complexity","authors":"Josef Dick (Guest Editors),&nbsp;Michael Gnewuch,&nbsp;Erich Novak,&nbsp;Leszek Plaskota,&nbsp;Jan Vybíral","doi":"10.1016/j.jco.2025.102002","DOIUrl":"10.1016/j.jco.2025.102002","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"93 ","pages":"Article 102002"},"PeriodicalIF":1.8,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145617993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Statistical analysis of prediction in functional polynomial quantile regression","authors":"Hongzhi Tong","doi":"10.1016/j.jco.2025.101995","DOIUrl":"10.1016/j.jco.2025.101995","url":null,"abstract":"<div><div>We consider in this paper the quantile regression in a functional polynomial model, where the conditional quantile of a scalar response is modeled by a polynomial of functional predictor. It extends beyond the standard functional linear setting to accommodate more general functional polynomial model. A Tikhonov regularized functional polynomial quantile regression approach is introduced and investigated. By utilizing some techniques of empirical processes, we establish the explicit convergence rates of the prediction error of the proposed estimator under mild assumptions.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101995"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalization bounds of adversarial bipartite ranking with pairwise perturbation","authors":"Wen Wen ,&nbsp;Han Li ,&nbsp;Yutao Hu ,&nbsp;Lingjuan Wu ,&nbsp;Hong Chen","doi":"10.1016/j.jco.2025.101991","DOIUrl":"10.1016/j.jco.2025.101991","url":null,"abstract":"<div><div>Investigating the generalization and robustness of adversarial learning is an active research topic due to its implications in designing robust models for a wide range of machine learning tasks. In this paper, we aim to investigate the adversarially robust generalization of bipartite ranking against pairwise perturbation attacks from the lens of learning theory. We establish high-probability generalization error bounds of linear hypotheses and multi-layer neural networks for bipartite ranking under adversarial attacks, by developing Rademacher complexity over i.i.d. sample blocks and covering numbers. Our results provide a theoretical characterization of the interplay between generalization error and perturbation-related factors, revealing the important impact of feature dimension and weight regularization for achieving good generalization performance. Experimental results on real-world datasets validate the effectiveness of our theoretical findings.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101991"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145157435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Jonathan Siegel is the winner of the 2025 Joseph F. Traub Information-Based Complexity Young Researcher Award","authors":"Matthieu Dolbeault,&nbsp;Erich Novak,&nbsp;Kateryna Pozharska,&nbsp;Mathias Sonnleitner,&nbsp;Henryk Woźniakowski","doi":"10.1016/j.jco.2025.101996","DOIUrl":"10.1016/j.jco.2025.101996","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101996"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145332348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal prediction of vector-valued functions from point samples","authors":"Simon Foucart","doi":"10.1016/j.jco.2025.101981","DOIUrl":"10.1016/j.jco.2025.101981","url":null,"abstract":"<div><div>Predicting the value of a function <em>f</em> at a new point given its values at old points is an ubiquitous scientific endeavor, somewhat less developed when <em>f</em> produces several values depending on one another, e.g. when it outputs a probability vector. Considering the points as fixed (not random) entities and focusing on the worst-case, this article uncovers a prediction procedure that is optimal relatively to some model-set information about the vector-valued function <em>f</em>. When the model sets are convex, this procedure turns out to be an affine map constructed by solving a convex optimization program. The theoretical result is specified in the two practical frameworks of (reproducing kernel) Hilbert spaces and of spaces of continuous functions.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101981"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144867183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Deep learning from strongly mixing observations: Sparse-penalized regularization and minimax optimality","authors":"William Kengne ,&nbsp;Modou Wade","doi":"10.1016/j.jco.2025.101978","DOIUrl":"10.1016/j.jco.2025.101978","url":null,"abstract":"<div><div>This paper considers deep learning from strongly mixing observations and performs a sparse-penalized regularization for deep neural networks (DNN) predictors. In a general framework that includes regression and classification, oracle inequalities for the expected excess risk are established, and upper bounds on the class of Hölder smooth functions and composition structured Hölder functions are provided. For nonparametric autoregression with the Gaussian and Laplace errors, and the Huber loss function, it is shown that the sparse-penalized DNN estimator proposed is optimal (up to a logarithmic factor) in the minimax sense. Based on the lower bound established in Alquier and Kengne (2024), we show that the proposed DNN estimator for the classification task with the logistic loss on strongly mixing observations achieves (up to a logarithmic factor), the minimax optimal convergence rate.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101978"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144748669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sampling and entropy numbers in the uniform norm","authors":"Mario Ullrich","doi":"10.1016/j.jco.2025.101992","DOIUrl":"10.1016/j.jco.2025.101992","url":null,"abstract":"<div><div>We prove a sharp bound between sampling numbers and entropy numbers in the uniform norm for bounded convex sets of bounded functions.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101992"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117737","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Borell's inequality and mean width of random polytopes via discrete inequalities","authors":"David Alonso-Gutiérrez,&nbsp;Luis C. García-Lirola","doi":"10.1016/j.jco.2025.101993","DOIUrl":"10.1016/j.jco.2025.101993","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Borell's inequality states the existence of a positive absolute constant &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; such that for every &lt;span&gt;&lt;math&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;〈&lt;/mo&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;〉&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; whenever &lt;em&gt;X&lt;/em&gt; is a random vector uniformly distributed on any convex body &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt; is the standard canonical basis in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we will prove a discrete version of this inequality, which will hold whenever &lt;em&gt;X&lt;/em&gt; is a random vector uniformly distributed on &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;∩&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for any convex body &lt;span&gt;&lt;math&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;⊆&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; containing the origin in its interior. We will also make use of such discrete version to obtain discrete inequalities from which we can recover the estimate &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;∼&lt;/mo&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for any convex body &lt;em&gt;K&lt;/em&gt; containing the origin in its interior, where &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is the centrally symmetric random polytope &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mtext&gt;conv&lt;/mtext&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101993"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145117738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Accelerated convergence of error quantiles using robust randomized quasi Monte Carlo methods","authors":"Emmanuel Gobet ,&nbsp;Matthieu Lerasle ,&nbsp;David Métivier","doi":"10.1016/j.jco.2025.101989","DOIUrl":"10.1016/j.jco.2025.101989","url":null,"abstract":"<div><div>We aim to calculate an expectation <span><math><mi>μ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>=</mo><mi>E</mi><mrow><mo>(</mo><mi>F</mi><mo>(</mo><mi>U</mi><mo>)</mo><mo>)</mo></mrow></math></span> for functions <span><math><mi>F</mi><mo>:</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup><mo>↦</mo><mi>R</mi></math></span> using a family of estimators <span><math><msub><mrow><mover><mrow><mi>μ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>B</mi></mrow></msub></math></span> with a budget of <em>B</em> evaluation points. The standard Monte Carlo method achieves a root mean squared risk of order <span><math><mn>1</mn><mo>/</mo><msqrt><mrow><mi>B</mi></mrow></msqrt></math></span>, both for a fixed square integrable function <em>F</em> and for the worst-case risk over the class <span><math><mi>F</mi></math></span> of functions with <span><math><msub><mrow><mo>‖</mo><mi>F</mi><mo>‖</mo></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><mo>≤</mo><mn>1</mn></math></span>. Using a sequence of Randomized Quasi Monte Carlo (RQMC) methods, in contrast, we achieve faster convergence <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>≪</mo><mn>1</mn><mo>/</mo><msqrt><mrow><mi>B</mi></mrow></msqrt></math></span> for the risk <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>B</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo></math></span> when fixing a function <em>F</em>, compared to the worst-case risk which is still of order <span><math><mn>1</mn><mo>/</mo><msqrt><mrow><mi>B</mi></mrow></msqrt></math></span>. We address the convergence of quantiles of the absolute error, namely, for a given confidence level <span><math><mn>1</mn><mo>−</mo><mi>δ</mi></math></span> this is the minimal <em>ε</em> such that <span><math><mi>P</mi><mo>(</mo><mo>|</mo><msub><mrow><mover><mrow><mi>μ</mi></mrow><mrow><mo>ˆ</mo></mrow></mover></mrow><mrow><mi>B</mi></mrow></msub><mo>(</mo><mi>F</mi><mo>)</mo><mo>−</mo><mi>μ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>|</mo><mo>&gt;</mo><mi>ε</mi><mo>)</mo><mo>≤</mo><mi>δ</mi></math></span> holds. We show that a judicious choice of a robust aggregation method coupled with RQMC methods allows reaching improved convergence rates for <em>ε</em> depending on <em>δ</em> and <em>B</em> when fixing a function <em>F</em>. This study includes a review on concentration bounds for the empirical mean as well as sub-Gaussian mean estimates and is supported by numerical experiments, ranging from bounded <em>F</em> to heavy-tailed <span><math><mi>F</mi><mo>(</mo><mi>U</mi><mo>)</mo></math></span>, the latter being well suited to functions <em>F</em> with a singularity. The different methods we have tested are available in a Julia package.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"92 ","pages":"Article 101989"},"PeriodicalIF":1.8,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144931571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}