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Rademacher learning rates for iterated random functions 迭代随机函数的Rademacher学习率
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-06-19 DOI: 10.1016/j.jco.2025.101971
Nikola Sandrić
{"title":"Rademacher learning rates for iterated random functions","authors":"Nikola Sandrić","doi":"10.1016/j.jco.2025.101971","DOIUrl":"10.1016/j.jco.2025.101971","url":null,"abstract":"<div><div>Most supervised learning methods assume training data is drawn from an i.i.d. sample. However, real-world problems often exhibit temporal dependence and strong correlations between marginals of the data-generating process, rendering the i.i.d. assumption unrealistic. Such cases naturally involve time-series processes and Markov chains. The learning rates typically obtained in these settings remain independent of the data distribution, potentially leading to restrictive hypothesis classes and suboptimal sample complexities. We consider training data generated by an iterated random function that need not be irreducible or aperiodic. Assuming the governing function is contractive in its first argument and subject to certain regularity conditions on the hypothesis class, we first establish uniform convergence for the sample error. We then prove learnability of approximate empirical risk minimization and derive its learning rate bound. Both bounds depend explicitly on the data distribution through the Rademacher complexities of the hypothesis class, thereby better capturing properties of the data-generating distribution.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101971"},"PeriodicalIF":1.8,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144321580","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}
引用次数: 0
Geo-indistinguishable location obfuscation with inference error bounds 具有推理误差界的地理不可区分位置混淆
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-06-11 DOI: 10.1016/j.jco.2025.101970
Shun Zhang , Benfei Duan , Zhili Chen , Hong Zhong
{"title":"Geo-indistinguishable location obfuscation with inference error bounds","authors":"Shun Zhang ,&nbsp;Benfei Duan ,&nbsp;Zhili Chen ,&nbsp;Hong Zhong","doi":"10.1016/j.jco.2025.101970","DOIUrl":"10.1016/j.jco.2025.101970","url":null,"abstract":"<div><div>Geo-indistinguishability and expected inference error are two complementary statistical notions for location privacy. The joint guarantee of differential privacy (indistinguishability) and distortion privacy (inference error) limits the information leakage. This paper analyzes the dynamic location obfuscation mechanism called PIVE by Yu, Liu and Pu (NDSS 2017), and shows that PIVE fails to offer either of the privacy guarantees on adaptive Protection Location Sets (PLSs) as claimed. Specifically, we demonstrate that different PLSs could intersect with one another due to the defined search algorithm, and different apriori locations in the same PLS could have different protection diameters which causes the problematic proof of local differential privacy for PIVE. Besides, the condition introduced in PIVE is confirmed to be not sufficient for bounding expected inference errors against Bayesian attacks. To address these issues, we introduce a relaxed definition of geo-indistinguishability, propose a couple of correction approaches, and analyze their satisfied privacy characteristics.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101970"},"PeriodicalIF":1.8,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144364448","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}
引用次数: 0
A sixth-order bi-parametric iterative method for nonlinear systems: Theory, stability and computational complexity 非线性系统的六阶双参数迭代法：理论、稳定性和计算复杂度
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-06-06 DOI: 10.1016/j.jco.2025.101960
G Thangkhenpau, Sunil Panday
{"title":"A sixth-order bi-parametric iterative method for nonlinear systems: Theory, stability and computational complexity","authors":"G Thangkhenpau,&nbsp;Sunil Panday","doi":"10.1016/j.jco.2025.101960","DOIUrl":"10.1016/j.jco.2025.101960","url":null,"abstract":"<div><div>In this paper, we propose a new bi-parametric three-step iterative method with sixth-order convergence for solving systems of nonlinear equations. The method is formulated using the composition technique, which we uniquely apply twice within a single method - an approach that, to our knowledge, is the first of its kind in the literature. This allows us to incorporate two free disposable parameters, offering flexibility with enhanced performance, stability and adaptability. The local convergence behaviour is rigorously analysed within the framework of Banach spaces, where we establish theoretical bounds for the convergence radius and demonstrate uniqueness conditions under Lipschitz-continuous Fréchet derivative assumptions. The theoretical outcomes are supported by numerical experiments. Lastly, we evaluate the method's efficiency by exploring its basins of attraction and applying it to systems of nonlinear equations and boundary value problems.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101960"},"PeriodicalIF":1.8,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144255436","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}
引用次数: 0
On upper and lower bounds for pathwise approximation of scalar SDEs with reflection 带反射的标量SDEs路径逼近的上界和下界
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-05-16 DOI: 10.1016/j.jco.2025.101959
Mario Hefter , André Herzwurm , Klaus Ritter
{"title":"On upper and lower bounds for pathwise approximation of scalar SDEs with reflection","authors":"Mario Hefter ,&nbsp;André Herzwurm ,&nbsp;Klaus Ritter","doi":"10.1016/j.jco.2025.101959","DOIUrl":"10.1016/j.jco.2025.101959","url":null,"abstract":"<div><div>For scalar SDEs with a one-sided reflection we study pathwise approximation, globally on a compact time interval or at a single time point. We consider algorithms based on sequential evaluations of the driving Brownian motion and establish upper and lower bounds for the minimal errors. Exploiting the relation to a reflected Ornstein-Uhlenbeck process, we also provide a new upper bound for a Cox-Ingersoll-Ross process.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101959"},"PeriodicalIF":1.8,"publicationDate":"2025-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144115566","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}
引用次数: 0
Skewness of a randomized quasi-Monte Carlo estimate 随机拟蒙特卡罗估计的偏度
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-05-05 DOI: 10.1016/j.jco.2025.101956
Zexin Pan, Art B. Owen
{"title":"Skewness of a randomized quasi-Monte Carlo estimate","authors":"Zexin Pan,&nbsp;Art B. Owen","doi":"10.1016/j.jco.2025.101956","DOIUrl":"10.1016/j.jco.2025.101956","url":null,"abstract":"<div><div>Some recent work on confidence intervals for randomized quasi-Monte Carlo (RQMC) sampling found a surprising result: ordinary Student's <em>t</em> 95% confidence intervals based on a modest number of replicates were seen to be very effective and even more reliable than some bootstrap <em>t</em> intervals that were expected to be best. One potential explanation is that those RQMC estimates have small skewness. In this paper we give conditions under which the skewness is <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> for any <span><math><mi>ϵ</mi><mo>&gt;</mo><mn>0</mn></math></span>, so ‘almost <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math></span>’. Under a random generator matrix model, we can improve this rate to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> with very high probability. We also improve some probabilistic bounds on the distribution of the quality parameter <em>t</em> for a digital net in a prime base under random sampling of generator matrices.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101956"},"PeriodicalIF":1.8,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143908058","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}
引用次数: 0
Bounds for the sampling discretization error and their applications to the universal sampling discretization 抽样离散误差的界限及其在通用抽样离散中的应用
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-05-05 DOI: 10.1016/j.jco.2025.101958
E.D. Kosov , V.N. Temlyakov
{"title":"Bounds for the sampling discretization error and their applications to the universal sampling discretization","authors":"E.D. Kosov ,&nbsp;V.N. Temlyakov","doi":"10.1016/j.jco.2025.101958","DOIUrl":"10.1016/j.jco.2025.101958","url":null,"abstract":"<div><div>In the first part of the paper we study absolute error of sampling discretization of the integral <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-norm for function classes of continuous functions. We use basic approaches from chaining technique to provide general upper bounds for the error of sampling discretization of the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-norm on a given function class in terms of entropy numbers in the uniform norm of this class. As an example we apply these general results to obtain new error bounds for sampling discretization of the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-norms on classes of multivariate functions with mixed smoothness. In the second part of the paper we apply our general bounds to study the problem of universal sampling discretization.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101958"},"PeriodicalIF":1.8,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143936949","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}
引用次数: 0
Average case tractability of multivariate approximation with Gevrey type kernels Gevrey型核多元逼近的平均情况可跟踪性
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-05-05 DOI: 10.1016/j.jco.2025.101957
Wanting Lu , Heping Wang
{"title":"Average case tractability of multivariate approximation with Gevrey type kernels","authors":"Wanting Lu ,&nbsp;Heping Wang","doi":"10.1016/j.jco.2025.101957","DOIUrl":"10.1016/j.jco.2025.101957","url":null,"abstract":"<div><div>We consider multivariate approximation problems in the average case setting with a zero mean Gaussian measure whose covariance kernel is a periodic Gevrey kernel. We investigate various notions of algebraic tractability and exponential tractability, and obtain necessary and sufficient conditions in terms of the parameters of the problem.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101957"},"PeriodicalIF":1.8,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143923221","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}
引用次数: 0
Tractability results for integration in subspaces of the Wiener algebra 维纳代数子空间积分的可溯性结果
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-04-16 DOI: 10.1016/j.jco.2025.101948
Josef Dick , Takashi Goda , Kosuke Suzuki
{"title":"Tractability results for integration in subspaces of the Wiener algebra","authors":"Josef Dick ,&nbsp;Takashi Goda ,&nbsp;Kosuke Suzuki","doi":"10.1016/j.jco.2025.101948","DOIUrl":"10.1016/j.jco.2025.101948","url":null,"abstract":"<div><div>In this paper, we present some new (in-)tractability results related to the integration problem in subspaces of the Wiener algebra over the <em>d</em>-dimensional unit cube. We show that intractability holds for multivariate integration in the standard Wiener algebra in the deterministic setting, in contrast to polynomial tractability in an unweighted subspace of the Wiener algebra recently shown by Goda (2023). Moreover, we prove that multivariate integration in the subspace of the Wiener algebra introduced by Goda is strongly polynomially tractable if we switch to the randomized setting, where we obtain a better <em>ε</em>-exponent than the one implied by the standard Monte Carlo method. We also identify subspaces in which multivariate integration in the deterministic setting are (strongly) polynomially tractable and we compare these results with the bound which can be obtained via Hoeffding's inequality.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"90 ","pages":"Article 101948"},"PeriodicalIF":1.8,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869597","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}
引用次数: 0
Takashi Goda is the winner of the 2025 Joseph F. Traub Prize for Achievement in Information-Based Complexity 后田隆史是 2025 年约瑟夫-特劳布信息复杂性成就奖（Joseph F. Traub Prize for Achievement in Information-Based Complexity）的获得者。
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-04-10 DOI: 10.1016/j.jco.2025.101947
Erich Novak
{"title":"Takashi Goda is the winner of the 2025 Joseph F. Traub Prize for Achievement in Information-Based Complexity","authors":"Erich Novak","doi":"10.1016/j.jco.2025.101947","DOIUrl":"10.1016/j.jco.2025.101947","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"89 ","pages":"Article 101947"},"PeriodicalIF":1.8,"publicationDate":"2025-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817477","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}
引用次数: 0
Constructions of normal numbers with infinite digit sets 具有无限位集的正常数的构造
IF 1.8 2区 数学
Journal of Complexity Pub Date : 2025-04-04 DOI: 10.1016/j.jco.2025.101945
Aafko Boonstra , Charlene Kalle
{"title":"Constructions of normal numbers with infinite digit sets","authors":"Aafko Boonstra ,&nbsp;Charlene Kalle","doi":"10.1016/j.jco.2025.101945","DOIUrl":"10.1016/j.jco.2025.101945","url":null,"abstract":"<div><div>Let <span><math><mi>L</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>d</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> be any ordered probability sequence, i.e., satisfying <span><math><mn>0</mn><mo>&lt;</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> for each <span><math><mi>d</mi><mo>∈</mo><mi>N</mi></math></span> and <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>d</mi><mo>∈</mo><mi>N</mi></mrow></msub><msub><mrow><mi>L</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><mn>1</mn></math></span>. We construct sequences <span><math><mi>A</mi><mo>=</mo><msub><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>N</mi></mrow></msub></math></span> on the countably infinite alphabet <span><math><mi>N</mi></math></span> in which each possible block of digits <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>∈</mo><mi>N</mi></math></span>, <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span>, occurs with frequency <span><math><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>d</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><msub><mrow><mi>L</mi></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></msub></math></span>. In other words, we construct <em>L</em>-normal sequences. These sequences can then be projected to normal numbers in various affine number systems, such as real numbers <span><math><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span> that are normal in GLS number systems that correspond to the sequence <em>L</em> or higher dimensional variants. In particular, this construction provides a family of numbers that have a normal Lüroth expansion.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"89 ","pages":"Article 101945"},"PeriodicalIF":1.8,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143817478","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}
引用次数: 0
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