{"title":"Nonexact oracle inequalities, r-learnability, and fast rates","authors":"Daniel Z. Zanger","doi":"10.1016/j.jco.2023.101804","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101804","url":null,"abstract":"<div><p>As an extension of the standard paradigm in statistical learning theory, we introduce the concept of <em>r</em>-learnability, <span><math><mn>0</mn><mo><</mo><mi>r</mi><mo>≤</mo><mn>1</mn></math></span>, which is a notion very closely related to that of nonexact oracle inequalities (see Lecue and Mendelson (2012) <span>[7]</span>). The <em>r</em>-learnability concept can enable so-called fast learning rates (along with corresponding sample complexity-type bounds) to be established at the cost of multiplying the approximation error term by an extra <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>r</mi><mo>)</mo></math></span>-factor in the learning error estimate. We establish a new, general <em>r</em>-learning bound (nonexact oracle inequality) yielding fast learning rates in probability (up to at most a logarithmic factor) for proper learning in the general setting of an agnostic model, essentially only assuming a uniformly bounded squared loss function and a hypothesis class of finite VC-dimension (that is, finite pseudo-dimension).</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101804"},"PeriodicalIF":1.7,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887911","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":"High-order lifting for polynomial Sylvester matrices","authors":"Clément Pernet , Hippolyte Signargout , Gilles Villard","doi":"10.1016/j.jco.2023.101803","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101803","url":null,"abstract":"<div><p>A new algorithm is presented for computing the resultant of two generic bivariate polynomials over an arbitrary field. For <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span> in <span><math><mi>K</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span> of degree <em>d</em> in <em>x</em> and <em>n</em> in <em>y</em>, the resultant with respect to <em>y</em> is computed using <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1.458</mn></mrow></msup><mi>d</mi><mo>)</mo></math></span> arithmetic operations if <span><math><mi>d</mi><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span>. For <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span>, the complexity estimate is therefore reconciled with the estimates of Neiger et al. 2021 for the related problems of modular composition and characteristic polynomial in a univariate quotient algebra. The 3/2 barrier in the exponent of <em>n</em> is crossed for the first time for the resultant. The problem is related to that of computing determinants of structured polynomial matrices. We identify new advanced aspects of structure for a polynomial Sylvester matrix. This enables to compute the determinant by mixing the baby steps/giant steps approach of Kaltofen and Villard 2005, until then restricted to the case <span><math><mi>d</mi><mo>=</mo><mn>1</mn></math></span> for characteristic polynomials, and the high-order lifting strategy of Storjohann 2003 usually reserved for dense polynomial matrices.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101803"},"PeriodicalIF":1.7,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X23000729/pdfft?md5=72b813e3258f79c8cf5a380cd73b1e8f&pid=1-s2.0-S0885064X23000729-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91959851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Numerical weighted integration of functions having mixed smoothness","authors":"Dinh Dũng","doi":"10.1016/j.jco.2023.101757","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101757","url":null,"abstract":"<div><p>We investigate the approximation of weighted integrals over <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span><span> for integrands<span><span> from weighted Sobolev spaces of mixed smoothness. We prove </span>upper and lower bounds of the convergence rate of optimal quadratures with respect to </span></span><em>n</em> integration nodes for functions from these spaces. In the one-dimensional case <span><math><mo>(</mo><mi>d</mi><mo>=</mo><mn>1</mn><mo>)</mo></math></span>, we obtain the right convergence rate of optimal quadratures. For <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"78 ","pages":"Article 101757"},"PeriodicalIF":1.7,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50198402","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":"Discrepancy bounds for normal numbers generated by necklaces in arbitrary base","authors":"Roswitha Hofer, Gerhard Larcher","doi":"10.1016/j.jco.2023.101767","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101767","url":null,"abstract":"<div><p>Mordechay B. Levin (1999) has constructed a number <em>λ</em> which is normal in base 2, and such that the sequence <span><math><msub><mrow><mo>(</mo><mrow><mo>{</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mi>λ</mi><mo>}</mo></mrow><mo>)</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo></mrow></msub></math></span> has very small discrepancy <span><math><mi>N</mi><mo>⋅</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>N</mi></mrow></msub><mo>=</mo><mi>O</mi><mrow><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>N</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></math></span>. This construction technique was generalized by Becher and Carton (2019), who generated normal numbers via nested perfect necklaces, for which the same upper discrepancy estimate holds. In this paper we derive an upper discrepancy bound for so-called semi-perfect nested necklaces and show that for Levin's normal number in arbitrary prime base <em>p</em> this upper bound for the discrepancy is best possible. This result generalizes a previous result by the authors (2022) in base 2.</p><p>Our result for Levin's normal number in any prime base might support the guess that <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>N</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> is the best order in <em>N</em> that can be achieved by a normal number, while generalizing the class of known normal numbers by introducing semi-perfect necklaces on the other hand might help for the search of normal numbers that satisfy smaller discrepancy bounds.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"78 ","pages":"Article 101767"},"PeriodicalIF":1.7,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50198397","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}
Gunther Leobacher , Friedrich Pillichshammer , Adrian Ebert
{"title":"Tractability of L2-approximation and integration in weighted Hermite spaces of finite smoothness","authors":"Gunther Leobacher , Friedrich Pillichshammer , Adrian Ebert","doi":"10.1016/j.jco.2023.101768","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101768","url":null,"abstract":"<div><p>In this paper we consider integration and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-approximation for functions over <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> from weighted Hermite spaces. The first part of the paper is devoted to a comparison of several weighted Hermite spaces that appear in literature, which is interesting on its own. Then we study tractability of the integration and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-approximation problem for the introduced Hermite spaces, which describes the growth rate of the information complexity when the error threshold <em>ε</em> tends to 0 and the problem dimension <em>s</em> grows to infinity. Our main results are characterizations of tractability in terms of the involved weights, which model the importance of the successive coordinate directions for functions from the weighted Hermite spaces.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"78 ","pages":"Article 101768"},"PeriodicalIF":1.7,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50198403","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":"Expected multivolumes of random amoebas","authors":"Turgay Bayraktar , Ali Ulaş Özgür Kişisel","doi":"10.1016/j.jco.2023.101802","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101802","url":null,"abstract":"<div><p>We compute the expected multivolume of the amoeba of a random half dimensional complete intersection in <span><math><msup><mrow><mi>CP</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>. We also give a relative generalization of our result to the toric case.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101802"},"PeriodicalIF":1.7,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49886730","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":"A strongly monotonic polygonal Euler scheme","authors":"Tim Johnston, Sotirios Sabanis","doi":"10.1016/j.jco.2023.101801","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101801","url":null,"abstract":"<div><p>In recent years tamed schemes have become an important technique for simulating SDEs and SPDEs whose continuous coefficients display superlinear growth. The taming method involves curbing the growth of the coefficients as a function of stepsize, but so far has not been adapted to preserve the monotonicity of the coefficients. This has arisen as an issue in <span>[4]</span>, where the lack of a strongly monotonic tamed scheme forces strong conditions on the setting. In this article we give a novel and explicit method for truncating monotonic functions in separable real Hilbert spaces, and show how this can be used to define a polygonal (tamed) Euler scheme on finite dimensional space, preserving the monotonicity of the drift coefficient, and converging to the true solution at the same rate as the classical Euler scheme for Lipschitz coefficients. Our construction is the first explicit method for truncating monotone functions we are aware of, and the first in infinite dimensions.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101801"},"PeriodicalIF":1.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887910","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":"On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting","authors":"Jiaxin Geng, Heping Wang","doi":"10.1016/j.jco.2023.101790","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101790","url":null,"abstract":"<div><p>We study multivariate approximation of periodic functions in the worst case setting with the error measured in the <span><math><msub><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> norm. We consider algorithms that use standard information <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>std</mi></mrow></msup></math></span> consisting of function values or general linear information <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>all</mi></mrow></msup></math></span> consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>std</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>all</mi></mrow></msup></math></span> under the absolute or normalized error criterion, and show that the power of <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>std</mi></mrow></msup></math></span> is the same as the one of <span><math><msup><mrow><mi>Λ</mi></mrow><mrow><mi>all</mi></mrow></msup></math></span> for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) <span>[40]</span>.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101790"},"PeriodicalIF":1.7,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887908","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":"Changes of the Editorial Board","authors":"Erich Novak","doi":"10.1016/j.jco.2023.101792","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101792","url":null,"abstract":"","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"79 ","pages":"Article 101792"},"PeriodicalIF":1.7,"publicationDate":"2023-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49877024","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":"Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere","authors":"Congpei An , Hao-Ning Wu","doi":"10.1016/j.jco.2023.101789","DOIUrl":"https://doi.org/10.1016/j.jco.2023.101789","url":null,"abstract":"<div><p>This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree <em>n</em> via hyperinterpolation. Hyperinterpolation of degree <em>n</em> is a discrete approximation of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-orthogonal projection of the same degree with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2<em>n</em>. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz–Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as <em>unfettered hyperinterpolation</em>. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"80 ","pages":"Article 101789"},"PeriodicalIF":1.7,"publicationDate":"2023-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49887909","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}