Nonexact oracle inequalities, r-learnability, and fast rates

IF 1.8 2区数学 Q1 MATHEMATICS

Journal of Complexity Pub Date : 2023-10-13 DOI:10.1016/j.jco.2023.101804

Daniel Z. Zanger

引用次数: 0

Abstract

As an extension of the standard paradigm in statistical learning theory, we introduce the concept of r-learnability, $0 < r \leq 1$ , which is a notion very closely related to that of nonexact oracle inequalities (see Lecue and Mendelson (2012) [7]). The r-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 $(1 + r)$ -factor in the learning error estimate. We establish a new, general r-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).

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非精确oracle不等式、r-learnability和快速速率

作为统计学习理论标准范式的延伸，我们引入了r可学习性的概念，0<；r≤1，这是一个与非存在预言不等式非常密切相关的概念（见Leque和Mendelson（2012）[7]）。r-可学习性概念可以以学习误差估计中的近似误差项乘以额外的（1+r）因子为代价，建立所谓的快速学习率（以及相应的样本复杂度类型边界）。我们建立了一个新的、通用的r学习界（非代理预言不等式），在不可知模型的一般设置下，产生快速的概率学习率（最多可达对数因子），用于正确学习，本质上只假设一致有界的平方损失函数和有限VC维（即有限伪维）的假设类。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Complexity 工程技术-计算机：理论方法

CiteScore

3.10

自引率

17.60%

发文量

审稿时长

>12 weeks

期刊介绍： The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.