🤖 AI Summary
This work proposes Asymptotic Learning Theory (ALT), a novel framework designed to efficiently and accurately estimate unknown constants or parameters from known asymptotic forms, while providing rigorous guarantees on convergence and convergence rates. The core methodology introduces sliding Linear Least Squares (sLLSQ) and its Tikhonov-regularized variant (sT-LLSQ), integrating tools from optimization, asymptotic analysis, complex analysis, and analytic combinatorics. For the first time, a theoretical foundation for asymptotic learning is established, proving convergence properties of the proposed estimators and delineating their regimes of applicability. Numerical experiments in contexts such as analytic combinatorics validate the theoretical findings, demonstrating that the method significantly outperforms conventional techniques like the ratio method.
📝 Abstract
We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence---or even, counterintuitively, divergence---in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.