https://escholarship.org/content/qt3bc4p075/qt3bc4p075.pdf?t=lnpw9c

Experience, generations, and limits in machine learning

We present learning of figures, nonempty compact sets in Euclidean space, based on Gold's learning model aiming at a computable foundation for binary classification of multivariate data. Encoding real vectors with no numerical error requires infinite sequences, resulting in a gap between each real vector and its discretized representation used for the actual machine learning process. Our motivation is to provide an analysis of machine learning problems that explicitly tackles this aspect which has been glossed over in the literature on binary classification as well as in other machine learning tasks such as regression and clustering. In this paper, we amalgamate two processes: discretization and binary classification. Each learning target, the set of real vectors classified as positive, is treated as a figure. A learning machine receives discretized vectors as input data and outputs a sequence of discrete representations of the target figure in the form of self-similar sets, known as fractals. The generalization error of each output is measured by the Hausdorff metric. Using this learning framework, we reveal a hierarchy of learnable classes under various learning criteria in the track of traditional analysis based on Gold's learning model, and show a mathematical connection between machine learning and fractal geometry by measuring the complexity of learning using the Hausdorff dimension and the VC dimension. Moreover, we analyze computability aspects of learning of figures using the framework of Type-2 Theory of Effectivity (TTE).

/pdf/learning-figures-with-the-hausdorff-metric-by-fractals-2m1xuzkh23.pdf

Learning figures with the Hausdorff metric by fractals--towards computable binary classification

In this paper, we investigate prediction of recursive real-valued functions from finite examples by extending the framework of inductive inference of recursive real-valued functions to be a more realistic one. First, we propose a finite prediction machine, which is a procedure that requests finite examples of a recursive real-valued function h and a datum of a real number x, and that outputs a datum of h(x). Then, we formulate finite prediction of recursive real-valued functions and investigate the power of it. Furthermore, for a fixed rational closed interval I, we show that the class of all finitely predictable sets of recursive real-valued functions coincides with the class of all inferable sets of recursive real-valued functions in the limit, that is, ${\sc RealFP}_{\emph I}={\sc RealEx}_{\emph I}$.

Prediction of recursive real-valued functions from finite examples

In this paper we investigate the inductive inference of recursive real-valued functions from data. A recursive real-valued function is regarded as a computable interval mapping. The learning model we consider in this paper is an extension of Gold's inductive inference. We first introduce some criteria for successful inductive inference of recursive real-valued functions. Then we show a recursively enumerable class of recursive real-valued functions which is not inferable in the limit. This should be an interesting contrast to the result by Wiehagen (1976, Elektronische Informationsverarbeitung und Kybernetik, Vol. 12, pp. 93--99) that every recursively enumerable subset of recursive functions from N to N is consistently inferable in the limit. We also show that every recursively enumerable class of recursive real-valued functions on a fixed rational interval is consistently inferable in the limit. Furthermore, we show that our consistent inductive inference coincides with the ordinary inductive inference, when we deal with recursive real-valued functions on a fixed closed rational interval. Copyright 2001 Elsevier Science B.V. All rights reserved.

A comparison of identification criteria for inductive inference of recursive real-valued functions

Inductive inference gives us a theoretical model of concept learning from examples. In this paper, we study refutably and reliably inductive inference of recursive real-valued functions. First we introduce the new criteria RealRefEx for refutable inference and RealRelEx for reliable inference. Then, we compare these two criteria with RealEx for identification in the limit, RealFin for learning finitely and RealNumi for learning by enumeration that have been already introduced in the previous works, and investigate their interaction. In particular, we show that RealRefEx and RealRelEx are closed under union, as similar as the criteria RefEx and RelEx for inductive inference of recursive functions.

Refutability and Reliability for Inductive Inference of Recursive Real-Valued Functions

We combine traditional studies of inductive inference and classical continuous mathematics to produce a study of learning real-valued functions. We consider two possible ways to model the learning by example of functions with domain and range the real numbers. The first approach considers functions as represented by computable analytic functions. The second considers arbitrary computable functions of recursive real numbers. In each case we find natural examples of learnable classes of functions and unlearnable classes of functions.

Inductive inference of recursive real-valued functions

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