In this chapter we describe a framework of density-ratio estimation by density-ratio fitting under the Bregman divergence (Bregman, 1967). This framework is a natural extension of the least-squares approach described in Chapter 6, and it includes various existing approaches as special cases (Sugiyama et al., 2011a).

In Section 7.1 we first describe the framework of density-ratio fitting under the Bregman divergence. Then, in Section 7.2, we show that various existing approaches can be accommodated in this framework, such as kernel mean matching (see Section 3.3), logistic regression (see Section 4.2), Kullback–Leibler importance estimation procedure (see Section 5.1), and least-squares importance fitting (see Section 6.1).We then show other views of the density-ratio fitting framework in Section 7.3. Furthermore, in Section 7.4, a robust density-ratio estimator is derived as an instance of the density-ratio fitting approach based on Basu's power divergence (Basu et al., 1998). The chapter is concluded in Section 7.5.

Basic Framework

A basic idea of density-ratio fitting is to fit a density ratio model r(x) to the true density-ratio function r*(x) under some divergence (Figure 7.1). At a glance, this density-ratio fitting problem may look equivalent to the regression problem, which is aimed at learning a real-valued function [see Section 1.1.1 and Figure 1.1(a)]. However, density-ratio fitting is essentially different from regression because samples of the true density-ratio function are not available. Here we employ the Bregman (BR) divergence (Bregman, 1967) for measuring the discrepancy between the true density-ratio function and the density-ratio model.

Machine learning is aimed at developing systems that learn. The mathematical foundation of machine learning and its real-world applications have been extensively explored in the last decades. Various tasks of machine learning, such as regression and classification, typically can be solved by estimating probability distributions behind data. However, estimating probability distributions is one of the most difficult problems in statistical data analysis, and thus solving machine learning tasks without going through distribution estimation is a key challenge in modern machine learning.

So far, various algorithms have been developed that do not involve distribution estimation but solve target machine learning tasks directly. The support vector machine is a successful example that follows this line – it does not estimate data generating distributions but directly obtains the class-decision boundary that is sufficient for classification. However, developing such an excellent algorithm for each of the machine learning tasks could be highly costly and difficult.

To overcome these limitations of current machine learning research, we introduce and develop a novel paradigm called density-ratio estimation – instead of probability distributions, the ratio of probability densities is estimated for statistical data processing. The density-ratio approach covers various machine learning tasks, for example, non-stationarity adaptation, multi-task learning, outlier detection, two-sample tests, feature selection, dimensionality reduction, independent component analysis, causal inference, conditional density estimation, and probabilitic classification. Thus, density-ratio estimation is a versatile tool for machine learning. This book is aimed at introducing the mathematical foundation, practical algorithms, and applications of density-ratio estimation.

In this book we described a new approach to machine learning based on densityratio estimation. This density-ratio approach offers a novel research paradigm in the field of machine learning and data mining from theory and algorithms to application.

In Part II, various methods for density-ratio estimation were described, including methods based on separate estimations of numerator and denominator densities (Chapter 2), moment matching between numerator and denominator samples (Chapter 3), probabilistic classifications of numerator and denominator samples (Chapter 4), density fitting between numerator and denominator densities (Chapter 5), and direct fitting of a density-ratio model to the true densityratio (Chapter 6). We also gave a unified framework of density-ratio estimation in Chapter 7, which accommodates the various methods described above and is substantially more general – as an example, a robust density-ratio estimator was derived. Finally, in Chapter 8, we described methods that combine density-ratio estimation with dimensionality reduction. Among various density-ratio estimators, the unconstrained least-squares importance fitting (uLSIF) method described in Chapter 6 would be most useful practically because of its high computational efficiency by an analytic-form solution, the availability of cross-validation for model selection, its wide applicability to various machine learning tasks (Part III), and its superior convergence and numerical properties (Part IV).

In Part III we covered the usage of density-ratio estimators in various machine learning tasks that were categorized into four groups. In Chapter 9 we described applications of density ratios to importance sampling tasks such as non-stationarity/domain adaptation and multi-task learning.

The data explosion

We live in a world that is rich in data, ever increasing in scale. This data comes from many different sources in science (bioinformatics, astronomy, physics, environmental monitoring) and commerce (customer databases, financial transactions, engine monitoring, speech recognition, surveillance, search). Possessing the knowledge as to how to process and extract value from such data is therefore a key and increasingly important skill. Our society also expects ultimately to be able to engage with computers in a natural manner so that computers can ‘talk’ to humans, ‘understand’ what they say and ‘comprehend’ the visual world around them. These are difficult large-scale information processing tasks and represent grand challenges for computer science and related fields. Similarly, there is a desire to control increasingly complex systems, possibly containing many interacting parts, such as in robotics and autonomous navigation. Successfully mastering such systems requires an understanding of the processes underlying their behaviour. Processing and making sense of such large amounts of data from complex systems is therefore a pressing modern-day concern and will likely remain so for the foreseeable future.

Machine learning

Machine learning is the study of data-driven methods capable of mimicking, understanding and aiding human and biological information processing tasks. In this pursuit, many related issues arise such as how to compress data, interpret and process it. Often these methods are not necessarily directed to mimicking directly human processing but rather to enhancing it, such as in predicting the stock market or retrieving information rapidly. In this probability theory is key since inevitably our limited data and understanding of the problem forces us to address uncertainty.