This book by Felipe Cucker and Ding-Xuan Zhou provides solid mathematical foundations and new insights into the subject called learning theory.

Some years ago, Felipe and I were trying to find something about brain science and artificial intelligence starting from literature on neural nets. It was in this setting that we encountered the beautiful ideas and fast algorithms of learning theory. Eventually we were motivated to write on the mathematical foundations of this new area of science.

I have found this arena to with its new challenges and growing number of application, be exciting. For example, the unification of dynamical systems and learning theory is a major problem.Another problem is to develop a comparative study of the useful algorithms currently available and to give unity to these algorithms. How can one talk about the “best algorithm” or find the most appropriate algorithm for a particular task when there are so many desirable features, with their associated trade-offs? How can one see the working of aspects of the human brain and machine vision in the same framework?

I know both authors well. I visited Felipe in Barcelona more than 13 years ago for several months, and when I took a position in Hong Kong in 1995, I asked him to join me. There Lenore Blum, Mike Shub, Felipe, and I finished a book on real computation and complexity. I returned to the USA in 2001, but Felipe continues his job at the City University of Hong Kong. Despite the distance we have continued to write papers together. I came to know Ding-Xuan as a colleague in the math department at City University. We have written…

Portfolio Selection

This chapter is devoted to the application of the ideas described in Chapter 9 to the problem of sequential investment. Imagine a market of m assets (stocks) in which, in each trading period (day), the price of a stock may vary in an arbitrary way. An investor operates on this market for n days with the goal of maximizing his final wealth. At the beginning of each day, on the basis of the past behavior of the market, the investor redistributes his current wealth among the m assets. Following the approach developed in the previous chapters, we avoid any statistical assumptions about the nature of the stock market, and evaluate the investor's wealth relative to the performance achieved by the best strategy in a class of reference investment strategie (the “experts”).

In the idealized stock market we assume that there are no transaction costs and the amount of each stock that can be bought at any trading period is only limited by the investor's wealth at that time. Similarly, the investor can sell any quantity of the stocks he possesses at any time at the actual market price.

The model may be formalized as follows. A market vectorx = (x1, …, xm) for m assets is a vector of nonnegative real numbers representing price relatives for a given trading period. In other words, the quantity xi ≥ 0 denotes the ratio of closing to opening price of the ith asset for that period.