On-line learning of rectangles

This paper solves the following open problem: Is there an algorithmfor on-line learning of rectangles <inline-equation><f><pr align="c"><ll>i=1</ll><ul>d</ul></pr></f><?Pub Caret></inline-equation><?Pub Fmt italic>a<subscrpt>i</subscrpt>,a<subscrpt>i</subscrpt>+1,…,b<subscrpt>i</subscrpt>}<?Pub Fmt /italic>over a discrete domain<?Pub Fmt italic>{1,…,n}<supscrpt>d</supscrpt><?Pub Fmt /italic>whose error bound is polylogarithmic in the size<?Pub Fmt italic>n<supscrpt>d</supscrpt><?Pub Fmt /italic> of the domain(i.e. polynomial in <?Pub Fmt italic>d<?Pub Fmt /italic> and log<?Pub Fmt italic>n<?Pub Fmt /italic> )? We give a positive solution byintroducing a new design technique that appears to be of some intereston its own. The new learning algorithm for rectangles consists of<?Pub Fmt italic>2d<?Pub Fmt /italic> separate search strategies thatsearch for the parameters <?Pub Fmt italic>a<subscrpt>1</subscrpt>,b<subscrpt>1</subscrpt>,…,a<subscrpt>d</subscrpt>,b<subscrpt>d</subscrpt><?Pub Fmt /italic>of the target rectangle. A learning algorithm with this type of modulardesign ends to fail because of the well known “credit assignmentproblem”: Which of the <?Pub Fmt italic>2d<?Pub Fmt /italic> localsearch strategies should be “blamed” when the globalalgorithm makes an error? We overcome this difficulty by employing localsearch strategies (“error tolerant binary search”) that areable to tolerate certain types of wrong credit assignments. Section 4 contains another application of this design technique: analgorithm for learning the union of two rectangles in the plane.