Online learning of binary and n-ary relations over clustered domains

We consider the online learning problem for binary relations defined over two finite sets, each clustered into a relatively small number k, l of types (such a relation is termed a (k, l)-binary relation), extending the models of S. Goldman, R. Rivest, and R. Schapire (1993, SIAM J. Comput. 22, 1006-1034). We investigate the learning complexity of (k, l)-binary relations with respect to both the self-directed and adversary-directed learning models. We also generalize this problem to the learning problem for (k1, . . ., kd)-d-ary relations. In the self-directed model, we exhibit an efficient learning algorithm which makes at most kl + (n - k)log k + (m -l)log l mistakes, where n and m are the number of rows and columns, roughly twice the lower bound we show for this problem, 1/4⌊log k⌋ ⌊log l⌋ + 1/2(n - k) ⌊log k⌋ + 1/2(m -l) ⌊log l⌋. In the adversary-directed model, we exhibit an efficient algorithm for the (2,2)-binary relations, which makes at most n + m + 2 mistakes, only two more than the lower bound we show for this problem, n + m. As for (k1, . . ., kd)-d-ary relations, we obtain lower bounds and upper bounds on the number of mistakes in the self-directed model, teacher-directed model, and adversary-directed model. Finally we show that, although the sample consistency problem for (2,2)-binary relations is solvable in polynomial time, the same problem for (2,2,2)-ternary relations is already NP-complete. © 2002 Elsevier Science (USA).