## The Rademacher Complexity of Co-Regularized Kernel Classes

**David S. Rosenberg, Peter L. Bartlett**;
JMLR W&P 2:396-403, 2007.

### Abstract

In the multi-view approach to semisupervised learning, we choose one predictor from each of multiple hypothesis classes, and we "co-regularize" our choices by penalizing disagreement among the predictors on the unlabeled data. We examine the co-regularization method used in the coregularized least squares (CoRLS) algorithm [12], in which the views are reproducing kernel Hilbert spaces (RKHS's), and the disagreement penalty is the average squared diffrence in predictions. The final predictor is the pointwise average of the predictors from each view. We call the set of predictors that can result from this procedure the co-regularized hypothesis class. Our main result is a tight bound on the Rademacher complexity of the co-regularized hypothesis class in terms of the kernel matrices of each RKHS. We find that the co-regularization reduces the Rademacher complexity by an amount that depends on the distance between the two views, as measured by a data dependent metric. We then use standard techniques to bound the gap between training error and test error for the CoRLS algorithm. Experimentally, we find that the amount of reduction in complexity introduced by co-regularization correlates with the amount of improvement that co-regularization gives in the CoRLS algorithm.