AbstractAdaptive learning in a world of projections The task of parameter/function estimation has been at the center of scientific attention for a long time and it comes under different names such as filtering, prediction, beamforming, curve fitting, classification, regression. In this talk, the estimation task is treated in the context of set theoretic estimation arguments. Instead of a single optimal point, we are searching for a set of solutions that are in agreement with the available information, which is provided to us in the form of a set of training points and a set of constraints. Each point in the training data set, as well as each one of the constraints, is associated with a convex set, constructed according to a (convex) loss function (differentiable or not). The goal of this talk is to present a general tool for parameter/function estimation, under a set of convex constraints, both for classification as well as regression tasks, in a time adaptive setting appropriate for operation in (infinite dimensional) Reproducing Kernel Hilbert spaces (RKHS). The algorithmic scheme consists of a sequence of projections, of linear complexity with respect to the number of unknown parameters. Our theory proves that such a scheme converges to the intersection of all (with the possible exception of a finite number of) the convex sets, where the required solution lies. The performance of the methodology is demonstrated in the context of regression under nontrivial convex constraints, such as those met in robust beamforming, and in the context of sparsity-aware learning in the presence of constraints related to the l1 norm regularization. The latter task also reveals that our method can cope with problems under time varying constraints. The work has been carried out in cooperation with Kostas Slavakis and Isao Yamada. |