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Learning methods in Banach spaces are often formulated as regularizationproblems which minimize the sum of a data fidelity term in a Banach norm and aregularization term in another Banach norm. Due to the infinite dimensionalnature of the space, solving such regularization problems is challenging. Weconstruct a direct sum space based on the Banach spaces for the data fidelityterm and the regularization term, and then recast the objective function as thenorm of a suitable quotient space of the direct sum space. In this way, weexpress the original regularized problem as an unregularized problem on thedirect sum space, which is in turn reformulated as a dual optimization problemin the dual space of the direct sum space. The dual problem is to find themaximum of a linear function on a convex polytope, which may be solved bylinear programming. A solution of the original problem is then obtained byusing related extremal properties of norming functionals from a solution of thedual problem. Numerical experiments are included to demonstrate that theproposed duality approach leads to an implementable numerical method forsolving the regularization learning problems.

A Duality Approach to Regularized Learning Problems in Banach Spaces