Learning methods in Banach spaces are often formulated as regularization problems which minimize the sum of a data fidelity term in a Banach norm and a regularization term in another Banach norm. Due to the infinite dimensional nature of the space, solving such regularization problems is challenging. We construct a direct sum space based on the Banach spaces for the data fidelity term and the regularization term, and then recast the objective function as the norm of a suitable quotient space of the direct sum space. In this way, we express the original regularized problem as an unregularized problem on the direct sum space, which is in turn reformulated as a dual optimization problem in the dual space of the direct sum space. The dual problem is to find the maximum of a linear function on a convex polytope, which may be solved by linear programming. A solution of the original problem is then obtained by using related extremal properties of norming functionals from a solution of the dual problem. Numerical experiments are included to demonstrate that the proposed duality approach leads to an implementable numerical method for solving the regularization learning problems.

翻译：巴拿赫空间中的学习方法通常被表述为正则化问题，即在巴拿赫范数下最小化数据保真项与另一巴拿赫范数下正则化项之和。由于空间的无限维特性，求解此类正则化问题具有挑战性。我们基于数据保真项和正则化项的巴拿赫空间构造了一个直和空间，然后将目标函数重新表述为该直和空间适商空间的范数。通过这种方式，我们将原始正则化问题转化为直和空间上的非正则化问题，该问题进一步被重构为直和对偶空间中的对偶优化问题。对偶问题是在凸多面体上求线性函数的最大值，可通过线性规划求解。随后，利用范数泛函的相关极值性质，从对偶问题的解中获取原始问题的解。数值实验表明，所提出的对偶方法能为求解正则化学习问题提供可实现的数值方法。