非凸径向基神经网络优化问题的规范对偶解
Canonical dual solutions to nonconvex radial basis neural network optimization problem
摘要：
径向基函数神经网络（RBFNN）是在回归问题中广泛使用的工具。它们的主要缺点之一是，与中心和权重都有监督的训练相对应的公式是一个高度非凸的优化问题，这导致传统优化理论和方法的一些根本困难。本文提出了一个广义正则对偶理论来解决这个具有挑战性的问题。我们证明了，通过连续的正则对偶变换，RBFNN的非凸优化问题可以重新表述为一个正则对偶问题（没有对偶间隙）。全局最优解和局部极值都可以分类。最常用的径向基函数之一，高斯函数的几个应用程序，说明。结果表明，即使在一维情况下，非凸问题的全局极小值也不一定是RBFNN的最优解，而正则对偶理论是解决一般神经网络训练问题的有效工具.
英文摘要：
Radial Basis Functions Neural Networks (RBFNNs) are tools widely used in regression problems. One of their principal drawbacks is that the formulation corresponding to the training with the supervision of both the centers and the weights is a highly non-convex optimization problem, which leads to some fundamentally difficulties for traditional optimization theory and methods. This paper presents a generalized canonical duality theory for solving this challenging problem. We demonstrate that by sequential canonical dual transformations, the nonconvex optimization problem of the RBFNN can be reformulated as a canonical dual problem (without duality gap). Both global optimal solution and local extrema can be classified. Several applications to one of the most used Radial Basis Functions, the Gaussian function, are illustrated. Our results show that even for one-dimensional case, the global minimizer of the nonconvex problem may not be the best solution to the RBFNNs, and the canonical dual theory is a promising tool for solving general neural networks training problems.
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