凸优化的双正交贪婪算法
Biorthogonal Greedy Algorithms in Convex Optimization
摘要：
随着贪婪算法被广泛应用于构造关于给定元素集的凸函数的稀疏极小问题，凸优化中的贪婪逼近问题成为一个很有前途的研究方向。本文给出了巴拿赫空间上极小化凸函数的一类贪婪型算法的统一分析方法。具体来说，我们定义了一类弱双正交贪婪算法的凸优化，其中包含了广泛的贪婪算法。我们分析了引入类的算法，并建立了收敛性，收敛速率，和数值稳定性，这是理解的意义上说，该算法的步骤被允许不精确地执行，但控制计算的不准确性。我们证明了下列著名的凸优化算法--弱Chebyshev贪婪算法（co）和带自由松弛的弱贪婪算法（co）--属于这一类，并引入了一个新的算法--重标弱松弛贪婪算法（co）.提出的数值实验证明了上述贪婪算法的实际性能相比，优化与正则化，这是传统的方法构造稀疏极小值的凸最小化的设置。
英文摘要：
The study of greedy approximation in the context of convex optimization is becoming a promising research direction as greedy algorithms are actively being employed to construct sparse minimizers for convex functions with respect to given sets of elements. In this paper we propose a unified way of analyzing a certain kind of greedy-type algorithms for the minimization of convex functions on Banach spaces. Specifically, we define the class of Weak Biorthogonal Greedy Algorithms for convex optimization that contains a wide range of greedy algorithms. We analyze the introduced class of algorithms and establish the properties of convergence, rate of convergence, and numerical stability, which is understood in the sense that the steps of the algorithm are allowed to be performed not precisely but with controlled computational inaccuracies. We show that the following well-known algorithms for convex optimization -- the Weak Chebyshev Greedy Algorithm (co) and the Weak Greedy Algorithm with Free Relaxation (co) -- belong to this class, and introduce a new algorithm -- the Rescaled Weak Relaxed Greedy Algorithm (co). Presented numerical experiments demonstrate the practical performance of the aforementioned greedy algorithms in the setting of convex minimization as compared to optimization with regularization, which is the conventional approach of constructing sparse minimizers.
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