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Adaptive Alternating Minimization Algorithms

机译：自适应交替最小化算法

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摘要

The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables. The iterative nature and simplicity of the algorithm has led to its application in many areas such as signal processing, information theory, control, and finance. A general set of sufficient conditions for the convergence and correctness of the algorithm are known when the underlying problem parameters are fixed. In many practical situations, however, the underlying problem parameters are changing over time, and the use of an adaptive algorithm is more appropriate. In this paper, we study such an adaptive version of the alternating minimization algorithm. More precisely, we consider the impact of having a slowly time-varying domain over which the minimization takes place. As a main result of this paper, we provide a general set of sufficient conditions for the convergence and correctness of the adaptive algorithm. Perhaps somewhat surprisingly, these conditions seem to be the minimal ones one would expect in such an adaptive setting. We present applications of our results to adaptive decomposition of mixtures, adaptive log-optimal portfolio selection, and adaptive filter design.
机译：经典的交替最小化（或投影）算法已成功解决了两个变量的优化问题。该算法的迭代性质和简单性使其在许多领域得到了应用，例如信号处理，信息论，控制和金融。当基本问题参数固定时，就可以找到算法收敛和正确性的一组一般条件。但是，在许多实际情况下，潜在的问题参数会随着时间而变化，因此自适应算法的使用更为合适。在本文中，我们研究了交替最小化算法的自适应版本。更确切地说，我们考虑在最小化范围内具有缓慢时变域的影响。作为本文的主要结果，我们为自适应算法的收敛性和正确性提供了一组充分的条件。也许有些令人惊讶，这些条件似乎是人们在这种适应性环境中所期望的最小条件。我们介绍了将结果应用于混合物的自适应分解，自适应对数最优投资组合选择和自适应滤波器设计的应用。

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