论文标题
通用闭合形式的最佳步进尺寸
A Generic Closed-form Optimal Step-size for ADMM
论文作者
论文摘要
在这项工作中，我们为ADMM类型近端算法提供了一个通用的阶梯尺寸选择。它承认封闭形式的表达，并且相对于最坏的案例收敛速率结合了理论上的最佳状态。仅由双重和原始解决方案的欧几里得规范（即$ ||λ^\ star ||）给出。 / || {x}^\ star || $。数值测试表明，其实际性能一般是最佳的。唯一的挑战是，这种比率尚不清楚，我们提供了两种策略来解决它。我们的步进选择的推导基于使用近端操作员研究ADMM的定点结构。但是，我们证明了经典的近端操作员定义包含输入缩放问题。这导致了缩放的阶梯大小优化问题，该问题将产生一个错误的解决方案。我们提出的对近端运营商的新定义自然避免了这样的问题。建立了一系列属性。
In this work, we present a generic step-size choice for the ADMM type proximal algorithms. It admits a closed-form expression and is theoretically optimal with respect to a worst-case convergence rate bound. It is simply given by the ratio of Euclidean norms of the dual and primal solutions, i.e., $ ||λ^\star|| / ||{x}^\star||$. Numerical tests show that its practical performance is near-optimal in general. The only challenge is that such a ratio is not known a priori and we provide two strategies to address it. The derivation of our step-size choice is based on studying the fixed-point structure of ADMM using the proximal operator. However, we demonstrate that the classical proximal operator definition contains an input scaling issue. This leads to a scaled step-size optimization problem which would yield a false solution. Such an issue is naturally avoided by our proposed new definition of the proximal operator. A series of its properties is established.
