This paper presents a comprehensive convergence analysis for the mirror descent (MD) method, a widely used algorithm in convex optimization. The key feature of this algorithm is that it provides a generalization of classical gradient-based methods via the use of generalized distance-like functions, which are formulated using the Bregman divergence. Establishing convergence rate bounds for this algorithm is in general a non-trivial problem due to the lack of monotonicity properties in the composite nonlinearities involved. In this paper, we show that the Bregman divergence from the optimal solution, which is commonly used as a Lyapunov function for this algorithm, is a special case of Lyapunov functions that follow when the Popov criterion is applied to an appropriate reformulation of the MD dynamics. This is then used as a basis to construct an integral quadratic constraint (IQC) framework through which convergence rate bounds with reduced conservatism can be deduced. We also illustrate via examples that the convergence rate bounds derived can be tight.

中文翻译：

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镜像下降法的收敛速率边界：IQC、Popov 准则和 Bregman 发散


本文对 Mirror descent （MD） 方法进行了全面的收敛分析，该方法是凸优化中广泛使用的算法。该算法的主要特点是，它通过使用广义距离函数（使用 Bregman 散度表示）提供了经典基于梯度的方法的泛化。为该算法建立收敛速率边界通常是一个重要的问题，因为所涉及的复合非线性中缺乏单调性。在本文中，我们表明，与最优解的 Bregman 背离（通常用作该算法的 Lyapunov 函数）是 Lyapunov 函数的一种特例，当 Popov 准则应用于 MD 动力学的适当重新表述时，遵循该函数。然后将其用作构建积分二次约束 （IQC） 框架的基础，通过该框架可以推断出具有降低保守性的收敛速率边界。我们还通过示例说明，得出的收敛速率边界可能很严格。