In this paper, we study the Kurdyka-{\L}ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly nonconvex and nonsmooth) functions formed from functions with known KL exponents. In addition, we show that the well-studied Luo-Tseng error bound together with a mild assumption on the separation of stationary values implies that the KL exponent is $\frac12$. The Luo-Tseng error bound is known to hold for a large class of concrete structured optimization problems, and thus we deduce the KL exponent of a large class of functions whose exponents were previously unknown. Building upon this and the calculus rules, we are then able to show that for many convex or nonconvex optimization models for applications such as sparse recovery, their objective function's KL exponent is $\frac12$. This includes the least squares problem with smoothly clipped absolute deviation (SCAD) regularization or minimax concave penalty (MCP) regularization and the logistic regression problem with $\ell_1$ regularization. Since many existing local convergence rate analysis for first-order methods in the nonconvex scenario relies on the KL exponent, our results enable us to obtain explicit convergence rate for various first-order methods when they are applied to a large variety of practical optimization models. Finally, we further illustrate how our results can be applied to establishing local linear convergence of the proximal gradient algorithm and the inertial proximal algorithm with constant step-sizes for some specific models that arise in sparse recovery.
翻译:在本文中, 我们研究 Kurdyka- {L} ojasiewicz (KL) exponent, 这是分析一级方法趋同率的重要数量。 具体地说, 我们开发了各种计算规则, 以推断由已知 KL Exponent 函数构成的新功能( 可能非 convex 和非 smooth) 的 KL Exponent 。 此外, 我们显示, 精心研究的 Luo- Tseng 错误, 加上对固定值分离的轻度假设, 意味着 KL exponent 是 $\ frac12$。 Luo- Tseng 错误组合是已知的, 用来控制大量具体结构优化问题的 KLLL Excregional, 因此我们推导出大量功能的 KLLL exponalent expregionalal 。 以此方法可以显示, 对于许多配置或非conforal 优化的应用程序来说, 它们的目标函数是首次应用 $\\ conforxxxxxxxx 。 。 rolation rocal rolation rolation rolation rolation 。 。 最后, rolal rolational rolation rolation rolal rolal rolation rolation roal a fal 和 rolation rogal rolation rolup rolation 。