论文标题
高阶算法的效率，以最大程度地减少复合函数
Efficiency of higher-order algorithms for minimizing composite functions
论文作者
论文摘要
综合最小化涉及以非平滑方式汇总的功能集合。它涵盖了一种特殊的情况，最小值游戏的平滑近似，最小化最小型函数以及简单的复合最小化问题，而目标函数具有非平滑的组件。我们为完全复合问题（可能是非convex）设计了一个高阶主要化算法框架。我们的框架用高阶替代替代了每个组件，以使相应的误差函数具有更高阶的Lipschitz连续导数。我们为我们的（非）凸和（非）平稳目标函数的复合优化问题提供了融合保证。特别是，我们证明了一般非covex（可能是非滑动）问题的固定点收敛保证，并且在目标函数的kurdyka-lojasiewicz（kl）属性下，我们得出根据KL参数提高了速率。对于凸（可能是非平滑）问题，我们还提供了均方根收敛率。
Composite minimization involves a collection of functions which are aggregated in a nonsmooth manner. It covers, as a particular case, smooth approximation of minimax games, minimization of max-type functions, and simple composite minimization problems, where the objective function has a nonsmooth component. We design a higher-order majorization algorithmic framework for fully composite problems (possibly nonconvex). Our framework replaces each component with a higher-order surrogate such that the corresponding error function has a higher-order Lipschitz continuous derivative. We present convergence guarantees for our method for composite optimization problems with (non)convex and (non)smooth objective function. In particular, we prove stationary point convergence guarantees for general nonconvex (possibly nonsmooth) problems and under Kurdyka-Lojasiewicz (KL) property of the objective function we derive improved rates depending on the KL parameter. For convex (possibly nonsmooth) problems we also provide sublinear convergence rates.
