论文标题
复合凸优化的自适应三阶方法
Adaptive Third-Order Methods for Composite Convex Optimization
论文作者
论文摘要
在本文中，我们提出了三阶方法，用于复合凸优化问题，其中平滑部分是连续三阶连续的三阶函数连续的三阶导数。这些方法是自适应的，因为它们不需要Lipschitz常数的知识。试验点是由模型的不精确最小化计算得出的，该模型由物镜的非平滑部分以及平滑部分的三阶泰勒多项式的四阶泰勒多项式的四分之一正则化。具体而言，通过使用Bregman梯度方法作为内求解器获得辅助问题的近似解决方案。与现有的高阶方法的现有自适应方法不同，在我们的新方案中，考虑到内求器的进度，对正则化参数进行了调整。通过这种技术，我们表明基本方法找到了目标函数的$ε$ - 最小化，最多最多$ \ Mathcal {o} \ left（| \ log（| \ log（ε）|ε^{ - \ frac {1} {1} {3}}}}}} \ right）$ itserations ninen solver的$ iterations。还出现了加速自适应的三阶方法，总迭代复杂性为$ \ MATHCAL {O} \ left（| \ log（ε）|ε^{ - \ frac {1} {1} {4}}}}}} \ right）$。
In this paper we propose third-order methods for composite convex optimization problems in which the smooth part is a three-times continuously differentiable function with Lipschitz continuous third-order derivatives. The methods are adaptive in the sense that they do not require the knowledge of the Lipschitz constant. Trial points are computed by the inexact minimization of models that consist in the nonsmooth part of the objective plus a quartic regularization of third-order Taylor polynomial of the smooth part. Specifically, approximate solutions of the auxiliary problems are obtained by using a Bregman gradient method as inner solver. Different from existing adaptive approaches for high-order methods, in our new schemes the regularization parameters are adjusted taking into account the progress of the inner solver. With this technique, we show that the basic method finds an $ε$-approximate minimizer of the objective function performing at most $\mathcal{O}\left(|\log(ε)|ε^{-\frac{1}{3}}\right)$ iterations of the inner solver. An accelerated adaptive third-order method is also presented with total inner iteration complexity of $\mathcal{O}\left(|\log(ε)|ε^{-\frac{1}{4}}\right)$.
