论文标题
具有动态精度的不精确张量方法
Inexact Tensor Methods with Dynamic Accuracies
论文作者
论文摘要
在本文中，我们研究了与复合物镜解决凸优化问题的不可过热的高阶张量方法。在此类方法的每个步骤中，我们都使用辅助问题的近似解决方案，该解决方案由功能值的残差界限定义。我们提出了选择内部精度的两种动态策略：第一个策略降低为$ 1/k^{p + 1} $，其中$ p \ geq 1 $是方法的顺序，$ k $是迭代计数器，第二种方法是使用目标目标的内部精度。我们表明，具有这些策略的不精确张量方法达到了与无错误情况相同的全局收敛率。对于第二种方法，我们还建立了当地的超线性利率（对于$ p \ geq 2 $），并提出了加速计划。最后，我们为多种方法和不同准确性策略的各种机器学习问题提供了计算结果。
In this paper, we study inexact high-order Tensor Methods for solving convex optimization problems with composite objective. At every step of such methods, we use approximate solution of the auxiliary problem, defined by the bound for the residual in function value. We propose two dynamic strategies for choosing the inner accuracy: the first one is decreasing as $1/k^{p + 1}$, where $p \geq 1$ is the order of the method and $k$ is the iteration counter, and the second approach is using for the inner accuracy the last progress in the target objective. We show that inexact Tensor Methods with these strategies achieve the same global convergence rate as in the error-free case. For the second approach we also establish local superlinear rates (for $p \geq 2$), and propose the accelerated scheme. Lastly, we present computational results on a variety of machine learning problems for several methods and different accuracy policies.
