论文标题
混合甲骨文中的张量方法，以解决最小的问题
Tensor methods inside mixed oracle for min-min problems
论文作者
论文摘要
在本文中，我们考虑了两组变量的最小问题或最小化。如果凸优化中的某些变量具有不同的维度或这些组具有不同的域，则可能会发生最小问题。这种问题结构使我们能够将主要任务分解为子问题，并允许用混合的甲壳解决。但是，有关此主题的现有文章仅涵盖零件和一阶oracles，在我们的工作中，我们考虑了解决内部问题和快速梯度方法来解决外部问题的高阶方法。我们假设外部问题被限制为某些凸紧密的集合，对于内部问题，我们考虑了不受约束的情况，并且被约束至某些凸形紧凑型集合。根据定义，张量方法使用高级导数，因此该方法的每次迭代的时间在很大程度上取决于其解决的问题的维度。因此，我们建议，内部问题变量的尺寸不大于1000。此外，我们需要一些特定的假设才能使用混合的甲壳。首先，我们假设这两个变量及其梯度在两个变量中都是凸的，这是Lipschitz的连续。其次，我们假设内部问题是强烈的凸，其梯度是Lipschitz的连续。另外，由于我们将使用张量方法来解决内部问题，因此我们需要它为$ p $ - 订购Lipschitz的连续（$ P> 1 $）。最后，我们假设外部问题的强凸度能够使用快速梯度方法来强烈凸出函数。我们需要强调，我们使用超快速张量方法在无约束的情况下解决内部子问题。当我们解决紧凑型集合中的内部问题时，我们会使用加速的高阶复合近端方法。
In this article we consider min-min type of problems or minimization by two groups of variables. Min-min problems may occur in case if some groups of variables in convex optimization have different dimensions or if these groups have different domains. Such problem structure gives us an ability to split the main task to subproblems, and allows to tackle it with mixed oracles. However existing articles on this topic cover only zeroth and first order oracles, in our work we consider high-order tensor methods to solve inner problem and fast gradient method to solve outer problem. We assume, that outer problem is constrained to some convex compact set, and for the inner problem we consider both unconstrained case and being constrained to some convex compact set. By definition, tensor methods use high-order derivatives, so the time per single iteration of the method depends a lot on the dimensionality of the problem it solves. Therefore, we suggest, that the dimension of the inner problem variable is not greater than 1000. Additionally, we need some specific assumptions to be able to use mixed oracles. Firstly, we assume, that the objective is convex in both groups of variables and its gradient by both variables is Lipschitz continuous. Secondly, we assume the inner problem is strongly convex and its gradient is Lipschitz continuous. Also, since we are going to use tensor methods for inner problem, we need it to be $p$-th order Lipschitz continuous ($p > 1$). Finally, we assume strong convexity of the outer problem to be able to use fast gradient method for strongly convex functions. We need to emphasize, that we use superfast tensor method to tackle inner subproblem in unconstrained case. And when we solve inner problem on compact set, we use accelerated high-order composite proximal method.
