论文标题
改进的双重优化方法
An Improved Unconstrained Approach for Bilevel Optimization
论文作者
论文摘要
在本文中，我们专注于非convex-rong-convex双重优化问题（BLO）。在此BLO中，高层问题的目标函数是非convex和可能非平滑的，而对于基本变量$ y $，下层问题是平稳且强烈凸的。我们表明，Blo的可行区域是Riemannian歧管。然后，我们将BLO转换为其相应的无约束约束解决问题（CDB），其目标函数是从BLO中的目标函数明确提出的。我们证明BLO等同于无约束的优化问题CDB。因此，可以通过CDB直接应用于BLO的各种有效的无约束方法以及它们的理论结果。我们提出了一个统一的框架，用于开发基于亚基的CDB方法。值得注意的是，我们表明几种现有的有效算法可以符合统一的框架，并将其解释为CDB的下降算法。这些例子进一步证明了我们提出的方法的巨大潜力。
In this paper, we focus on the nonconvex-strongly-convex bilevel optimization problem (BLO). In this BLO, the objective function of the upper-level problem is nonconvex and possibly nonsmooth, and the lower-level problem is smooth and strongly convex with respect to the underlying variable $y$. We show that the feasible region of BLO is a Riemannian manifold. Then we transform BLO to its corresponding unconstrained constraint dissolving problem (CDB), whose objective function is explicitly formulated from the objective functions in BLO. We prove that BLO is equivalent to the unconstrained optimization problem CDB. Therefore, various efficient unconstrained approaches, together with their theoretical results, can be directly applied to BLO through CDB. We propose a unified framework for developing subgradient-based methods for CDB. Remarkably, we show that several existing efficient algorithms can fit the unified framework and be interpreted as descent algorithms for CDB. These examples further demonstrate the great potential of our proposed approach.
