关于简单信赖域算法的通用性
On the Universality of Simple Trust-Region Algorithms
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中文总结 AI 辅助
研究为二次信赖域方法建立通用复杂度保证，基于新模型减少估计确定通用行为机制。证明基本方法在凸性下通用，给出复杂度界，还表明相关变体在多情况通用，揭示信赖域机制适应性，为其实际成功提供理论支持。
中文摘要 AI 辅助
我们基于信赖域文献中一种似乎全新的函数间隙模型减少估计，为二次信赖域方法建立了通用复杂度保证，并确定了其在凸性下通用行为的共同机制。首先，证明了具有不精确子问题求解的基本信赖域方法在凸性下是通用的。在\(\nu\)-Hölder连续黑塞矩阵下，计算\(\varepsilon\)-近似极小值的全局复杂度界为\(\mathcal{O}(\varepsilon^{-1/(1+\nu)})\)。在非凸情况下，该方法在通常的附加有界黑塞矩阵假设下保持经典的\(\mathcal{O}(\varepsilon^{-2})\)一阶复杂度界。其次，表明相同的凸通用机制适用于具有精确子问题求解和接受率简单修改的信赖域变体。该变体在非凸、凸和局部情况下同时通用，获得最优非凸一阶复杂度\(\mathcal{O}(\varepsilon^{-(2+\nu)/(1+\nu)})\)，同时保留通用凸复杂度和局部牛顿率。结果表明信赖域机制在非凸、凸和局部强凸情况下具有内在适应性，为信赖域方法的实际成功提供了进一步的理论支持。
英文摘要
We establish universal complexity guarantees for quadratic trust-region methods and identify a common mechanism underlying their universal behavior under convexity, based on a function-gap model-decrease estimate that appears to be new in the trust-region literature. First, we prove that the basic trust-region method with inexact subproblem solves is universal under convexity. Under a $ν$-Hölder-continuous Hessian, it attains the global complexity bound $\mathcal{O}(\varepsilon^{-1/(1+ν)})$ for computing an $\varepsilon$-approximate minimizer, without knowledge of $ν\in[0,1]$ or the corresponding Hölder constant. In the nonconvex regime, the method retains the classical $\mathcal{O}(\varepsilon^{-2})$ first-order complexity bound under the usual additional bounded-Hessian assumption. With suitably vanishing inexactness, it also recovers Q-superlinear local convergence for $ν=0$ and convergence of order $1+ν$ for $ν\in(0,1]$. Second, we show that the same convex-universal mechanism applies to a trust-region variant with exact subproblem solves and a simple modification of the acceptance ratio. This variant is universal simultaneously in the nonconvex, convex, and local regimes: it attains the optimal nonconvex first-order complexity $\mathcal{O}(\varepsilon^{-(2+ν)/(1+ν)})$, while preserving the universal convex complexity and the local Newton rates. These guarantees require no knowledge of $ν$ or its Hölder constant. Both methods use the usual quadratic trust-region model and the classical radius-update mechanism, without gradient-dependent radii or model modifications such as cubic, gradient, or tensor regularization. The results show that the trust-region mechanism is inherently adaptive across nonconvex, convex, and locally strongly convex regimes, providing further theoretical support for the practical success of trust-region methods.
