通过分析不定矩阵上的CG实现非孤立极小值信赖域的快速收敛
Fast convergence of trust-regions for non-isolated minima via analysis of CG on indefinite matrices
AI总结：
本文针对非孤立极小值下信赖域方法收敛慢的问题，从理论上证明了TR-tCG在PŁ条件下的超线性收敛性，并开发了分析不定矩阵上CG动力学的新工具。
摘要：
信赖域方法（TR）可以在Hessian矩阵正定的极小值处实现二次收敛。然而，如果极小值不是孤立的，那么该处的Hessian矩阵就不可能正定。较弱的Polyak–Łojasiewicz（PŁ）条件与非孤立极小值兼容，且足以使许多算法保持良好的局部行为。但是，使用精确子问题求解器的TR在PŁ条件下甚至缺乏诸如捕获定理等基本特性。在实践中，一种流行的不精确子问题求解器是截断共轭梯度法（tCG）。经验表明，TR-tCG在PŁ条件下表现出超线性收敛。我们在理论上证实了这一点。主要的数学障碍在于，在PŁ条件下，在任意接近极小值的点处，Hessian矩阵具有趋于零的、可能为负的特征值。因此，tCG被应用于病态不定系统。然而，tCG背后的核心理论是共轭梯度法（CG）的理论，它假设算子是正定的。因此，我们开发了新工具来分析在任意符号小特征值存在时CG的动力学，适用于TR-tCG感兴趣的区间。
英文摘要：
Trust-region methods (TR) can converge quadratically to minima where the Hessian is positive definite. However, if the minima are not isolated, then the Hessian there cannot be positive definite. The weaker Polyak$\unicode{x2013}$Łojasiewicz (PŁ) condition is compatible with non-isolated minima, and it is enough for many algorithms to preserve good local behavior. Yet, TR with an $\textit{exact}$ subproblem solver lacks even basic features such as a capture theorem under PŁ. In practice, a popular $\textit{inexact}$ subproblem solver is the truncated conjugate gradient method (tCG). Empirically, TR-tCG exhibits super-linear convergence under PŁ. We confirm this theoretically. The main mathematical obstacle is that, under PŁ, at points arbitrarily close to minima, the Hessian has vanishingly small, possibly negative eigenvalues. Thus, tCG is applied to ill-conditioned, indefinite systems. Yet, the core theory underlying tCG is that of CG, which assumes a positive definite operator. Accordingly, we develop new tools to analyze the dynamics of CG in the presence of small eigenvalues of any sign, for the regime of interest to TR-tCG.
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