It is very well-known that when the exact line search gradient descent method is applied to a convex quadratic objective, the worst case rate of convergence (among all seed vectors) deteriorates as the condition number of the Hessian of the objective grows. By an elegant analysis by H. Akaike, it is generally believed -- but not proved -- that in the ill-conditioned regime the ROC for almost all initial vectors, and hence also the average ROC, is close to the worst case ROC. We complete Akaike's analysis using the theorem of center and stable manifolds. Our analysis also makes apparent the effect of an intermediate eigenvalue in the Hessian by establishing the following somewhat amusing result: In the absence of an intermediate eigenvalue, the average ROC gets arbitrarily fast -- not slow -- as the Hessian gets increasingly ill-conditioned. We discuss in passing some contemporary applications of exact line search GD to polynomial optimization problems arising from imaging and data sciences.

翻译：众所周知，当精确线搜索梯度下降法应用于凸二次目标函数时，最坏情况下的收敛速度（在所有初始向量中）会随着目标函数Hessian矩阵的条件数增大而恶化。通过H. Akaike的优雅分析，人们普遍认为——但尚未证明——在病态区域中，几乎所有初始向量的收敛速度（因此也包括平均收敛速度）都接近最坏情况收敛速度。我们利用中心流形和稳定流形定理完成了Akaike的分析。我们的分析还通过建立以下略带趣味的结果揭示了Hessian矩阵中间特征值的影响：在不存在中间特征值的情况下，随着Hessian矩阵病态程度加剧，平均收敛速度会任意快——而非慢。我们顺便讨论了精确线搜索GD在成像和数据科学中产生的多项式优化问题中的一些当代应用。