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，最坏情况收敛速率（在所有初始向量中）会随着目标函数Hessian矩阵条件数的增大而恶化。根据赤池弘次的一个优雅分析，学界普遍认为——但未得到证明——在病态条件下，几乎所有初始向量的收敛速率，因而平均收敛速率，都接近最坏情况收敛速率。我们利用中心流形与稳定流形定理完善了赤池的分析。我们的分析还通过建立以下略显有趣的结果，清晰地展示了Hessian矩阵中间特征值的影响：当不存在中间特征值时，随着Hessian矩阵病态程度加剧，平均收敛速率会变得任意快——而非任意慢。我们顺便讨论了精确线搜索梯度下降法在成像与数据科学中出现的多项式优化问题的一些当代应用。