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

The Average Rate of Convergence of the Exact Line Search Gradient Descent Method