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Nesterov's accelerated gradient (AG) method for minimizing a smooth stronglyconvex function $f$ is known to reduce $f({\bf x}_k)-f({\bf x}^*)$ by a factorof $\epsilon\in(0,1)$ after $k=O(\sqrt{L/\ell}\log(1/\epsilon))$ iterations,where $\ell,L$ are the two parameters of smooth strong convexity. Furthermore,it is known that this is the best possible complexity in the function-gradientoracle model of computation. Modulo a line search, the geometric descent (GD)method of Bubeck, Lee and Singh has the same bound for this class of functions.The method of linear conjugate gradients (CG) also satisfies the samecomplexity bound in the special case of strongly convex quadratic functions,but in this special case it can be faster than the AG and GD methods.  Despite similarities in the algorithms and their asymptotic convergencerates, the conventional analysis of the running time of CG is mostly disjointfrom that of AG and GD. The analyses of the AG and GD methods are also ratherdistinct.  Our main result is analyses of the three methods that share several commonthreads: all three analyses show a relationship to a certain "idealizedalgorithm", all three establish the convergence rate through the use of theBubeck-Lee-Singh geometric lemma, and all three have the same potential that iscomputable at run-time and exhibits decrease by a factor of $1-\sqrt{\ell/L}$or better per iteration.  One application of these analyses is that they open the possibility of hybridor intermediate algorithms. One such algorithm is proposed herein and is shownto perform well in computational tests.

A single potential governing convergence of conjugate gradient, accelerated gradient and geometric descent