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

Abstract

Nesterov's accelerated gradient (AG) method for minimizing a smooth strongly convex function $f$ is known to reduce $f({\bf x}_k)-f({\bf x}^*)$ by a factor of $\epsilon\in(0,1)$ 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-gradient oracle 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 same complexity 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 convergence rates, the conventional analysis of the running time of CG is mostly disjoint from that of AG and GD. The analyses of the AG and GD methods are also rather distinct. Our main result is analyses of the three methods that share several common threads: all three analyses show a relationship to a certain "idealized algorithm", all three establish the convergence rate through the use of the Bubeck-Lee-Singh geometric lemma, and all three have the same potential that is computable 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 hybrid or intermediate algorithms. One such algorithm is proposed herein and is shown to perform well in computational tests.