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We show that accelerated gradient descent, averaged gradient descent and theheavy-ball method for non-strongly-convex problems may be reformulated asconstant parameter second-order difference equation algorithms, where stabilityof the system is equivalent to convergence at rate O(1/n 2), where n is thenumber of iterations. We provide a detailed analysis of the eigenvalues of thecorresponding linear dynamical system , showing various oscillatory andnon-oscillatory behaviors, together with a sharp stability result with explicitconstants. We also consider the situation where noisy gradients are available,where we extend our general convergence result, which suggests an alternativealgorithm (i.e., with different step sizes) that exhibits the good aspects ofboth averaging and acceleration.

From Averaging to Acceleration, There is Only a Step-size