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In this work, we propose an adaptive variation on the classical Heavy-ballmethod for convex quadratic minimization. The adaptivity crucially relies onso-called "Polyak step-sizes", which consists in using the knowledge of theoptimal value of the optimization problem at hand instead of problem parameterssuch as a few eigenvalues of the Hessian of the problem. This method happens toalso be equivalent to a variation of the classical conjugate gradient method,and thereby inherits many of its attractive features, including its finite-timeconvergence, instance optimality, and its worst-case convergence rates.  The classical gradient method with Polyak step-sizes is known to behave verywell in situations in which it can be used, and the question of whetherincorporating momentum in this method is possible and can improve the methoditself appeared to be open. We provide a definitive answer to this question forminimizing convex quadratic functions, a arguably necessary first step fordeveloping such methods in more general setups.

Quadratic minimization: from conjugate gradient to an adaptive Heavy-ball method with Polyak step-sizes