  We develop a novel framework to study smooth and strongly convex optimization
algorithms, both deterministic and stochastic. Focusing on quadratic functions
we are able to examine optimization algorithms as a recursive application of
linear operators. This, in turn, reveals a powerful connection between a class
of optimization algorithms and the analytic theory of polynomials whereby new
lower and upper bounds are derived. Whereas existing lower bounds for this
setting are only valid when the dimensionality scales with the number of
iterations, our lower bound holds in the natural regime where the
dimensionality is fixed. Lastly, expressing it as an optimal solution for the
corresponding optimization problem over polynomials, as formulated by our
framework, we present a novel systematic derivation of Nesterov's well-known
Accelerated Gradient Descent method. This rather natural interpretation of AGD
contrasts with earlier ones which lacked a simple, yet solid, motivation.
