  Optimization algorithms are increasingly being used in applications with
limited time budgets. In many real-time and embedded scenarios, only a few
iterations can be performed and traditional convergence metrics cannot be used
to evaluate performance in these non-asymptotic regimes. In this paper, we
examine the transient behavior of accelerated first-order optimization
algorithms. For convex quadratic problems, we employ tools from linear systems
theory to show that transient growth arises from the presence of non-normal
dynamics. We identify the existence of modes that yield an algebraic growth in
early iterations and quantify the transient excursion from the optimal solution
caused by these modes. For strongly convex smooth optimization problems, we
utilize the theory of integral quadratic constraints (IQCs) to establish an
upper bound on the magnitude of the transient response of Nesterov's
accelerated algorithm. We show that both the Euclidean distance between the
optimization variable and the global minimizer and the rise time to the
transient peak are proportional to the square root of the condition number of
the problem. Finally, for problems with large condition numbers, we demonstrate
tightness of the bounds that we derive up to constant factors.
