Tradeoffs between convergence rate and noise amplification for momentum-based accelerated optimization algorithms

We study momentum-based first-order optimization algorithms in which the iterations utilize information from the two previous steps and are subject to an additive white noise. This class of algorithms includes Polyak's heavy-ball and Nesterov's accelerated methods as special cases and noise accounts for uncertainty in either gradient evaluation or iteration updates. For strongly convex quadratic problems, we use the steady-state variance of the error in the optimization variable to quantify noise amplification and identify fundamental stochastic performance tradeoffs. Our approach utilizes the Jury stability criterion to provide a novel geometric characterization of conditions for linear convergence, and it clarifies the relation between the noise amplification and convergence rate as well as their dependence on the condition number and the constant algorithmic parameters. This geometric insight leads to simple alternative proofs of standard convergence results and allows us to establish analytical lower bounds on the product between the settling time and noise amplification that scale quadratically with the condition number. Our analysis also identifies a key difference between the gradient and iterate noise models: while the amplification of gradient noise can be made arbitrarily small by sufficiently decelerating the algorithm, the best achievable variance amplification for the iterate noise model increases linearly with the settling time in decelerating regime. Furthermore, we introduce two parameterized families of algorithms that strike a balance between noise amplification and settling time while preserving order-wise Pareto optimality for both noise models. Finally, by analyzing a class of accelerated gradient flow dynamics, whose suitable discretization yields the two-step momentum algorithm, we establish that stochastic performance tradeoffs also extend to continuous time.