  This paper investigates online algorithms for smooth time-varying
optimization problems, focusing first on methods with constant step-size,
momentum, and extrapolation-length. Assuming strong convexity, precise results
for the tracking iterate error (the limit supremum of the norm of the
difference between the optimal solution and the iterates) for online gradient
descent are derived. The paper then considers a general first-order framework,
where a universal lower bound on the tracking iterate error is established.
Furthermore, a method using "long-steps" is proposed and shown to achieve the
lower bound up to a fixed constant. This method is then compared with online
gradient descent for specific examples. Finally, the paper analyzes the effect
of regularization when the cost is not strongly convex. With regularization, it
is possible to achieve a non-regret bound. The paper ends by testing the
accelerated and regularized methods on synthetic time-varying least-squares and
logistic regression problems, respectively.
