  Gradient descent is arguably one of the most popular online optimization
methods with a wide array of applications. However, the standard implementation
where agents simultaneously update their strategies yields several undesirable
properties; strategies diverge away from equilibrium and regret grows over
time. In this paper, we eliminate these negative properties by introducing a
different implementation to obtain finite regret via arbitrary fixed step-size.
We obtain this surprising property by having agents take turns when updating
their strategies. In this setting, we show that an agent that uses gradient
descent obtains bounded regret -- regardless of how their opponent updates
their strategies. Furthermore, we show that in adversarial settings that
agents' strategies are bounded and cycle when both are using the alternating
gradient descent algorithm.
