  In this work we consider the online control of a known linear dynamic system
with adversarial disturbance and adversarial controller cost. The goal in
online control is to minimize the regret, defined as the difference between
cumulative cost over a period $T$ and the cumulative cost for the best policy
from a comparator class. For the setting we consider, we generalize the
previously proposed online Disturbance Response Controller (DRC) to the
adaptive gradient online Disturbance Response Controller. Using the modified
controller, we present novel regret guarantees that improves the established
regret guarantees for the same setting. We show that the proposed online
learning controller is able to achieve intermediate intermediate regret rates
between $\sqrt{T}$ and $\log{T}$ for intermediate convex conditions, while it
recovers the previously established regret results for general convex
controller cost and strongly convex controller cost.
