  We consider the online convex optimization (OCO) problem with quadratic and
linear switching cost in the limited information setting, where an online
algorithm can choose its action using only gradient information about the
previous objective function. For $L$-smooth and $\mu$-strongly convex objective
functions, we propose an online multiple gradient descent (OMGD) algorithm and
show that its competitive ratio for the OCO problem with quadratic switching
cost is at most $4(L + 5) + \frac{16(L + 5)}{\mu}$. The competitive ratio upper
bound for OMGD is also shown to be order-wise tight in terms of $L,\mu$. In
addition, we show that the competitive ratio of any online algorithm is
$\max\{\Omega(L), \Omega(\frac{L}{\sqrt{\mu}})\}$ in the limited information
setting when the switching cost is quadratic. We also show that the OMGD
algorithm achieves the optimal (order-wise) dynamic regret in the limited
information setting. For the linear switching cost, the competitive ratio upper
bound of the OMGD algorithm is shown to depend on both the path length and the
squared path length of the problem instance, in addition to $L, \mu$, and is
shown to be order-wise, the best competitive ratio any online algorithm can
achieve. Consequently, we conclude that the optimal competitive ratio for the
quadratic and linear switching costs are fundamentally different in the limited
information setting.
