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.

在线凸优化问题中，我们考虑带有二次和线性切换成本的有限信息环境下的问题，通过使用关于先前目标函数的梯度信息，我们提出了在线多梯度下降算法 (Online Multiple Gradient Descent, OMGD)，并证明了其在二次切换成本的 OCO 问题的竞争比为至多 4 (L + 5) + (16 (L + 5))/μ。对于有界信息环境中的在线算法，其竞争比的上界和下界分别为 max {Ω(L), Ω(L/√μ)}。此外，还证明了 OMGD 算法实现了有限信息环境下的动态最优（按顺序）遗憾，并且对于线性切换成本，OMGD 算法的竞争比的上界取决于问题实例的路径长度、平方路径长度以及 L、μ，并且被证明是任何在线算法能够达到的最佳竞争比。因此，我们得出结论，在有限信息环境中，二次和线性切换成本的最优竞争比基本上是不同的。