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.

翻译：我们研究了有限信息设置下具有二次和线性切换成本的在线凸优化问题，其中在线算法仅能通过前一个目标函数的梯度信息选择其动作。针对$L$-光滑且$\mu$-强凸的目标函数，我们提出了一种在线多重梯度下降算法，并证明其在二次切换成本下该算法竞争比至多为$4(L + 5) + \frac{16(L + 5)}{\mu}$。同时表明该上界关于$L,\mu$是阶次紧的。此外，我们证明在二次切换成本下，有限信息设置中任何在线算法的竞争比下界为$\max\{\Omega(L), \Omega(\frac{L}{\sqrt{\mu}})\}$。进一步，我们证明了OMGD算法在有限信息设置中能够实现（阶次）最优的动态遗憾。对于线性切换成本，OMGD算法的竞争比上界不仅依赖于$L,\mu$，还取决于问题实例的路径长度和平方路径长度，且从阶次意义上看，该上界是任何在线算法所能达到的最优竞争比。由此我们得出结论：在有限信息设置下，二次与线性切换成本的最优竞争比存在本质差异。