We study online convex optimization in a setting where the learner seeks to minimize the sum of a per-round hitting cost and a movement cost which is incurred when changing decisions between rounds. We prove a new lower bound on the competitive ratio of any online algorithm in the setting where the costs are $m$-strongly convex and the movement costs are the squared $\ell_2$ norm. This lower bound shows that no algorithm can achieve a competitive ratio that is $o(m^{-1/2})$ as $m$ tends to zero. No existing algorithms have competitive ratios matching this bound, and we show that the state-of-the-art algorithm, Online Balanced Decent (OBD), has a competitive ratio that is $\Omega(m^{-2/3})$. We additionally propose two new algorithms, Greedy OBD (G-OBD) and Regularized OBD (R-OBD) and prove that both algorithms have an $O(m^{-1/2})$ competitive ratio. The result for G-OBD holds when the hitting costs are quasiconvex and the movement costs are the squared $\ell_2$ norm, while the result for R-OBD holds when the hitting costs are $m$-strongly convex and the movement costs are Bregman Divergences. Further, we show that R-OBD simultaneously achieves constant, dimension-free competitive ratio and sublinear regret when hitting costs are strongly convex.
翻译:在学习者试图将每轮打击成本和流动成本的总和最小化的情况下,我们研究在线 convex优化,学习者在这种环境下力求将每轮打击成本和每轮改变决策时发生的移动成本降低。我们证明,在成本为百万美元的环境下,任何在线算法的竞争性比率都有了新的较低约束,在成本为百万美元的环境下,移动成本为平方美元=2美元的标准。这一较低约束表明,没有任何一种算法能够达到以美元(m ⁇ -1/2)为单位的竞争性比率,而现有的算法没有与这一约束相匹配的竞争性比率,而且我们表明,在成本为百万分数的状态下,最先进的算法成本为百万分之五,而稳定的OB-Obx流动成本则持续不变。