In this paper, we revisit the problem of smoothed online learning, in which the online learner suffers both a hitting cost and a switching cost, and target two performance metrics: competitive ratio and dynamic regret with switching cost. To bound the competitive ratio, we assume the hitting cost is known to the learner in each round, and investigate the simple idea of balancing the two costs by an optimization problem. Surprisingly, we find that minimizing the hitting cost alone is $\max(1, \frac{2}{\alpha})$-competitive for $\alpha$-polyhedral functions and $1 + \frac{4}{\lambda}$-competitive for $\lambda$-quadratic growth functions, both of which improve state-of-the-art results significantly. Moreover, when the hitting cost is both convex and $\lambda$-quadratic growth, we reduce the competitive ratio to $1 + \frac{2}{\sqrt{\lambda}}$ by minimizing the weighted sum of the hitting cost and the switching cost. To bound the dynamic regret with switching cost, we follow the standard setting of online convex optimization, in which the hitting cost is convex but hidden from the learner before making predictions. We modify Ader, an existing algorithm designed for dynamic regret, slightly to take into account the switching cost when measuring the performance. The proposed algorithm, named as Smoothed Ader, attains an optimal $O(\sqrt{T(1+P_T)})$ bound for dynamic regret with switching cost, where $P_T$ is the path-length of the comparator sequence. Furthermore, if the hitting cost is accessible in the beginning of each round, we obtain a similar guarantee without the bounded gradient condition, and establish an $\Omega(\sqrt{T(1+P_T)})$ lower bound to confirm the optimality.
翻译:在本文中,我们重新审视了平滑在线学习的问题, 在线学习者在其中既要付出打击成本,又要付出转换成本, 并且针对两个性能衡量标准: 竞争性比率和对转换成本的动态遗憾。 为了约束竞争比率, 我们假设每个回合的学习者都知道打击成本, 并且调查以优化问题来平衡两种成本的简单想法。 令人惊讶的是, 我们发现, 仅将打击成本最小化为$( 1,\ frac, 3, 2, 26, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 3, 4, 4, 3, 4, 3, 4, 4, 3, 4, 4, 3, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4,, 4, 4, 4, 4, 4, 4, 4,, 4,,,,,,,,,, 4, 4, 3,,,,,,,,,,,,, 4, 4, 3, 4, 4, 4, 4, 3, 4, 4, 4,,, 4, 4,,, 4, 4,,,,,,,,,,,, 4, 4,,,,,,,,,,,