We study the smoothed online quadratic optimization (SOQO) problem where, at each round $t$, a player plays an action $x_t$ in response to a quadratic hitting cost and an additional squared $\ell_2$-norm cost for switching actions. This problem class has strong connections to a wide range of application domains including smart grid management, adaptive control, and data center management, where switching-efficient algorithms are highly sought after. We study the SOQO problem in both adversarial and stochastic settings, and in this process, perform the first stochastic analysis of this class of problems. We provide the online optimal algorithm when the minimizers of the hitting cost function evolve as a general stochastic process, which, for the case of martingale process, takes the form of a distribution-agnostic dynamic interpolation algorithm (LAI). Next, we present the stochastic-adversarial trade-off by proving an $\Omega(T)$ expected regret for the adversarial optimal algorithm in the literature (ROBD) with respect to LAI and, a sub-optimal competitive ratio for LAI in the adversarial setting. Finally, we present a best-of-both-worlds algorithm that obtains a robust adversarial performance while simultaneously achieving a near-optimal stochastic performance.

翻译：我们研究平滑在线二次优化（Smoothed Online Quadratic Optimization, SOQO）问题，其中在每个轮次 $t$，玩家根据二次冲击成本和额外的切换动作的 $\ell_2$ 范数平方成本选择动作 $x_t$。该问题类别与智能电网管理、自适应控制和数据中心管理等广泛应用领域紧密相关，其中对切换高效的算法需求旺盛。我们在对抗性和随机两种设置下研究SOQO问题，并在此过程中首次对该问题进行随机分析。当冲击成本函数的最小化器演变为一般随机过程时，我们提供了在线最优算法；对于鞅过程的情况，该算法表现为一种与分布无关的动态插值算法（LAI）。接着，我们通过证明文献中对抗性最优算法（ROBD）相对于LAI的 $\Omega(T)$ 期望遗憾，以及LAI在对抗性设置中次优的竞争比，展示了随机-对抗性权衡。最后，我们提出一种两全其美算法，该算法在获得稳健对抗性性能的同时，实现接近最优的随机性能。