This paper investigates the problem of controlling a linear system under possibly unbounded and degenerate noise with unknown cost functions, known as an online control problem. In contrast to the existing work, which assumes the boundedness of noise, we reveal that for convex costs, an $ \widetilde{O}(\sqrt{T}) $ regret bound can be achieved even for unbounded noise, where $ T $ denotes the time horizon. Moreover, when the costs are strongly convex, we establish an $ O({\rm poly} (\log T)) $ regret bound without the assumption that noise covariance is non-degenerate, which has been required in the literature. The key ingredient in removing the rank assumption on noise is a system transformation associated with the noise covariance. This simultaneously enables the parameter reduction of an online control algorithm.

翻译：本文研究了在成本函数未知且噪声可能无界且退化的条件下控制线性系统的问题，即在线控制问题。与现有假设噪声有界的工作不同，我们揭示了对于凸成本函数，即使噪声无界，仍可实现 $\widetilde{O}(\sqrt{T})$ 的遗憾界，其中 $T$ 表示时间范围。此外，当成本函数为强凸时，我们建立了 $O({\rm poly} (\log T))$ 的遗憾界，且无需文献中通常要求的噪声协方差非退化假设。消除噪声秩假设的关键在于一种与噪声协方差相关的系统变换，该变换同时实现了在线控制算法的参数降维。