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.

对于具有未知成本函数和可能无界和退化噪声的线性系统控制问题，本文研究了在线控制问题。通过研究发现，对于凸代价函数，即使存在无界噪声，也可以达到约等于O(根号T)的后悔界，其中T是时间跨度。此外，当成本函数是强凸时，在文献中需要的噪声协方差非退化假设下，我们得到了约等于O(多项式(log T))的后悔界。去除对噪声秩的假设的关键是与噪声协方差相关的系统变换，这同时实现了在线控制算法的参数减少。

无界退化噪声下的在线线性系统控制