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

Online Control of Linear Systems with Unbounded and Degenerate Noise

This paper considers an online control problem over a linear time-invariant
system with unknown dynamics, bounded disturbance, and adversarial cost. We
propose a data-driven strategy to reduce the regret of the controller. Unlike
model-based methods, our algorithm does not identify the system model, instead,
it leverages a single noise-free trajectory to calculate the accumulation of
disturbance and makes decisions using the accumulated disturbance action
controller we design, whose parameters are updated by online gradient descent.
We prove that the regret of our algorithm is $\mathcal{O}(\sqrt{T})$ under mild
assumptions, suggesting that its performance is on par with model-based
methods.

该论文研究在线控制问题，通过使用单一无噪声轨迹计算干扰累积并通过在线梯度下降更新参数，提出了一种数据驱动的策略来减小控制器的后悔。