π€ AI Summary
This work addresses the problem of distributed online control for networked linear time-invariant systems subject to adversarial disturbances and time-varying convex costs, where each agent has access only to its local cost information. To this end, it introduces spectral control into the distributed online learning framework for the first time: each agent constructs a spectral controller by convolving historical disturbances with the leading eigenvector of a Hankel matrix derived from local observations and neighbor communication, and collaboratively updates its parameters via distributed online gradient descent. The approach establishes a regret analysis framework based on spectral parameterization and, under standard assumptions, proves a sublinear regret bound of $O(\frac{\sqrt{T}\,\text{poly}(\log T)}{\gamma^3})$, explicitly characterizing the dependence on the time horizon $T$, the systemβs stability margin $\gamma$, and the network size and connectivity.
π Abstract
This paper studies the distributed online control problem over a network of linear time-invariant (LTI) systems in the presence of adversarial disturbances and time-varying convex costs. The network cost is characterized by the summation of local cost functions, where each local function is sequentially revealed only to the corresponding agent. The goal of each agent is to generate a control sequence, using only local observations and neighbor communication, that competes with the best {\it centralized} linear policy in hindsight. We extend the recently proposed Online Spectral Control framework from the centralized setting to the distributed setting. In particular, each agent applies a spectral controller obtained by convolving past disturbances with the leading eigenvectors of a Hankel matrix, while the controller parameters are updated through a distributed online gradient descent step over the local surrogate costs. We formulate this problem this problem as a {\it regret} minimization problem based on the spectral parameterization, and under standard assumptions, we establish a sublinear regret bound of $O(\frac{\sqrt{T}\text{poly}(\log T)}{Ξ³^3})$, where $T$ is the time horizon and $Ξ³$ denotes the stability margin. The resulting bound also captures the dependence on the network size and connectivity.