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Representation learning is a powerful tool that enables learning over large multitudes of agents or domains by enforcing that all agents operate on a shared set of learned features. However, many robotics or controls applications that would benefit from collaboration operate in settings with changing environments and goals, whereas most guarantees for representation learning are stated for static settings. Toward rigorously establishing the benefit of representation learning in dynamic settings, we analyze the regret of multi-task representation learning for linear-quadratic control. This setting introduces unique challenges. Firstly, we must account for and balance the misspecification introduced by an approximate representation. Secondly, we cannot rely on the parameter update schemes of single-task online LQR, for which least-squares often suffices, and must devise a novel scheme to ensure sufficient improvement. We demonstrate that for settings where exploration is benign, the regret of any agent after T timesteps scales with the square root of T/H, where H is the number of agents. In settings with difficult exploration, the regret scales as the square root of the input dimension times the parameter dimension multiplied by T, plus a term which scales with T to the three quarters divided by H to the one fifth. In both cases, by comparing to the minimax single-task regret, we see a benefit of a large number of agents. Notably, in the difficult exploration case, by sharing a representation across tasks, the effective task-specific parameter count can often be small. Lastly, we validate the trends we predict. 

Abstract We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the nine finite graphs from the classical result of Beineke together with a 10th infinite graph associated with the equivalence relation$$\mathbb {E}_0$$on the Cantor space. As a corollary, we prove a partial converse to the Feldman–Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers. 

Graph signal processing (GSP) has emerged as a powerful tool for practical network applications, including power system monitoring. Recent research has focused on developing GSP-based methods for state estimation, attack detection, and topology identification using the representation of the power system voltages as smooth graph signals. Within this framework, efficient methods have been developed for detecting false data injection (FDI) attacks, which until now were perceived as nonsmooth with respect to the graph Laplacian matrix. Consequently, these methods may not be effective against smooth FDI attacks. In this paper, we propose a graph FDI (GFDI) attack that minimizes the Laplacian-based graph total variation (TV) under practical constraints. We present the GFDI attack as the solution for a non-convex constrained optimization problem. The solution to the GFDI attack problem is obtained through approximating it using ℓ1 relaxation. A series of quadratic programming problems that are classified as convex optimization problems are solved to obtain the final solution. We then propose a protection scheme that identifies the minimal set of measurements necessary to constrain the GFDI output to a high graph TV, thereby enabling its detection by existing GSP-based detectors. Our numerical simulations on the IEEE-57 and IEEE-118 bus test cases reveal the potential threat posed by well-designed GSP-based FDI attacks. Moreover, we demonstrate that integrating the proposed protection design with GSP-based detection can lead to significant hardware cost savings compared to previous designs of protection methods against FDI attacks.