Regret Analysis of Multi-task Representation Learning for Linear-Quadratic Adaptive Control

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 textitmisspecification 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 as tilde O(√T/H), where H is the number of agents. In settings with "difficult" exploration, the regret scales as tilde O(√du dθ √T + T3/4/H1/5), where dx is the state-space dimension, du is the input dimension, and dθ is the task-specific parameter count. In both cases, by comparing to the minimax single-task regret O(√dx du2√T), 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 dθ < dx du. Lastly, we provide numerical validation of the trends we predict.