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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 Tornado motion changes occurring with major internal rear-flank momentum surges are examined in three significant tornado-producing supercells. The analysis primarily uses fixed-site Doppler radar data, but also utilizes in situ and videographic observations when available. In the cases examined, the peak lowest-level remotely sensed or in situ rear-flank surge wind speeds ranged from 48 to at least 63 m s −1 . Contemporaneous with major surges impacting the tornadoes and their parent low-level mesocyclones, longer-duration tornado heading changes were leftward and ranged from 30° to 55°. In all cases, the tornado speed increased substantially upon surge impact, with tornado speeds approximately doubling in two of the events. A storm-relative change in the hook echo orientation accompanied the major surges and provided a signal that a marked leftward heading change for an ongoing tornado was under way. Concurrent with the surge interaction, the hook echo tip and associated low-level mesocyclone turned leftward while also moving in a storm-relative downshear direction. The major rear-flank internal surges influenced tornado motion such that a generally favorable storm updraft-relative position was maintained. In all cases, the tornado lasted well beyond (≥21 min) the time of the surge-associated left turn with no evident marked loss of intensity until well down-track of the turn. The local momentum balance between outflow and inflow that bounds the tornado or its parent circulation, especially the directionality evolution of the bounding momentum, is the most apparent explanation for tornado down-track or off-track accelerations in the featured events.