Most modern reinforcement learning algorithms optimize a cumulative single-step cost along a trajectory. The optimized motions are often unnatural, representing, for example, behaviors with sudden accelerations that waste energy and lack predictability. In this work, we present a novel paradigm of controlling nonlinear systems via the minimization of the Koopman spectrum cost: a cost over the Koopman operator of the controlled dynamics. This induces a broader class of dynamical behaviors that evolve over stable manifolds such as nonlinear oscillators, closed loops, and smooth movements. We demonstrate that some dynamics realizations that are not possible with a cumulative cost are feasible in this paradigm. Moreover, we present a provably efficient online learning algorithm for our problem that enjoys a sub-linear regret bound under some structural assumptions.
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