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A key assumption in the theory of nonlinear adaptive control is that theuncertainty of the system can be expressed in the linear span of a set of knownbasis functions. While this assumption leads to efficient algorithms, it limitsapplications to very specific classes of systems. We introduce a novelnonparametric adaptive algorithm that estimates an infinite-dimensional densityover parameters online to learn an unknown dynamics in a reproducing kernelHilbert space. Surprisingly, the resulting control input admits an analyticalexpression that enables its implementation despite its underlyinginfinite-dimensional structure. While this adaptive input is rich andexpressive - subsuming, for example, traditional linear parameterizations - itscomputational complexity grows linearly with time, making it comparatively moreexpensive than its parametric counterparts. Leveraging the theory of randomFourier features, we provide an efficient randomized implementation thatrecovers the complexity of classical parametric methods while provablyretaining the expressivity of the nonparametric input. In particular, ourexplicit bounds only depend polynomially on the underlying parameters of thesystem, allowing our proposed algorithms to efficiently scale tohigh-dimensional systems. As an illustration of the method, we demonstrate theability of the randomized approximation algorithm to learn a predictive modelof a 60-dimensional system consisting of ten point masses interacting throughNewtonian gravitation. By reinterpretation as a gradient flow on a specificloss, we conclude with a natural extension of our kernel-based adaptivealgorithms to deep neural networks. We show empirically that the extraexpressivity afforded by deep representations can lead to improved performanceat the expense of closed-loop stability that is rigorously guaranteed andconsistently observed for kernel machines.

Nonparametric adaptive control and prediction: theory and randomized algorithms