非参数适应控制与预测：理论与随机算法
Nonparametric adaptive control and prediction: theory and randomized algorithms
摘要：
非线性适应控制理论中的一个关键假设是系统的不确定性可以用一组已知基函数的线性跨度来表示。虽然这个假设导致了有效的算法，但它将应用限制在非常特定的系统类别。我们介绍了一种新的非参数自适应算法，估计一个无限维密度的参数在线学习一个未知的动态再生核希尔伯特空间。令人惊讶的是，所得到的控制输入承认的分析表达式，使其实施，尽管其底层的无限维结构。虽然这种自适应输入是丰富和富有表现力的-例如，包含传统的线性参数化-但其计算复杂性随时间线性增长，使得其相对于其参数化对应物更昂贵。利用随机傅立叶特征的理论，我们提供了一个有效的随机实现，恢复经典参数方法的复杂性，同时可证明地保留了非参数输入的表达能力。特别是，我们的显式边界只依赖于多项式的系统的底层参数，使我们提出的算法有效地扩展到高维系统。作为该方法的一个例子，我们展示了随机近似算法学习一个60维系统的预测模型的能力，该系统由通过牛顿引力相互作用的10个点质量组成。通过将其重新解释为特定损失的梯度流，我们得出结论，将基于内核的自适应算法自然扩张到深度神经网络。我们的经验表明，深度表示提供的额外表现力可以提高性能，但代价是闭环稳定性，这是严格保证和一致观察内核机器。
英文摘要：
A key assumption in the theory of nonlinear adaptive control is that the uncertainty of the system can be expressed in the linear span of a set of known basis functions. While this assumption leads to efficient algorithms, it limits applications to very specific classes of systems. We introduce a novel nonparametric adaptive algorithm that estimates an infinite-dimensional density over parameters online to learn an unknown dynamics in a reproducing kernel Hilbert space. Surprisingly, the resulting control input admits an analytical expression that enables its implementation despite its underlying infinite-dimensional structure. While this adaptive input is rich and expressive - subsuming, for example, traditional linear parameterizations - its computational complexity grows linearly with time, making it comparatively more expensive than its parametric counterparts. Leveraging the theory of random Fourier features, we provide an efficient randomized implementation that recovers the complexity of classical parametric methods while provably retaining the expressivity of the nonparametric input. In particular, our explicit bounds only depend polynomially on the underlying parameters of the system, allowing our proposed algorithms to efficiently scale to high-dimensional systems. As an illustration of the method, we demonstrate the ability of the randomized approximation algorithm to learn a predictive model of a 60-dimensional system consisting of ten point masses interacting through Newtonian gravitation. By reinterpretation as a gradient flow on a specific loss, we conclude with a natural extension of our kernel-based adaptive algorithms to deep neural networks. We show empirically that the extra expressivity afforded by deep representations can lead to improved performance at the expense of closed-loop stability that is rigorously guaranteed and consistently observed for kernel machines.
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