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We study the problem of estimating the derivatives of the regression function, which has a wide range of applications as a key nonparametric functional of unknown functions. Standard analysis may be tailored to specific derivative orders, and parameter tuning remains a daunting challenge particularly for high-order derivatives. In this article, we propose a simple plug-in kernel ridge regression (KRR) estimator in nonparametric regression with random design that is broadly applicable for multi-dimensional support and arbitrary mixed-partial derivatives. We provide a non-asymptotic analysis to study the behavior of the proposed estimator, leading to two error bounds for a general class of kernels under the strong $L_infty$ norm. In a concrete example specialized to kernels with polynomially decaying eigenvalues, the proposed estimator recovers the minimax optimal rate up to a logarithmic factor for estimating derivatives of functions in Holder class. Interestingly, the proposed estimator achieves the optimal rate of convergence with the same choice of tuning parameter for any order of derivatives. Hence, the proposed estimator enjoys a remarkable textit{plug-in property} for derivatives in that it automatically adapts to the order of derivatives to be estimated, enabling easy tuning in practice. Our simulation studies show favorable finite sample performance of the proposed method relative to several existing methods.
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Machine learning models have traditionally been developed under the assumption that the training and test distributions match exactly. However, recent success in few-shot learning and related problems are encouraging signs that these models can be adapted to more realistic settings where train and test distributions differ. Unfortunately, there is severely limited theoretical support for these algorithms and little is known about the difficulty of these problems. In this work, we provide novel information-theoretic lower-bounds on minimax rates of convergence for algorithms that are trained on data from multiple sources and tested on novel data. Our bounds depend intuitively on the information shared between sources of data, and characterize the difficulty of learning in this setting for arbitrary algorithms. We demonstrate these bounds on a hierarchical Bayesian model of meta-learning, computing both upper and lower bounds on parameter estimation via maximum-a-posteriori inference.
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