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Register a new userConvergence of Gaussian process regression: Optimality, robustness, and relationship with kernel ridge regression
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Added by
Wenjia Wang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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In this work, we investigate Gaussian process regression used to recover a function based on noisy observations. We derive upper and lower error bounds for Gaussian process regression with possibly misspecified correlation functions. The optimal convergence rate can be attained even if the smoothness of the imposed correlation function exceeds that of the true correlation function and the sampling scheme is quasi-uniform. As byproducts, we also obtain convergence rates of kernel ridge regression with misspecified kernel function, where the underlying truth is a deterministic function. The convergence rates of Gaussian process regression and kernel ridge regression are closely connected, which is aligned with the relationship between sample paths of Gaussian process and the corresponding reproducing kernel Hilbert space.
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In this paper we introduce a novel model for Gaussian process (GP) regression in the fully Bayesian setting. Motivated by the ideas of sparsification, localization and Bayesian additive modeling, our model is built around a recursive partitioning (RP) scheme. Within each RP partition, a sparse GP (SGP) regression model is fitted. A Bayesian additive framework then combines multiple layers of partitioned SGPs, capturing both global trends and local refinements with efficient computations. The model addresses both the problem of efficiency in fitting a full Gaussian process regression model and the problem of prediction performance associated with a single SGP. Our approach mitigates the issue of pseudo-input selection and avoids the need for complex inter-block correlations in existing methods. The crucial trade-off becomes choosing between many simpler local model components or fewer complex global model components, which the practitioner can sensibly tune. Implementation is via a Metropolis-Hasting Markov chain Monte-Carlo algorithm with Bayesian back-fitting. We compare our model against popular alternatives on simulated and real datasets, and find the performance is competitive, while the fully Bayesian procedure enables the quantification of model uncertainties.
Kernel ridge regression is an important nonparametric method for estimating smooth functions. We introduce a new set of conditions, under which the actual rates of convergence of the kernel ridge regression estimator under both the L_2 norm and the norm of the reproducing kernel Hilbert space exceed the standard minimax rates. An application of this theory leads to a new understanding of the Kennedy-OHagan approach for calibrating model parameters of computer simulation. We prove that, under certain conditions, the Kennedy-OHagan calibration estimator with a known covariance function converges to the minimizer of the norm of the residual function in the reproducing kernel Hilbert space.
Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression function, and (iii) the adaptability (or qualification) of the regularization. A simple consequence of this upper bound is that the risk of the regularization-based estimators matches the minimax rate in a variety of settings. The general bound also illustrates how some regularization techniques are more adaptable than others to favorable regularity properties that the true regression function may possess. This, in particular, demonstrates a striking difference between kernel ridge regression and kernel principal component regression. Our theoretical results are supported by numerical experiments.
We propose statistical inferential procedures for panel data models with interactive fixed effects in a kernel ridge regression framework.Compared with traditional sieve methods, our method is automatic in the sense that it does not require the choice of basis functions and truncation parameters.Model complexity is controlled by a continuous regularization parameter which can be automatically selected by generalized cross validation. Based on empirical processes theory and functional analysis tools, we derive joint asymptotic distributions for the estimators in the heterogeneous setting. These joint asymptotic results are then used to construct confidence intervals for the regression means and prediction intervals for the future observations, both being the first provably valid intervals in literature. Marginal asymptotic normality of the functional estimators in homogeneous setting is also obtained. Simulation and real data analysis demonstrate the advantages of our method.
In this paper we study multi-task kernel ridge regression and try to understand when the multi-task procedure performs better than the single-task one, in terms of averaged quadratic risk. In order to do so, we compare the risks of the estimators with perfect calibration, the emph{oracle risk}. We are able to give explicit settings, favorable to the multi-task procedure, where the multi-task oracle performs better than the single-task one. In situations where the multi-task procedure is conjectured to perform badly, we also show the oracle does so. We then complete our study with simulated examples, where we can compare both oracle risks in more natural situations. A consequence of our result is that the multi-task ridge estimator has a lower risk than any single-task estimator, in favorable situations.
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