Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on these models. However, existing error bounds require restrictive assumptions. In this paper, we employ the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions. Furthermore, we demonstrate how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of our bound. Finally, we derive safety conditions for the control of unknown dynamical systems based on Gaussian process models and evaluate them in simulations of a robotic manipulator.
Regularized kernel regression is a method which extends many ideas from scattered data interpolation to noisy observations and it is highly related to Gaussian process regression as pointed out in @cite_25 . In fact, the GP posterior mean function is identical to kernel ridge regression with squared cost function @cite_30 . Many error bounds such as @cite_12 depend on the empirical @math covering number and the norm of the unknown function in the RKHS attached to the regression kernel. In @cite_34 the effective dimension of the feature space, in which regression is performed, is employed to derive a uniform error bound. The effect of approximations of the kernel, e.g. with the Nystr "om method, on the regression error is analyzed in @cite_10 . Tight error bounds using empirical @math covering numbers are derived under mild assumptions in @cite_1 . Finally, error bounds for general regularization are developed in @cite_4 , which depend on regularization and the RKHS norm of the function.
@cite_30 @cite_4 @cite_1 @cite_34 @cite_10 @cite_25 @cite_12
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In this paper, a novel approximate solution of the criterion used in non-linear class-specific discriminant subspace learning is proposed. We build on the class-specific kernel spectral regression method, which is a two-step process formed by an eigenanalysis step and a kernel regression step. Based on the structure of the intra-class and out-of-class scatter matrices, we provide a fast solution for the first step. For the second step, we propose the use of approximate kernel space definitions. We analytically show that the adoption of randomized and class-specific kernels has the effect of regularization and Nystrom-based approximation, respectively. We evaluate the proposed approach in face verification problems and compare it with the existing approaches. Experimental results show the effectiveness and efficiency of the proposed approximate class-specific kernel spectral regression method, since it can provide satisfactory performance and scale well with the size of the data. One approach to improving the running time of kernel-based methods is to build a small sketch of the kernel matrix and use it in lieu of the full matrix in the machine learning task of interest. Here, we describe a version of this approach that comes with running time guarantees as well as improved guarantees on its statistical performance. By extending the notion of statistical leverage scores to the setting of kernel ridge regression, we are able to identify a sampling distribution that reduces the size of the sketch (i.e., the required number of columns to be sampled) to the effective dimensionality of the problem. This latter quantity is often much smaller than previous bounds that depend on the maximal degrees of freedom. We give an empirical evidence supporting this fact. Our second contribution is to present a fast algorithm to quickly compute coarse approximations to these scores in time linear in the number of samples. More precisely, the running time of the algorithm is O(np2) with p only depending on the trace of the kernel matrix and the regularization parameter. This is obtained via a variant of squared length sampling that we adapt to the kernel setting. Lastly, we discuss how this new notion of the leverage of a data point captures a fine notion of the difficulty of the learning problem. Given a collection of r ≥ 2 linear regression problems in p dimensions, suppose that the regression coefficients share partially common supports of size at most s. This set-up suggests the use of l1 l∞-regularized regression for joint estimation of the p×r matrix of regression coefficients. We analyze the high-dimensional scaling of l1 l∞-regularized quadratic programming, considering both consistency rates in l∞-norm, and how the minimal sample size n required for consistent variable selection scales with model dimension, sparsity, and overlap between the supports. We first establish bounds on the l∞-error as well sufficient conditions for exact variable selection for fixed design matrices, as well as for designs drawn randomly from general Gaussian distributions. Specializing to the case r = 2 linear regression problems with standard Gaussian designs whose supports overlap in a fraction α ∈ [0,1] of their entries, we prove that l1 l∞-regularized method undergoes a phase transition characterized by the rescaled sample size θ1,∞(n, p, s, α) = n (4 - 3 α) s log(p-(2- α) s) . An implication is that the use of l1 l∞-regularization yields improved statistical efficiency if the overlap parameter is large enough ( α >; 2 3), but has worse statistical efficiency than a naive Lasso-based approach for moderate to small overlap (α <; 2 3 ). Empirical simulations illustrate the close agreement between theory and actual behavior in practice. These results show that caution must be exercised in applying l1 l∞ block regularization: if the data does not match its structure very closely, it can impair statistical performance relative to computationally less expensive schemes. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations.