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We propose a data-driven approach to quantify the uncertainty of modelsconstructed by kernel methods. Our approach minimizes the needed distributionalassumptions, hence, instead of working with, for example, Gaussian processes orexponential families, it only requires knowledge about some mild regularity ofthe measurement noise, such as it is being symmetric or exchangeable. We show,by building on recent results from finite-sample system identification, that byperturbing the residuals in the gradient of the objective function, informationcan be extracted about the amount of uncertainty our model has. Particularly,we provide an algorithm to build exact, non-asymptotically guaranteed,distribution-free confidence regions for ideal, noise-free representations ofthe function we try to estimate. For the typical convex quadratic problems andsymmetric noises, the regions are star convex centered around a given nominalestimate, and have efficient ellipsoidal outer approximations. Finally, weillustrate the ideas on typical kernel methods, such as LS-SVC, KRR,$\varepsilon$-SVR and kernelized LASSO.

Distribution-Free Uncertainty Quantification for Kernel Methods by Gradient Perturbations