Distribution-Free Uncertainty Quantification for Kernel Methods by Gradient Perturbations
This provides a distribution-free uncertainty quantification method for kernel-based models, addressing a key limitation in reliability for practitioners in machine learning and statistics, though it is incremental in building on existing system identification results.
The paper tackles the problem of quantifying uncertainty for kernel methods with minimal distributional assumptions, proposing a gradient perturbation approach that constructs exact, non-asymptotically guaranteed confidence regions for noise-free function representations, with efficient ellipsoidal approximations for convex quadratic problems under symmetric noise.
We propose a data-driven approach to quantify the uncertainty of models constructed by kernel methods. Our approach minimizes the needed distributional assumptions, hence, instead of working with, for example, Gaussian processes or exponential families, it only requires knowledge about some mild regularity of the measurement noise, such as it is being symmetric or exchangeable. We show, by building on recent results from finite-sample system identification, that by perturbing the residuals in the gradient of the objective function, information can 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 of the function we try to estimate. For the typical convex quadratic problems and symmetric noises, the regions are star convex centered around a given nominal estimate, and have efficient ellipsoidal outer approximations. Finally, we illustrate the ideas on typical kernel methods, such as LS-SVC, KRR, $\varepsilon$-SVR and kernelized LASSO.