arXiv subjects
Publications and source records attributed to Nicole Mücke.
We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples.
While stochastic gradient descent (SGD) is one of the major workhorses in machine learning, the learning properties of many practically used variants are poorly understood. In this paper, we consider least squares learning in a nonparametric setting and contribute to filling this gap by focusing on the effect and interplay of multiple passes, mini-batching and averaging, and in particular tail averaging. Our results show how these different variants of SGD can be combined to achieve optimal learning errors, hence providing practical insights. In particular, we show for the first time in the literature that tail averaging allows faster convergence rates than uniform averaging in the nonparametric setting. Finally, we show that a combination of tail-averaging and minibatching allows more aggressive step-size choices than using any one of said components.
In the setting of supervised learning using reproducing kernel methods, we propose a data-dependent regularization parameter selection rule that is adaptive to the unknown regularity of the target function and is optimal both for the least-square (prediction) error and for the reproducing kernel Hilbert space (reconstruction) norm error. It is based on a modified Lepskii balancing principle using a varying family of norms.
We study generalization properties of kernel regularized least squares regression based on a partitioning approach. We show that optimal rates of convergence are preserved if the number of local sets grows sufficiently slowly with the sample size. Moreover, the partitioning approach can be efficiently combined with local Nyström subsampling, improving computational cost twofold.
We address the problem of {\it adaptivity} in the framework of reproducing kernel Hilbert space (RKHS) regression. More precisely, we analyze estimators arising from a linear regularization scheme $g_\lam$. In practical applications, an important task is to choose the regularization parameter $\lam$ appropriately, i.e. based only on the given data and independently on unknown structural assumptions on the regression function. An attractive approach avoiding data-splitting is the {\it Lepskii Principle} (LP), also known as the {\it Balancing Principle} is this setting. We show that a modified parameter choice based on (LP) is minimax optimal adaptive, up to $\log\log(n)$. A convenient result is the fact that balancing in $L^2(ν)-$ norm, which is easiest, automatically gives optimal balancing in all stronger norms, interpolating between $L^2(ν)$ and the RKHS. An analogous result is open for other classical approaches to data dependent choices of the regularization parameter, e.g. for Hold-Out.
We consider a distributed learning approach in supervised learning for a large class of spectral regularization methods in an RKHS framework. The data set of size n is partitioned into $m=O(n^α)$ disjoint subsets. On each subset, some spectral regularization method (belonging to a large class, including in particular Kernel Ridge Regression, $L^2$-boosting and spectral cut-off) is applied. The regression function $f$ is then estimated via simple averaging, leading to a substantial reduction in computation time. We show that minimax optimal rates of convergence are preserved if m grows sufficiently slowly (corresponding to an upper bound for $α$) as $n \to \infty$, depending on the smoothness assumptions on $f$ and the intrinsic dimensionality. In spirit, our approach is classical.
We investigate if kernel regularization methods can achieve minimax convergence rates over a source condition regularity assumption for the target function. These questions have been considered in past literature, but only under specific assumptions about the decay, typically polynomial, of the spectrum of the the kernel mapping covariance operator. In the perspective of distribution-free results, we investigate this issue under much weaker assumption on the eigenvalue decay, allowing for more complex behavior that can reflect different structure of the data at different scales.
We consider a statistical inverse learning problem, where we observe the image of a function $f$ through a linear operator $A$ at i.i.d. random design points $X_i$, superposed with an additive noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of $Af$) and the inverse (estimation of $f$) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations $n$ grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in $n$ but also in the explicit dependency of the constant factor in the variance of the noise and the radius of the source condition set.