1,721,202 research outputs found
Stochastic Forward–Backward Splitting for Monotone Inclusions
No full textWe propose and analyze the convergence of a novel stochastic algorithm for monotone inclusions that are sum of a maximal monotone operator and a single-valued cocoercive operator. The algorithm we propose is a natural stochastic extension of the classical forward–backward method. We provide a non-asymptotic error analysis in expectation for the strongly monotone case, as well as almost sure convergence under weaker assumptions. For minimization problems, we recover rates matching those obtained by stochastic extensions of the so-called accelerated methods. Stochastic quasi-Fejér’s sequences are a key technical tool to prove almost sure convergence
Discretization error analysis for Tikhonov regularization
No full textWe study the discretization of inverse problems defined by a Carleman operator. In particular, we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples, as well as the discretization of integral equations, can be analyzed in our settin
Adaptive Kernel Methods via the Balancing Principle
No full textThe regularization parameter choice is a fundamental problem in Learn- ing Theory since the performance of most supervised algorithms crucially depends on the choice of one or more of such parameters. In particular a main theoretical issue regards the amount of prior knowledge needed to choose the regularization pa- rameter in order to obtain good learning rates. In this paper we present a parameter choice strategy, called the balancing principle, to choose the regularization parameter without knowledge of the regularity of the target function. Such a choice adaptively achieves the best error rate. Our main result applies to regularization algorithms in reproducing kernel Hilbert space with the square loss, though we also study how a similar principle can be used in other situations
Spectral Regularization for Support Estimation.
No full textIn this paper we consider the problem of learning from data the support of a probability distribution when the distribution does not have a density (with respect to some reference measure). We propose a new class of regularized spectral estimators based on a new notion of reproducing kernel Hilbert space, which we call “completely regular”. Completely regular kernels allow to capture the relevant geometric and topological properties of an arbitrary probability space. In particular, they are the key ingredient to prove the universal consistency of the spectral estimators and in this respect they are the analogue of universal kernels for supervised problems. Numerical experiments show that spectral estimators compare favorably to state of the art machine learning algorithms for density support estimation
Learning sets with separating kernels
Full text linkWe consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of regularized learning algorithms which are provably universally consistent and stable with respect to random sampling. Numerical experiments show that the proposed approach is competitive, and often better, than other state of the art techniques
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