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Many machine learning problems can be characterized by mutual contamination models. In these problems, one observes several random samples from different convex combinations of a set of unknown base distributions and the goal is to infer these base distributions. This paper considers the general setting where the base distributions are defined on arbitrary probability spaces. We examine three popular machine learning problems that arise in this general setting: multiclass classification with label noise, demixing of mixed membership models, and classification with partial labels. In each case, we give sufficient conditions for identifiability and present algorithms for the infinite and finite sample settings, with associated performance guarantees.
Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound.
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.
The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which takes into account both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration this modification is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.
Gilles Blanchards Vortrag gewährt Einblicke in seine Arbeiten zur Entwicklung und Analyse statistischer Eigenschaften von Lernalgorithmen. In vielen modernen Anwendungen, beispielsweise bei der Schrifterkennung oder dem Spam- Filtering, kann ein Computerprogramm auf der Basis vorgegebener Beispiele automatisch lernen, relevante Vorhersagen für weitere Fälle zu treffen. Mit der mathematischen Analyse der Eigenschaften solcher Methoden beschäftigt sich die Lerntheorie, die mit der Statistik eng zusammenhängt. Dabei spielt der Begriff der Komplexität der erlernten Vorhersageregel eine wichtige Rolle. Ist die Regel zu einfach, wird sie wichtige Einzelheiten ignorieren. Ist sie zu komplex, wird sie die vorgegebenen Beispiele "auswendig" lernen und keine Verallgemeinerungskraft haben. Blanchard wird erläutern, wie Mathematische Werkzeuge dabei helfen, den richtigen Kompromiss zwischen diesen beiden Extremen zu finden.
The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which corrects both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration it is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.
We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding p-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the p-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables.
We derive an upper bound on the local Rademacher complexity of l(p)-norm multiple kernel learning, which yields a tighter excess risk bound than global approaches. Previous local approaches analyzed the case p - 1 only while our analysis covers all cases 1 <= p <= infinity, assuming the different feature mappings corresponding to the different kernels to be uncorrelated. We also show a lower bound that shows that the bound is tight, and derive consequences regarding excess loss, namely fast convergence rates of the order O( n(-)1+alpha/alpha where alpha is the minimum eigenvalue decay rate of the individual kernels.
We introduce extensions of stability selection, a method to stabilise variable selection methods introduced by Meinshausen and Buhlmann (J R Stat Soc 72:417-473, 2010). We propose to apply a base selection method repeatedly to random subsamples of observations and subsets of covariates under scrutiny, and to select covariates based on their selection frequency. We analyse the effects and benefits of these extensions. Our analysis generalizes the theoretical results of Meinshausen and Buhlmann (J R Stat Soc 72:417-473, 2010) from the case of half-samples to subsamples of arbitrary size. We study, in a theoretical manner, the effect of taking random covariate subsets using a simplified score model. Finally we validate these extensions on numerical experiments on both synthetic and real datasets, and compare the obtained results in detail to the original stability selection method.