Online learning with exponential weights in metric spaces

This paper addresses the problem of online learning in metric spaces using exponential weights. We extend the analysis of the exponentially weighted average forecaster, traditionally studied in a Euclidean settings, to a more abstract framework. Our results rely on the notion of barycenters, a suitable version of Jensen's inequality and a synthetic notion of lower curvature bound in metric spaces known as the measure contraction property. We also adapt the online-to-batch conversion principle to apply our results to a statistical learning framework.

The exponentially weighted average forecaster in geodesic spaces of non-positive curvature

This paper addresses the problem of prediction with expert advice for outcomes in a geodesic space with non-positive curvature in the sense of Alexandrov. Via geometric considerations, and in particular the notion of barycenters, we extend to this setting the definition and analysis of the classical exponentially weighted average forecaster. We also adapt the principle of online to batch conversion to this setting. We shortly discuss the application of these results in the context of aggregation and for the problem of barycenter estimation.

A notion of stability for k-means clustering

In this paper, we define and study a new notion of stability for the $k$-means clustering scheme building upon the notion of quantization of a probability measure. We connect this notion of stability to a geometric feature of the underlying distribution of the data, named absolute margin condition, inspired by recent works on the subject.

On the Exponentially Weighted Aggregate with the Laplace Prior

In this paper, we study the statistical behaviour of the Exponentially Weighted Aggregate (EWA) in the problem of high-dimensional regression with fixed design. Under the assumption that the underlying regression vector is sparse, it is reasonable to use the Laplace distribution as a prior. The resulting estimator and, specifically, a particular instance of it referred to as the Bayesian lasso, was already used in the statistical literature because of its computational convenience, even though no thorough mathematical analysis of its statistical properties was carried out. The present work fills this gap by establishing sharp oracle inequalities for the EWA with the Laplace prior. These inequalities show that if the temperature parameter is small, the EWA with the Laplace prior satisfies the same type of oracle inequality as the lasso estimator does, as long as the quality of estimation is measured by the prediction loss. Extensions of the proposed methodology to the problem of prediction with low-rank matrices are considered.

On the prediction loss of the lasso in the partially labeled setting

In this paper we revisit the risk bounds of the lasso estimator in the context of transductive and semi-supervised learning. In other terms, the setting under consideration is that of regression with random design under partial labeling. The main goal is to obtain user-friendly bounds on the off-sample prediction risk. To this end, the simple setting of bounded response variable and bounded (high-dimensional) covariates is considered. We propose some new adaptations of the lasso to these settings and establish oracle inequalities both in expectation and in deviation. These results provide non-asymptotic upper bounds on the risk that highlight the interplay between the bias due to the mis-specification of the linear model, the bias due to the approximate sparsity and the variance. They also demonstrate that the presence of a large number of unlabeled features may have significant positive impact in the situations where the restricted eigenvalue of the design matrix vanishes or is very small.