Albert Thomas

Geovani Rizk, Igor Colin, Albert Thomas et al.

We propose the first regret-based approach to the Graphical Bilinear Bandits problem, where $n$ agents in a graph play a stochastic bilinear bandit game with each of their neighbors. This setting reveals a combinatorial NP-hard problem that prevents the use of any existing regret-based algorithm in the (bi-)linear bandit literature. In this paper, we fill this gap and present the first regret-based algorithm for graphical bilinear bandits using the principle of optimism in the face of uncertainty. Theoretical analysis of this new method yields an upper bound of $\tilde{O}(\sqrt{T})$ on the $α$-regret and evidences the impact of the graph structure on the rate of convergence. Finally, we show through various experiments the validity of our approach.

Igor Colin, Albert Thomas, Moez Draief

As cellular networks become denser, a scalable and dynamic tuning of wireless base station parameters can only be achieved through automated optimization. Although the contextual bandit framework arises as a natural candidate for such a task, its extension to a parallel setting is not straightforward: one needs to carefully adapt existing methods to fully leverage the multi-agent structure of this problem. We propose two approaches: one derived from a deterministic UCB-like method and the other relying on Thompson sampling. Thanks to its bayesian nature, the latter is intuited to better preserve the exploration-exploitation balance in the bandit batch. This is verified on toy experiments, where Thompson sampling shows robustness to the variability of the contexts. Finally, we apply both methods on a real base station network dataset and evidence that Thompson sampling outperforms both manual tuning and contextual UCB.

Stephan Clémençon, Albert Thomas

This paper aims at formulating the issue of ranking multivariate unlabeled observations depending on their degree of abnormality as an unsupervised statistical learning task. In the 1-d situation, this problem is usually tackled by means of tail estimation techniques: univariate observations are viewed as all the more `abnormal' as they are located far in the tail(s) of the underlying probability distribution. It would be desirable as well to dispose of a scalar valued `scoring' function allowing for comparing the degree of abnormality of multivariate observations. Here we formulate the issue of scoring anomalies as a M-estimation problem by means of a novel functional performance criterion, referred to as the Mass Volume curve (MV curve in short), whose optimal elements are strictly increasing transforms of the density almost everywhere on the support of the density. We first study the statistical estimation of the MV curve of a given scoring function and we provide a strategy to build confidence regions using a smoothed bootstrap approach. Optimization of this functional criterion over the set of piecewise constant scoring functions is next tackled. This boils down to estimating a sequence of empirical minimum volume sets whose levels are chosen adaptively from the data, so as to adjust to the variations of the optimal MV curve, while controling the bias of its approximation by a stepwise curve. Generalization bounds are then established for the difference in sup norm between the MV curve of the empirical scoring function thus obtained and the optimal MV curve.