Institut de Mathématiques de Toulouse

On considère l’opérateur elliptique <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:msub> <mml:mo>)</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>c</mml:mi> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> où les coefficients sont des fonctions mesurables appartenant à des espaces <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> convenables dans un ouvert borné <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> de <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi mathvariant="bold">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . Le but principal est d’étendre un résultat [par W. Littman, G. Stampacchia et H. Weinberger] sur les points réguliers pour le problème de Dirichlet à des équations plus générales (§10). Le paragraphe 3 contient aussi un principe de maximum pour les solutions faibles. Le paragraphe 4 contient des majorations a priori dans <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Ω</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> des solutions. Les paragraphes 5 et 7 sont consacrés à l’extension du théorème de Giorgi sur la continuité höldérienne des solutions ; l’extension au bord est aussi envisagée. La généralisation d’un théorème de Moser sur l’inégalité de Harnack est considérée dans le paragraphe 8. Le paragraphe 9 contient l’étude de la fonction de Green.

BACKGROUND: Variable selection on high throughput biological data, such as gene expression or single nucleotide polymorphisms (SNPs), becomes inevitable to select relevant information and, therefore, to better characterize diseases or assess genetic structure. There are different ways to perform variable selection in large data sets. Statistical tests are commonly used to identify differentially expressed features for explanatory purposes, whereas Machine Learning wrapper approaches can be used for predictive purposes. In the case of multiple highly correlated variables, another option is to use multivariate exploratory approaches to give more insight into cell biology, biological pathways or complex traits. RESULTS: A simple extension of a sparse PLS exploratory approach is proposed to perform variable selection in a multiclass classification framework. CONCLUSIONS: sPLS-DA has a classification performance similar to other wrapper or sparse discriminant analysis approaches on public microarray and SNP data sets. More importantly, sPLS-DA is clearly competitive in terms of computational efficiency and superior in terms of interpretability of the results via valuable graphical outputs. sPLS-DA is available in the R package mixOmics, which is dedicated to the analysis of large biological data sets.

We consider the Dirichlet Laplacian operator<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mo>−</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow></mml:math>on a curved quantum guide in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:msup><mml:mi>ℝ </mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mrow></mml:math>with an asymptotically straight reference curve. We give uniqueness results for the inverse problem associated to the reconstruction of the curvature by using either observations of spectral data or a boot-strapping method.

Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP) . This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones,

The stochastic multi-armed bandit model is a simple abstraction that has proven useful in many different contexts in statistics and machine learning. Whereas the achievable limit in terms of regret minimization is now well known, our aim is to contribute to a better understanding of the performance in terms of identifying the m best arms. We introduce generic notions of complexity for the two dominant frameworks considered in the literature: fixed-budget and fixed-confidence settings. In the fixed-confidence setting, we provide the first known distribution-dependent lower bound on the complexity that involves information-theoretic quantities and holds when m is larger than 1 under general assumptions. In the specific case of two armed-bandits, we derive refined lower bounds in both the fixed-confidence and fixed-budget settings, along with matching algorithms for Gaussian and Bernoulli bandit models. These results show in particular that the complexity of the fixed-budget setting may be smaller than the complexity of the fixed-confidence setting, contradicting the familiar behavior observed when testing fully specified alternatives. In addition, we also provide improved sequential stopping rules that have guaranteed error probabilities and shorter average running times. The proofs rely on two technical results that are of independent interest : a deviation lemma for self-normalized sums (Lemma 19) and a novel change of measure inequality for bandit models (Lemma 1).

International audience

The global estimation of microplastic afloat in the ocean is only approximately 1% of annual global plastic inputs. This reflects fundamental knowledge gaps in the transformation, fragmentation, and fates of microplastics in the ocean. In order to better understand microplastic fragmentation we proceeded to a thorough physicochemical characterization of samples collected from the North Artlantic subtropical gyre during the sea campaign Expedition seventh Continent in May 2014. The results were confronted with a mathematical approach. The introduction of mass distribution in opposition to the size distribution commonly proposed in this area clarify the fragmentation pattern. The mathematical analysis of the mass distribution points out a lack of debris with mass lighter than 1 mg. Characterization by means of microscopy, microtomography, and infrared microscopy gives a better understanding of the behavior of microplastic at sea. Flat pieces of debris (2 to 5 mm in length) typically have one face that is more photodegraded (due to exposure to the sun) and the other with more biofilm, suggesting that they float in a preferred orientation. Smaller debris, with a cubic shape (below 2 mm), seems to roll at sea. All faces are evenly photodegraded and they are less colonized. The breakpoint in the mathematical model and the experimental observation around 2 mm leads to the conclusion that there is a discontinuity in the rate of fragmentation: we hypothesized that the smaller microplastics, the cubic ones mostly, are fragmented much faster than the parallelepipeds.

Recent biotechnology advances allow for multiple types of omics data, such as transcriptomic, proteomic or metabolomic data sets to be integrated. The problem of feature selection has been addressed several times in the context of classification, but needs to be handled in a specific manner when integrating data. In this study, we focus on the integration of two-block data that are measured on the same samples. Our goal is to combine integration and simultaneous variable selection of the two data sets in a one-step procedure using a Partial Least Squares regression (PLS) variant to facilitate the biologists' interpretation. A novel computational methodology called ;;sparse PLS" is introduced for a predictive analysis to deal with these newly arisen problems. The sparsity of our approach is achieved with a Lasso penalization of the PLS loading vectors when computing the Singular Value Decomposition. Sparse PLS is shown to be effective and biologically meaningful. Comparisons with classical PLS are performed on a simulated data set and on real data sets. On one data set, a thorough biological interpretation of the obtained results is provided. We show that sparse PLS provides a valuable variable selection tool for highly dimensional data sets.

We describe a major update of our Matlab freeware GloptiPoly for parsing generalized problems of moments and solving them numerically with semidefinite programming.

A timely and comprehensive treatment of random field theory with applications across diverse areas of study Level Sets and Extrema of Random Processes and Fields discusses how to understand the properties of the level sets of paths as well as how to compute the probability distribution of its extremal values, which are two general classes of problems that arise in the study of random processes and fields and in related applications. This book provides a unified and accessible approach to these two topics and their relationship to classical theory and Gaussian processes and fields, and the most modern research findings are also discussed. The authors begin with an introduction to the basic concepts of stochastic processes, including a modern review of Gaussian fields and their classical inequalities. Subsequent chapters are devoted to Rice formulas, regularity properties, and recent results on the tails of the distribution of the maximum. Finally, applications of random fields to various areas of mathematics are provided, specifically to systems of random equations and condition numbers of random matrices. Throughout the book, applications are illustrated from various areas of study such as statistics, genomics, and oceanography while other results are relevant to econometrics, engineering, and mathematical physics. The presented material is reinforced by end-of-chapter exercises that range in varying degrees of difficulty. Most fundamental topics are addressed in the book, and an extensive, up-to-date bibliography directs readers to existing literature for further study. Level Sets and Extrema of Random Processes and Fields is an excellent book for courses on probability theory, spatial statistics, Gaussian fields, and probabilistic methods in real computation at the upper-undergraduate and graduate levels. It is also a valuable reference for professionals in mathematics and applied fields such as statistics, engineering, econometrics, mathematical physics, and biology.

In this paper we study the geometric properties, existence, regularity and related issues for a family of surfaces which are boundaries of sets minimizing certain integral norms. These surfaces can be interpreted as a non-infinitesimal version of classical minimal surfaces.

MOTIVATION: With the availability of many 'omics' data, such as transcriptomics, proteomics or metabolomics, the integrative or joint analysis of multiple datasets from different technology platforms is becoming crucial to unravel the relationships between different biological functional levels. However, the development of such an analysis is a major computational and technical challenge as most approaches suffer from high data dimensionality. New methodologies need to be developed and validated. RESULTS: integrOmics efficiently performs integrative analyses of two types of 'omics' variables that are measured on the same samples. It includes a regularized version of canonical correlation analysis to enlighten correlations between two datasets, and a sparse version of partial least squares (PLS) regression that includes simultaneous variable selection in both datasets. The usefulness of both approaches has been demonstrated previously and successfully applied in various integrative studies. AVAILABILITY: integrOmics is freely available from http://CRAN.R-project.org/ or from the web site companion (http://math.univ-toulouse.fr/biostat) that provides full documentation and tutorials. CONTACT: k.lecao@uq.edu.au SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Brain atrophy measured by magnetic resonance structural imaging has been proposed as a surrogate marker for the early diagnosis of Alzheimer's disease. Studies on large samples are still required to determine its practical interest at the individual level, especially with regards to the capacity of anatomical magnetic resonance imaging to disentangle the confounding role of the cognitive reserve in the early diagnosis of Alzheimer's disease. One hundred and thirty healthy controls, 122 subjects with mild cognitive impairment of the amnestic type and 130 Alzheimer's disease patients were included from the ADNI database and followed up for 24 months. After 24 months, 72 amnestic mild cognitive impairment had converted to Alzheimer's disease (referred to as progressive mild cognitive impairment, as opposed to stable mild cognitive impairment). For each subject, cortical thickness was measured on the baseline magnetic resonance imaging volume. The resulting cortical thickness map was parcellated into 22 regions and a normalized thickness index was computed using the subset of regions (right medial temporal, left lateral temporal, right posterior cingulate) that optimally distinguished stable mild cognitive impairment from progressive mild cognitive impairment. We tested the ability of baseline normalized thickness index to predict evolution from amnestic mild cognitive impairment to Alzheimer's disease and compared it to the predictive values of the main cognitive scores at baseline. In addition, we studied the relationship between the normalized thickness index, the education level and the timeline of conversion to Alzheimer's disease. Normalized thickness index at baseline differed significantly among all the four diagnosis groups (P < 0.001) and correctly distinguished Alzheimer's disease patients from healthy controls with an 85% cross-validated accuracy. Normalized thickness index also correctly predicted evolution to Alzheimer's disease for 76% of amnestic mild cognitive impairment subjects after cross-validation, thus showing an advantage over cognitive scores (range 63-72%). Moreover, progressive mild cognitive impairment subjects, who converted later than 1 year after baseline, showed a significantly higher education level than those who converted earlier than 1 year after baseline. Using a normalized thickness index-based criterion may help with early diagnosis of Alzheimer's disease at the individual level, especially for highly educated subjects, up to 24 months before clinical criteria for Alzheimer's disease diagnosis are met.

In this survey we consider the development and mathematical analysis of numerical methods for kinetic partial differential equations. Kinetic equations represent a way of describing the time evolution of a system consisting of a large number of particles. Due to the high number of dimensions and their intrinsic physical properties, the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity. Here we review the basic numerical techniques for dealing with such equations, including the case of semi-Lagrangian methods, discrete-velocity models and spectral methods. In addition we give an overview of the current state of the art of numerical methods for kinetic equations. This covers the derivation of fast algorithms, the notion of asymptotic-preserving methods and the construction of hybrid schemes.

The effectiveness of most cancer targeted therapies is short-lived. Tumors often develop resistance that might be overcome with drug combinations. However, the number of possible combinations is vast, necessitating data-driven approaches to find optimal patient-specific treatments. Here we report AstraZeneca's large drug combination dataset, consisting of 11,576 experiments from 910 combinations across 85 molecularly characterized cancer cell lines, and results of a DREAM Challenge to evaluate computational strategies for predicting synergistic drug pairs and biomarkers. 160 teams participated to provide a comprehensive methodological development and benchmarking. Winning methods incorporate prior knowledge of drug-target interactions. Synergy is predicted with an accuracy matching biological replicates for >60% of combinations. However, 20% of drug combinations are poorly predicted by all methods. Genomic rationale for synergy predictions are identified, including ADAM17 inhibitor antagonism when combined with PIK3CB/D inhibition contrasting to synergy when combined with other PI3K-pathway inhibitors in PIK3CA mutant cells.

BACKGROUND: Each omics platform is now able to generate a large amount of data. Genomics, proteomics, metabolomics, interactomics are compiled at an ever increasing pace and now form a core part of the fundamental systems biology framework. Recently, several integrative approaches have been proposed to extract meaningful information. However, these approaches lack of visualisation outputs to fully unravel the complex associations between different biological entities. RESULTS: The multivariate statistical approaches 'regularized Canonical Correlation Analysis' and 'sparse Partial Least Squares regression' were recently developed to integrate two types of highly dimensional 'omics' data and to select relevant information. Using the results of these methods, we propose to revisit few graphical outputs to better understand the relationships between two 'omics' data and to better visualise the correlation structure between the different biological entities. These graphical outputs include Correlation Circle plots, Relevance Networks and Clustered Image Maps. We demonstrate the usefulness of such graphical outputs on several biological data sets and further assess their biological relevance using gene ontology analysis. CONCLUSIONS: Such graphical outputs are undoubtedly useful to aid the interpretation of these promising integrative analysis tools and will certainly help in addressing fundamental biological questions and understanding systems as a whole. AVAILABILITY: The graphical tools described in this paper are implemented in the freely available R package mixOmics and in its associated web application.

We study degenerate complex Monge-Ampère equations of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis omega plus d d Superscript c Baseline phi right-parenthesis Superscript n Baseline equals e Superscript t phi Baseline mu"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi> ω </mml:mi> <mml:mo>+</mml:mo> <mml:mi>d</mml:mi> <mml:msup> <mml:mi>d</mml:mi> <mml:mi>c</mml:mi> </mml:msup> <mml:mi> φ </mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi> φ </mml:mi> </mml:mrow> </mml:msup> <mml:mi> μ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">(\omega +dd^c\varphi )^n = e^{t \varphi }\mu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a big semi-positive form on a compact Kähler manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t element-of double-struck upper R Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">t \in \mathbb {R}^+</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu equals f omega Superscript n"> <mml:semantics> <mml:mrow> <mml:mi> μ </mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:msup> <mml:mi> ω </mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu =f\omega ^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a positive measure with density <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of upper L Superscript p Baseline left-parenthesis upper X comma omega Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi> ω </mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f\in L^p(X,\omega ^n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p&gt;1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We prove the existence and unicity of bounded <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi> ω </mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition. In case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is projective and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega equals psi Superscript asterisk Baseline omega prime"> <mml:semantics> <mml:mrow> <mml:mi> ω </mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi> ψ </mml:mi>

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BACKGROUND: In the context of systems biology, few sparse approaches have been proposed so far to integrate several data sets. It is however an important and fundamental issue that will be widely encountered in post genomic studies, when simultaneously analyzing transcriptomics, proteomics and metabolomics data using different platforms, so as to understand the mutual interactions between the different data sets. In this high dimensional setting, variable selection is crucial to give interpretable results. We focus on a sparse Partial Least Squares approach (sPLS) to handle two-block data sets, where the relationship between the two types of variables is known to be symmetric. Sparse PLS has been developed either for a regression or a canonical correlation framework and includes a built-in procedure to select variables while integrating data. To illustrate the canonical mode approach, we analyzed the NCI60 data sets, where two different platforms (cDNA and Affymetrix chips) were used to study the transcriptome of sixty cancer cell lines. RESULTS: We compare the results obtained with two other sparse or related canonical correlation approaches: CCA with Elastic Net penalization (CCA-EN) and Co-Inertia Analysis (CIA). The latter does not include a built-in procedure for variable selection and requires a two-step analysis. We stress the lack of statistical criteria to evaluate canonical correlation methods, which makes biological interpretation absolutely necessary to compare the different gene selections. We also propose comprehensive graphical representations of both samples and variables to facilitate the interpretation of the results. CONCLUSION: sPLS and CCA-EN selected highly relevant genes and complementary findings from the two data sets, which enabled a detailed understanding of the molecular characteristics of several groups of cell lines. These two approaches were found to bring similar results, although they highlighted the same phenomenons with a different priority. They outperformed CIA that tended to select redundant information.

We consider optimal sequential allocation in the context of the so-called stochastic multi-armed bandit model. We describe a generic index policy, in the sense of Gittins [J. R. Stat. Soc. Ser. B Stat. Methodol. 41 (1979) 148–177], based on upper confidence bounds of the arm payoffs computed using the Kullback–Leibler divergence. We consider two classes of distributions for which instances of this general idea are analyzed: the kl-UCB algorithm is designed for one-parameter exponential families and the empirical KL-UCB algorithm for bounded and finitely supported distributions. Our main contribution is a unified finite-time analysis of the regret of these algorithms that asymptotically matches the lower bounds of Lai and Robbins [Adv. in Appl. Math. 6 (1985) 4–22] and Burnetas and Katehakis [Adv. in Appl. Math. 17 (1996) 122–142], respectively. We also investigate the behavior of these algorithms when used with general bounded rewards, showing in particular that they provide significant improvements over the state-of-the-art.