A fractional Brownian field indexed by L2L^2 and a varying Hurst parameter

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as L\'evy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and H\"older regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on L\'evy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local H\"older regularity on general indexing collections

Increment stationarity of L2L^2-indexed stochastic processes: spectral representation and characterization

We are interested in the increment stationarity property for $L^2$-indexed stochastic processes, which is a fairly general concern since many random fields can be interpreted as the restriction of a more generally defined $L^2$-indexed process. We first give a spectral representation theorem in the sense of \citet{Ito54}, and see potential applications on random fields, in particular on the $L^2$-indexed extension of the fractional Brownian motion. Then we prove that this latter process is characterized by its increment stationarity and self-similarity properties, as in the one-dimensional case

Some singular sample path properties of a multiparameter fractional Brownian motion

We prove a Chung-type law of the iterated logarithm for a multiparameter extension of the fractional Brownian motion which is not increment stationary. This multiparameter fractional Brownian motion behaves very differently at the origin and away from the axes, which also appears in the Hausdorff dimension of its range and in the measure of its pointwise H\"older exponents. A functional version of this Chung-type law is also provided.Comment: 21 pages. To appear in J. Theoret. Proba

Lipschitz Bandits: Regret Lower Bounds and Optimal Algorithms

We consider stochastic multi-armed bandit problems where the expected reward is a Lipschitz function of the arm, and where the set of arms is either discrete or continuous. For discrete Lipschitz bandits, we derive asymptotic problem specific lower bounds for the regret satisfied by any algorithm, and propose OSLB and CKL-UCB, two algorithms that efficiently exploit the Lipschitz structure of the problem. In fact, we prove that OSLB is asymptotically optimal, as its asymptotic regret matches the lower bound. The regret analysis of our algorithms relies on a new concentration inequality for weighted sums of KL divergences between the empirical distributions of rewards and their true distributions. For continuous Lipschitz bandits, we propose to first discretize the action space, and then apply OSLB or CKL-UCB, algorithms that provably exploit the structure efficiently. This approach is shown, through numerical experiments, to significantly outperform existing algorithms that directly deal with the continuous set of arms. Finally the results and algorithms are extended to contextual bandits with similarities.Comment: COLT 201

Lagrangian Pairs and Lagrangian Orthogonal Matroids

Represented Coxeter matroids of types $C_n$ and $D_n$, that is, symplectic and orthogonal matroids arising from totally isotropic subspaces of symplectic or (even-dimensional) orthogonal spaces, may also be represented in buildings of type $C_n$ and $D_n$, respectively. Indeed, the particular buildings involved are those arising from the flags or oriflammes, respectively, of totally isotropic subspaces. There are also buildings of type $B_n$ arising from flags of totally isotropic subspaces in odd-dimensional orthogonal space. Coxeter matroids of type $B_n$ are the same as those of type $C_n$ (since they depend only upon the reflection group, not the root system). However, buildings of type $B_n$ are distinct from those of the other types. The matroids representable in odd dimensional orthogonal space (and therefore in the building of type $B_n$) turn out to be a special case of symplectic (flag) matroids, those whose top component, or Lagrangian matroid, is a union of two Lagrangian orthogonal matroids. These two matroids are called a Lagrangian pair, and they are the combinatorial manifestation of the ``fork'' at the top of an oriflamme (or of the fork at the end of the Coxeter diagram of $D_n$). Here we give a number of equivalent characterizations of Lagrangian pairs, and prove some rather strong properties of them.Comment: Requires amssymb.sty; 12 pages, 2 LaTeX figure

Deriving GENERIC from a generalized fluctuation symmetry

Much of the structure of macroscopic evolution equations for relaxation to equilibrium can be derived from symmetries in the dynamical fluctuations around the most typical trajectory. For example, detailed balance as expressed in terms of the Lagrangian for the path-space action leads to gradient zero-cost flow. We find a new such fluctuation symmetry that implies GENERIC, an extension of gradient flow where a Hamiltonian part is added to the dissipative term in such a way as to retain the free energy as Lyapunov function

Local H\"older regularity for set-indexed processes

In this paper, we study the H\"older regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Kolmogorov-like H\"older-continuity Theorem, whose novelty is illustrated on an example which could not have been treated with anterior tools. Increments for set-indexed processes are usually not simply written as $X_U-X_V$, hence we considered different notions of H\"older-continuity. Then, the localization of these properties leads to various definitions of H\"older exponents, which we compare to one another. In the case of Gaussian processes, almost sure values are proved for these exponents, uniformly along the sample paths. As an application, the local regularity of the set-indexed fractional Brownian motion is proved to be equal to the Hurst parameter uniformly, with probability one.Comment: 32 page

Lagrangian Matroids: Representations of Type BnB_n

We introduce the concept of orientation for Lagrangian matroids represented in the flag variety of maximal isotropic subspaces of dimension N in the real vector space of dimension 2N+1. The paper continues the study started in math.CO/0209100.Comment: Requires amssymb.sty; 17 page