We consider exponential two-armed bandit problem in which incomes are described by exponential distribution densities. We develop Bayesian approach and present recursive equation for determination of Bayesian strategy and Bayesian risk. In the limiting case as the control horizon goes to infinity, we obtain the second order partial differential equation in the domain of "close distributions". Results are compared with Gaussian two-armed bandit. It turned out that exponential and Gaussian two-armed bandits have the same description in the limiting case. Since Gaussian two-armed bandit describes the batch processing, this means that in case of exponential two-armed bandit batch processing does not enlarge Bayesian risk in comparison with one-by-one optimal processing as the total number of processed data items goes to infinity.
We consider a continuous time two-armed bandit problem in which incomes are described by Poissonian processes. We develop Bayesian approach with arbitrary prior distribution. We present two versions of recursive equation for determination of Bayesian piece-wise constant strategy and Bayesian risk and partial differential equation in the limiting case. Unlike the previously considered Bayesian settings our description uses current history of the process and not evolution of the posterior distribution.
This habilitation thesis presents several contributions to (1) the PAC-Bayesian analysis of statistical learning, (2) the three aggregation problems: given d functions, how to predict as well as (i) the best of these d functions (model selection type aggregation), (ii) the best convex combination of these d functions, (iii) the best linear combination of these d functions, (3) the multi-armed bandit problems.
We consider a Kullback-Leibler-based algorithm for the stochastic multi-armed bandit problem in the case of distributions with finite supports (not necessarily known beforehand), whose asymptotic regret matches the lower bound of \cite{Burnetas96}. Our contribution is to provide a finite-time analysis of this algorithm; we get bounds whose main terms are smaller than the ones of previously known algorithms with finite-time analyses (like UCB-type algorithms).
Lai and Robbins (1985) and Lai (1987) provided efficient parametric solutions to the multi-armed bandit problem, showing that arm allocation via upper confidence bounds (UCB) achieves minimum regret. These bounds are constructed from the Kullback-Leibler information of the reward distributions, estimated from specified parametric families. In recent years there has been renewed interest in the multi-armed bandit problem due to new applications in machine learning algorithms and data analytics. Non-parametric arm allocation procedures like $ε$-greedy, Boltzmann exploration and BESA were studied…
We obtain the upper bound of the loss function for a strategy in the multi-armed bandit problem with Gaussian distributions of incomes. Considered strategy is an asymptotic generalization of the strategy proposed by J. Bather for the multi-armed bandit problem and using UCB rule, i.e. choosing the action corresponding to the maximum of the upper bound of the confidence interval of the current estimate of the expected value of one-step income. Results are obtained with the help of invariant description of the control on the unit horizon in the domain of close distributions because just there th…
Myopic strategy is one of the most important strategies when studying bandit problems. In this paper, we consider the two-armed bandit problem proposed by Feldman. With general distributions and utility functions, we obtain a necessary and sufficient condition for the optimality of the myopic strategy. As an application, we could solve Nouiehed and Ross's conjecture for Bernoulli two-armed bandit problems that myopic strategy stochastically maximizes the number of wins.
We consider the two-armed bandit problem as applied to data processing if there are two alternative processing methods available with different a priori unknown efficiencies. One should determine the most effective method and provide its predominant application. Gaussian two-armed bandit describes the batch, and possibly parallel, processing when the same methods are applied to sufficiently large packets of data and accumulated incomes are used for the control. If the number of packets is large enough then such control does not deteriorate the control performance, i.e. does not increase the mi…
We study a multi-armed bandit problem with covariates in a setting where there is a possible delay in observing the rewards. Under some mild assumptions on the probability distributions for the delays and using an appropriate randomization to select the arms, the proposed strategy is shown to be strongly consistent.
Princeton University（普林斯顿大学）；Université Paris Diderot（巴黎狄德罗大学）
We study the stochastic multi-armed bandit problem when one knows the value $μ^{(\star)}$ of an optimal arm, as a well as a positive lower bound on the smallest positive gap $Δ$. We propose a new randomized policy that attains a regret {\em uniformly bounded over time} in this setting. We also prove several lower bounds, which show in particular that bounded regret is not possible if one only knows $Δ$, and bounded regret of order $1/Δ$ is not possible if one only knows $μ^{(\star)}$
We consider stochastic bandit problems with $K$ arms, each associated with a bounded distribution supported on the range $[m,M]$. We do not assume that the range $[m,M]$ is known and show that there is a cost for learning this range. Indeed, a new trade-off between distribution-dependent and distribution-free regret bounds arises, which prevents from simultaneously achieving the typical $\ln T$ and $\sqrt{T}$ bounds. For instance, a $\sqrt{T}$}distribution-free regret bound may only be achieved if the distribution-dependent regret bounds are at least of order $\sqrt{T}$. We exhibit a strategy …
Laboratoire de Mathématiques d’Orsay（奥赛数学实验室）；Université Paris-Sud（巴黎南大学）
In the context of stochastic continuum-armed bandits, we present an algorithm that adapts to the unknown smoothness of the objective function. We exhibit and compute a polynomial cost of adaptation to the H{ö}lder regularity for regret minimization. To do this, we first reconsider the recent lower bound of Locatelli and Carpentier [20], and define and characterize admissible rate functions. Our new algorithm matches any of these minimal rate functions. We provide a finite-time analysis and a thorough discussion about asymptotic optimality.
I analyse the frequentist regret of the famous Gittins index strategy for multi-armed bandits with Gaussian noise and a finite horizon. Remarkably it turns out that this approach leads to finite-time regret guarantees comparable to those available for the popular UCB algorithm. Along the way I derive finite-time bounds on the Gittins index that are asymptotically exact and may be of independent interest. I also discuss some computational issues and present experimental results suggesting that a particular version of the Gittins index strategy is a modest improvement on existing algorithms with…
Université Paris Diderot（巴黎第七大学）；Princeton University（普林斯顿大学）
We consider a multi-armed bandit problem in a setting where each arm produces a noisy reward realization which depends on an observable random covariate. As opposed to the traditional static multi-armed bandit problem, this setting allows for dynamically changing rewards that better describe applications where side information is available. We adopt a nonparametric model where the expected rewards are smooth functions of the covariate and where the hardness of the problem is captured by a margin parameter. To maximize the expected cumulative reward, we introduce a policy called Adaptively Binn…
Continuum-armed bandits (a.k.a., black-box or $0^{th}$-order optimization) involves optimizing an unknown objective function given an oracle that evaluates the function at a query point, with the goal of using as few query points as possible. In the most well-studied case, the objective function is assumed to be Lipschitz continuous and minimax rates of simple and cumulative regrets are known in both noiseless and noisy settings. This paper studies continuum-armed bandits under more general smoothness conditions, namely Besov smoothness conditions, on the objective function. In both noiseless …
In $\mathcal{X}$-armed bandit problem an agent sequentially interacts with environment which yields a reward based on the vector input the agent provides. The agent's goal is to maximise the sum of these rewards across some number of time steps. The problem and its variations have been a subject of numerous studies, suggesting sub-linear and some times optimal strategies. The given paper introduces a novel variation of the problem. We consider an environment, which can abruptly change its behaviour an unknown number of times. To that end we propose a novel strategy and prove it attains sub-lin…
