One Arrow, Two Kills: An Unified Framework for Achieving Optimal Regret Guarantees in Sleeping Bandits

We address the problem of \emph{`Internal Regret'} in \emph{Sleeping Bandits} in the fully adversarial setup, as well as draw connections between different existing notions of sleeping regrets in the multiarmed bandits (MAB) literature and consequently analyze the implications: Our first contribution is to propose the new notion of \emph{Internal Regret} for sleeping MAB. We then proposed an algorithm that yields sublinear regret in that measure, even for a completely adversarial sequence of losses and availabilities. We further show that a low sleeping internal regret always implies a low external regret, and as well as a low policy regret for iid sequence of losses. The main contribution of this work precisely lies in unifying different notions of existing regret in sleeping bandits and understand the implication of one to another. Finally, we also extend our results to the setting of \emph{Dueling Bandits} (DB)--a preference feedback variant of MAB, and proposed a reduction to MAB idea to design a low regret algorithm for sleeping dueling bandits with stochastic preferences and adversarial availabilities. The efficacy of our algorithms is justified through empirical evaluations

One Arrow, Two Kills: An Unified Framework for Achieving Optimal Regret Guarantees in Sleeping Bandits

We address the problem of `Internal Regret' in Sleeping Bandits in the fully adversarial setup, as well as draw connections between different existing notions of sleeping regrets in the multiarmed bandits (MAB) literature and consequently analyze the implications: Our first contribution is to propose the new notion of Internal Regret for sleeping MAB. We then proposed an algorithm that yields sublinear regret in that measure, even for a completely adversarial sequence of losses and availabilities. We further show that a low sleeping internal regret always implies a low external regret, and as well as a low policy regret for iid sequence of losses. The main contribution of this work precisely lies in unifying different notions of existing regret in sleeping bandits and understand the implication of one to another. Finally, we also extend our results to the setting of Dueling Bandits (DB)--a preference feedback variant of MAB, and proposed a reduction to MAB idea to design a low regret algorithm for sleeping dueling bandits with stochastic preferences and adversarial availabilities. The efficacy of our algorithms is justified through empirical evaluations

Adaptive Bandits: Towards the best history-dependent strategy

Ce document a été accepté pour publication à AI&Statistics 2011. Je dois me référer à ce travail, je me réfère donc à ceci en attendant de publier la version camera-ready (le mois prochain).We consider multi-armed bandit games with possibly adaptive opponents. We introduce models Theta of constraints based on equivalence classes on the common history (information shared by the player and the opponent) which dene two learning scenarios: (1) The opponent is constrained, i.e. he provides rewards that are stochastic functions of equivalence classes dened by some model theta*\in Theta. The regret is measured with respect to (w.r.t.) the best history-dependent strategy. (2) The opponent is arbitrary and we measure the regret w.r.t. the best strategy among all mappings from classes to actions (i.e. the best history-class-based strategy) for the best model in Theta. This allows to model opponents (case 1) or strategies (case 2) which handles nite memory, periodicity, standard stochastic bandits and other situations. When Theta={theta}, i.e. only one model is considered, we derive tractable algorithms achieving a tight regret (at time T) bounded by ~O(sqrt(TAC)), where C is the number of classes of theta. Now, when many models are available, all known algorithms achieving a nice regret O(sqrt(T)) are unfortunately not tractable and scale poorly with the number of models |Theta|. Our contribution here is to provide tractable algorithms with regret bounded by T^{2/3}C^{1/3} log(|Theta|)^{1/2}

Improved sleeping bandits with stochastic action sets and adversarial rewards

International audienceIn this paper, we consider the problem of sleeping bandits with stochastic action sets and adversarial rewards. In this setting, in contrast to most work in bandits, the actions may not be available at all times. For instance, some products might be out of stock in item recommendation. The best existing efficient (i.e., polynomial-time) algorithms for this problem only guarantee an $O(T 2/3)$ upper-bound on the regret. Yet, inefficient algorithms based on EXP4 can achieve $O(√ T)$. In this paper , we provide a new computationally efficient algorithm inspired by EXP3 satisfying a regret of order $O(√ T)$ when the availabilities of each action $i ∈ A$ are independent. We then study the most general version of the problem where at each round available sets are generated from some unknown arbitrary distribution (i.e., without the independence assumption) and propose an efficient algorithm with $O(√ 2 K T)$ regret guarantee. Our theoretical results are corroborated with experimental evaluations

Dueling Bandits with Adversarial Sleeping

We introduce the problem of sleeping dueling bandits with stochastic preferences and adversarial availabilities (DB-SPAA). In almost all dueling bandit applications, the decision space often changes over time; eg, retail store management, online shopping, restaurant recommendation, search engine optimization, etc. Surprisingly, this `sleeping aspect' of dueling bandits has never been studied in the literature. Like dueling bandits, the goal is to compete with the best arm by sequentially querying the preference feedback of item pairs. The non-triviality however results due to the non-stationary item spaces that allow any arbitrary subsets items to go unavailable every round. The goal is to find an optimal `no-regret' policy that can identify the best available item at each round, as opposed to the standard `fixed best-arm regret objective' of dueling bandits. We first derive an instance-specific lower bound for DB-SPAA $\Omega( \sum_{i =1}^{K-1}\sum_{j=i+1}^K \frac{\log T}{\Delta(i,j)})$, where $K$ is the number of items and $\Delta(i,j)$ is the gap between items $i$ and $j$. This indicates that the sleeping problem with preference feedback is inherently more difficult than that for classical multi-armed bandits (MAB). We then propose two algorithms, with near optimal regret guarantees. Our results are corroborated empirically