We introduce the feasible arm identification problem, a pure exploration multi-armed bandit problem where the agent is given a set of $D$-dimensional arms and a polyhedron $P = \{x : A x \leq b \} \subset R^D$. Pulling an arm gives a random vector and the goal is to determine, using a fixed budget of $T$ pulls, which of the arms have means belonging to $P$. We propose three algorithms MD-UCBE, MD-SAR, and MD-APT and provide a unified analysis establishing upper bounds for each of them. We also establish a lower bound that matches up to constants the upper bounds of MD-UCBE and MD-APT. Finally, we demonstrate the effectiveness of our algorithms on synthetic and real-world datasets.

We study a variant of the stochastic $K$-armed bandit problem, which we call "bandits with delayed, aggregated anonymous feedback''. In this problem, when the player pulls an arm, a reward is generated, however it is not immediately observed. Instead, at the end of each round the player observes only the sum of a number of previously generated rewards which happen to arrive in the given round. The rewards are stochastically delayed and due to the aggregated nature of the observations, the information of which arm led to a particular reward is lost. The question is what is the cost of the information loss due to this delayed, aggregated anonymous feedback? Previous works have studied bandits with stochastic, non-anonymous delays and found that the regret increases only by an additive factor relating to the expected delay. In this paper, we show that this additive regret increase can be maintained in the harder delayed, aggregated anonymous feedback setting when the expected delay (or a bound on it) is known. We provide an algorithm that matches the worst case regret of the non-anonymous problem exactly when the delays are bounded, and up to logarithmic factors or an additive variance term for unbounded delays.

Regret bounds in online learning compare the player's performance to $L*$, the optimal performance in hindsight with a fixed strategy. Typically such bounds scale with the square root of the time horizon $T$. The more refined concept of first-order regret bound replaces this with a scaling $\sqrt{L*}$, which may be much smaller than $\sqrt{T}$.It is well known that minor variants of standard algorithms satisfy first-order regret bounds in the full information and multi-armed bandit settings. In a COLT 2017 open problem, Agarwal, Krishnamurthy, Langford, Luo, and Schapire raised the issue that existing techniques do not seem sufficient to obtain first-order regret bounds for the contextual bandit problem. In the present paper, we resolve this open problem by presenting a new strategy based on augmenting the policy space.

We study the application of the Thompson sampling (TS) methodology to the stochastic combinatorial multi-armed bandit (CMAB) framework. We analyze the standard TS algorithm for the general CMAB, and obtain the first distribution-dependent regret bound of $O(m\log T / \Delta_{\min}) $ for TS under general CMAB, where $m$ is the number of arms, $T$ is the time horizon, and $\Delta_{\min}$ is the minimum gap between the expected reward of the optimal solution and any non-optimal solution. We also show that one cannot use an approximate oracle in TS algorithm for even MAB problems. Then we expand the analysis to matroid bandit, a special case of CMAB and for which we could remove the independence assumption across arms and achieve a better regret bound. Finally, we use some experiments to show the comparison of regrets of CUCB and CTS algorithms.