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We study the problem of $K$-armed dueling bandit for both stochastic and adversarial environments, where the goal of the learner is to aggregate information through relative preferences of pair of decisions points queried in an online sequential manner. We first propose a novel reduction from any (general) dueling bandits to multi-armed bandits and despite the simplicity, it allows us to improve many existing results in dueling bandits. In particular, \emph{we give the first best-of-both world result for the dueling bandits regret minimization problem} -- a unified framework that is guaranteed to perform optimally for both stochastic and adversarial preferences simultaneously. Moreover, our algorithm is also the first to achieve an optimal $O(\sum_{i = 1}^K \frac{\log T}{\Delta_i})$ regret bound against the Condorcet-winner benchmark, which scales optimally both in terms of the arm-size $K$ and the instance-specific suboptimality gaps $\{\Delta_i\}_{i = 1}^K$. This resolves the long-standing problem of designing an instancewise gap-dependent order optimal regret algorithm for dueling bandits (with matching lower bounds up to small constant factors). We further justify the robustness of our proposed algorithm by proving its optimal regret rate under adversarially corrupted preferences -- this outperforms the existing state-of-the-art corrupted dueling results by a large margin. In summary, we believe our reduction idea will find a broader scope in solving a diverse class of dueling bandits setting, which are otherwise studied separately from multi-armed bandits with often more complex solutions and worse guarantees. The efficacy of our proposed algorithms is empirically corroborated against the existing dueling bandit methods.

In this paper, we study smooth stochastic multi-level composition optimization problems, where the objective function is a nested composition of $T$ functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an $\epsilon$-stationary point of the problem. We show that the first algorithm, which is a generalization of \cite{GhaRuswan20} to the $T$ level case, can achieve a sample complexity of $\mathcal{O}(1/\epsilon^6)$ by using mini-batches of samples in each iteration. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to $\mathcal{O}(1/\epsilon^4)$. {\color{black}This modification not only removes the requirement of having a mini-batch of samples in each iteration, but also makes the algorithm parameter-free and easy to implement}. To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multi-level setting, obtains the same sample complexity of the smooth single-level setting, under standard assumptions (unbiasedness and boundedness of the second moments) on the stochastic first-order oracle.

We consider a budgeted combinatorial multi-armed bandit setting where, in every round, the algorithm selects a super-arm consisting of one or more arms. The goal is to minimize the total expected regret after all rounds within a limited budget. Existing techniques in this literature either fix the budget per round or fix the number of arms pulled in each round. Our setting is more general where based on the remaining budget and remaining number of rounds, the algorithm can decide how many arms to be pulled in each round. First, we propose CBwK-Greedy-UCB algorithm, which uses a greedy technique, CBwK-Greedy, to allocate the arms to the rounds. Next, we propose a reduction of this problem to Bandits with Knapsacks (BwK) with a single pull. With this reduction, we propose CBwK-LPUCB that uses PrimalDualBwK ingeniously. We rigorously prove regret bounds for CBwK-LP-UCB. We experimentally compare the two algorithms and observe that CBwK-Greedy-UCB performs incrementally better than CBwK-LP-UCB. We also show that for very high budgets, the regret goes to zero.

We consider the problem introduced by \cite{Mason2020} of identifying all the $\varepsilon$-optimal arms in a finite stochastic multi-armed bandit with Gaussian rewards. In the fixed confidence setting, we give a lower bound on the number of samples required by any algorithm that returns the set of $\varepsilon$-good arms with a failure probability less than some risk level $\delta$. This bound writes as $T_{\varepsilon}^*(\mu)\log(1/\delta)$, where $T_{\varepsilon}^*(\mu)$ is a characteristic time that depends on the vector of mean rewards $\mu$ and the accuracy parameter $\varepsilon$. We also provide an efficient numerical method to solve the convex max-min program that defines the characteristic time. Our method is based on a complete characterization of the alternative bandit instances that the optimal sampling strategy needs to rule out, thus making our bound tighter than the one provided by \cite{Mason2020}. Using this method, we propose a Track-and-Stop algorithm that identifies the set of $\varepsilon$-good arms w.h.p and enjoys asymptotic optimality (when $\delta$ goes to zero) in terms of the expected sample complexity. Finally, using numerical simulations, we demonstrate our algorithm's advantage over state-of-the-art methods, even for moderate values of the risk parameter.

We consider the problem of combining and learning over a set of adversarial bandit algorithms with the goal of adaptively tracking the best one on the fly. The CORRAL algorithm of Agarwal et al. (2017) and its variants (Foster et al., 2020a) achieve this goal with a regret overhead of order $\widetilde{O}(\sqrt{MT})$ where $M$ is the number of base algorithms and $T$ is the time horizon. The polynomial dependence on $M$, however, prevents one from applying these algorithms to many applications where $M$ is poly$(T)$ or even larger. Motivated by this issue, we propose a new recipe to corral a larger band of bandit algorithms whose regret overhead has only \emph{logarithmic} dependence on $M$ as long as some conditions are satisfied. As the main example, we apply our recipe to the problem of adversarial linear bandits over a $d$-dimensional $\ell_p$ unit-ball for $p \in (1,2]$. By corralling a large set of $T$ base algorithms, each starting at a different time step, our final algorithm achieves the first optimal switching regret $\widetilde{O}(\sqrt{d S T})$ when competing against a sequence of comparators with $S$ switches (for some known $S$). We further extend our results to linear bandits over a smooth and strongly convex domain as well as unconstrained linear bandits.

Multiple-input multiple-output (MIMO) systems greatly increase the overall throughput of wireless systems since they are capable of transmitting multiple streams employing the same time-frequency resources. However, this gain requires an appropriate precoder design and a power allocation technique. In general, precoders and power allocation schemes are designed assuming perfect channel estate information (CSI). Nonetheless, this is an optimistic assumption since real systems only possess partial or imperfect CSI at the transmitter (CSIT). The imperfect CSIT originates residual inter-user interference, which is detrimental for wireless systems. In this paper, two adaptive power allocation algorithms are proposed, which are more robust against CSIT imperfections than conventional techniques. Both techniques employ the mean square error as the objective function. Simulation results show that the proposed techniques obtain a higher performance in terms of sum-rate than conventional approaches.

The greedy algorithm for monotone submodular function maximization subject to cardinality constraint is guaranteed to approximate the optimal solution to within a $1-1/e$ factor. Although it is well known that this guarantee is essentially tight in the worst case -- for greedy and in fact any efficient algorithm, experiments show that greedy performs better in practice. We observe that for many applications in practice, the empirical distribution of the budgets (i.e., cardinality constraints) is supported on a wide range, and moreover, all the existing hardness results in theory break under a large perturbation of the budget. To understand the effect of the budget from both algorithmic and hardness perspectives, we introduce a new notion of budget smoothed analysis. We prove that greedy is optimal for every budget distribution, and we give a characterization for the worst-case submodular functions. Based on these results, we show that on the algorithmic side, under realistic budget distributions, greedy and related algorithms enjoy provably better approximation guarantees, that hold even for worst-case functions, and on the hardness side, there exist hard functions that are fairly robust to all the budget distributions.

In this paper, we propose a novel neural-based exploration strategy in contextual bandits, EE-Net, distinct from the standard UCB-based and TS-based approaches. Contextual multi-armed bandits have been studied for decades with various applications. To solve the exploitation-exploration tradeoff in bandits, there are three main techniques: epsilon-greedy, Thompson Sampling (TS), and Upper Confidence Bound (UCB). In recent literature, linear contextual bandits have adopted ridge regression to estimate the reward function and combine it with TS or UCB strategies for exploration. However, this line of works explicitly assumes the reward is based on a linear function of arm vectors, which may not be true in real-world datasets. To overcome this challenge, a series of neural-based bandit algorithms have been proposed, where a neural network is assigned to learn the underlying reward function and TS or UCB are adapted for exploration. In this paper, we propose "EE-Net", a neural-based bandit approach with a novel exploration strategy. In addition to utilizing a neural network (Exploitation network) to learn the reward function, EE-Net adopts another neural network (Exploration network) to adaptively learn potential gains compared to currently estimated reward for making explorations. Then, a decision-maker is constructed to combine the outputs from the Exploitation and Exploration networks. We prove that EE-Net can achieve $\mathcal{O}(\sqrt{T\log T})$ regret, which is tighter than existing state-of-the-art neural bandit algorithms ($\mathcal{O}(\sqrt{T}\log T)$ for both UCB-based and TS-based). Through extensive experiments on four real-world datasets, we show that EE-Net outperforms existing linear and neural bandit approaches.

In this paper, we tackle the computational efficiency of kernelized UCB algorithms in contextual bandits. While standard methods require a O(CT^3) complexity where T is the horizon and the constant C is related to optimizing the UCB rule, we propose an efficient contextual algorithm for large-scale problems. Specifically, our method relies on incremental Nystrom approximations of the joint kernel embedding of contexts and actions. This allows us to achieve a complexity of O(CTm^2) where m is the number of Nystrom points. To recover the same regret as the standard kernelized UCB algorithm, m needs to be of order of the effective dimension of the problem, which is at most O(\sqrt(T)) and nearly constant in some cases.

Contextual multi-armed bandit (MAB) achieves cutting-edge performance on a variety of problems. When it comes to real-world scenarios such as recommendation system and online advertising, however, it is essential to consider the resource consumption of exploration. In practice, there is typically non-zero cost associated with executing a recommendation (arm) in the environment, and hence, the policy should be learned with a fixed exploration cost constraint. It is challenging to learn a global optimal policy directly, since it is a NP-hard problem and significantly complicates the exploration and exploitation trade-off of bandit algorithms. Existing approaches focus on solving the problems by adopting the greedy policy which estimates the expected rewards and costs and uses a greedy selection based on each arm's expected reward/cost ratio using historical observation until the exploration resource is exhausted. However, existing methods are hard to extend to infinite time horizon, since the learning process will be terminated when there is no more resource. In this paper, we propose a hierarchical adaptive contextual bandit method (HATCH) to conduct the policy learning of contextual bandits with a budget constraint. HATCH adopts an adaptive method to allocate the exploration resource based on the remaining resource/time and the estimation of reward distribution among different user contexts. In addition, we utilize full of contextual feature information to find the best personalized recommendation. Finally, in order to prove the theoretical guarantee, we present a regret bound analysis and prove that HATCH achieves a regret bound as low as $O(\sqrt{T})$. The experimental results demonstrate the effectiveness and efficiency of the proposed method on both synthetic data sets and the real-world applications.