We introduce efficient algorithms which achieve nearly optimal regrets for the problem of stochastic online shortest path routing with end-to-end feedback. The setting is a natural application of the combinatorial stochastic bandits problem, a special case of the linear stochastic bandits problem. We show how the difficulties posed by the large scale action set can be overcome by the networked structure of the action set. Our approach presents a novel connection between bandit learning and shortest path algorithms. Our main contribution is an adaptive exploration algorithm with nearly optimal instance-dependent regret for any directed acyclic network. We then modify it so that nearly optimal worst case regret is achieved simultaneously. Driven by the carefully designed Top-Two Comparison (TTC) technique, the algorithms are efficiently implementable. We further conduct extensive numerical experiments to show that our proposed algorithms not only achieve superior regret performances, but also reduce the runtime drastically.
A special case of linear bandits is combinatorial bandits where the action set is constrained to subset of @math In combinatorial stochastic bandits, it is often assumed that the reward loss vector is observed at all the coordinates sampled by the action taken. This is the so-called semi-bandit feedback setting @cite_32 . The authors of @cite_9 initiated the study of combinatorial stochastic bandits under semi-bandit feedback and a network-structured action set; while @cite_14 studied the general action set case. The authors of @cite_21 further characterized tight upper and lower bounds for this problem. Assuming the noise is independent across different coordinates, the authors of @cite_1 improved upon the results obtained in @cite_21 . For the bandit feedback case, the authors of @cite_8 gives algorithms that require brute-force search over the action space with instance dependent regret @math . For adversarial combinatorial bandits, the authors of @cite_24 presented the efficient and optimal algorithm for the semi-bandit feedback case while the authors of @cite_10 described an optimal algorithm for the bandit feedback case, but its computational complexity scales linearly with the number of actions.
@cite_14 @cite_8 @cite_9 @cite_21 @cite_1 @cite_32 @cite_24 @cite_10
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We investigate a nonstochastic bandit setting in which the loss of an action is not immediately charged to the player, but rather spread over at most d consecutive steps in an adversarial way. This implies that the instantaneous loss observed by the player at the end of each round is a sum of as many as d loss components of previously played actions. Hence, unlike the standard bandit setting with delayed feedback, here the player cannot observe the individual delayed losses, but only their sum. Our main contribution is a general reduction transforming a standard bandit algorithm into one that can operate in this harder setting. We also show how the regret of the transformed algorithm can be bounded in terms of the regret of the original algorithm. Our reduction cannot be improved in general: we prove a lower bound on the regret of any bandit algorithm in this setting that matches (up to log factors) the upper bound obtained via our reduction. Finally, we show how our reduction can be extended to more complex bandit settings, such as combinatorial linear bandits and online bandit convex optimization. This paper investigates stochastic and adversarial combinatorial multi-armed bandit problems. In the stochastic setting under semi-bandit feedback, we derive a problem-specific regret lower bound, and discuss its scaling with the dimension of the decision space. We propose ESCB, an algorithm that efficiently exploits the structure of the problem and provide a finite-time analysis of its regret. ESCB has better performance guarantees than existing algorithms, and significantly outperforms these algorithms in practice. In the adversarial setting under bandit feedback, we propose COMBEXP, an algorithm with the same regret scaling as state-of-the-art algorithms, but with lower computational complexity for some combinatorial problems. We address the online linear optimization problem when the actions of the forecaster are represented by binary vectors. Our goal is to understand the magnitude of the minimax regret for the worst possible set of actions. We study the problem under three different assumptions for the feedback: full information, and the partial information models of the so-called "semi-bandit", and "bandit" problems. We consider both @math -, and @math -type of restrictions for the losses assigned by the adversary. We formulate a general strategy using Bregman projections on top of a potential-based gradient descent, which generalizes the ones studied in the series of papers (2007), (2008), (2008), Cesa-Bianchi and Lugosi (2009), Helmbold and Warmuth (2009), (2010), (2010), (2010) and Audibert and Bubeck (2010). We provide simple proofs that recover most of the previous results. We propose new upper bounds for the semi-bandit game. Moreover we derive lower bounds for all three feedback assumptions. With the only exception of the bandit game, the upper and lower bounds are tight, up to a constant factor. Finally, we answer a question asked by (2010) by showing that the exponentially weighted average forecaster is suboptimal against @math adversaries. We address online linear optimization problems when the possible actions of the decision maker are represented by binary vectors. The regret of the decision maker is the difference between her realized loss and the minimal loss she would have achieved by picking, in hindsight, the best possible action. Our goal is to understand the magnitude of the best possible minimax regret. We study the problem under three different assumptions for the feedback the decision maker receives: full information, and the partial information models of the so-called semi-bandit and bandit problems. In the full information case we show that the standard exponentially weighted average forecaster is a provably suboptimal strategy. For the semi-bandit model, by combining the Mirror Descent algorithm and the INF Implicitely Normalized Forecaster strategy, we are able to prove the first optimal bounds. Finally, in the bandit case we discuss existing results in light of a new lower bound, and suggest a conjecture on the optimal regret in that case. We consider online learning problems under a a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem. We study combinatorial multi-armed bandit with probabilistically triggered arms (CMAB-T) and semi-bandit feedback. We resolve a serious issue in the prior CMAB-T studies where the regret bounds contain a possibly exponentially large factor of 1 p*, where p* is the minimum positive probability that an arm is triggered by any action. We address this issue by introducing a triggering probability modulated (TPM) bounded smoothness condition into the influence maximization bandit and combinatorial cascading bandit satisfy this TPM condition. As a result, we completely remove the factor of 1 p* from the regret bounds, achieving significantly better regret bounds for influence maximization and cascading bandits than before. Finally, we provide lower bound results showing that the factor 1 p* is unavoidable for general CMAB-T problems, suggesting that the TPM condition is crucial in removing this factor. We study the problem of stochastic combinatorial pure exploration (CPE), where an agent sequentially pulls a set of single arms (a.k.a. a super arm) and tries to find the best super arm. Among a variety of problem settings of the CPE, we focus on the full-bandit setting, where we cannot observe the reward of each single arm, but only the sum of the rewards. Although we can regard the CPE with full-bandit feedback as a special case of pure exploration in linear bandits, an approach based on linear bandits is not computationally feasible since the number of super arms may be exponential. In this paper, we first propose a polynomial-time bandit algorithm for the CPE under general combinatorial constraints and provide an upper bound of the sample complexity. Second, we design an approximation algorithm for the 0-1 quadratic maximization problem, which arises in many bandit algorithms with confidence ellipsoids. Based on our approximation algorithm, we propose novel bandit algorithms for the top-k selection problem, and prove that our algorithms run in polynomial time. Finally, we conduct experiments on synthetic and real-world datasets, and confirm the validity of our theoretical analysis in terms of both the computation time and the sample complexity. We present and study a partial-information model of online learning, where a decision maker repeatedly chooses from a finite set of actions, and observes some subset of the associated losses. This naturally models several situations where the losses of different actions are related, and knowing the loss of one action provides information on the loss of other actions. Moreover, it generalizes and interpolates between the well studied full-information setting (where all losses are revealed) and the bandit setting (where only the loss of the action chosen by the player is revealed). We provide several algorithms addressing different variants of our setting, and provide tight regret bounds depending on combinatorial properties of the information feedback structure.