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We explore a novel setting of the Multi-Armed Bandit (MAB) problem inspired from real world applications which we call bandits with stochastic delayed composite anonymous feedback (SDCAF). In SDCAF, the rewards on pulling arms are stochastic with respect to time but spread over a fixed number of time steps in the future after pulling the arm. The complexity of this problem stems from the anonymous feedback to the player and the stochastic generation of the reward. Due to the aggregated nature of the rewards, the player is unable to associate the reward to a particular time step from the past. We present two algorithms for this more complicated setting of SDCAF using phase based extensions of the UCB algorithm. We perform regret analysis to show sub-linear theoretical guarantees on both the algorithms.
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We study a constrained contextual linear bandit setting, where the goal of the agent is to produce a sequence of policies, whose expected cumulative reward over the course of $T$ rounds is maximum, and each has an expected cost below a certain threshold $tau$. We propose an upper-confidence bound algorithm for this problem, called optimistic pessimistic linear bandit (OPLB), and prove an $widetilde{mathcal{O}}(frac{dsqrt{T}}{tau-c_0})$ bound on its $T$-round regret, where the denominator is the difference between the constraint threshold and the cost of a known feasible action. We further specialize our results to multi-armed bandits and propose a computationally efficient algorithm for this setting. We prove a regret bound of $widetilde{mathcal{O}}(frac{sqrt{KT}}{tau - c_0})$ for this algorithm in $K$-armed bandits, which is a $sqrt{K}$ improvement over the regret bound we obtain by simply casting multi-armed bandits as an instance of contextual linear bandits and using the regret bound of OPLB. We also prove a lower-bound for the problem studied in the paper and provide simulations to validate our theoretical results.
We propose a generalization of the best arm identification problem in stochastic multi-armed bandits (MAB) to the setting where every pull of an arm is associated with delayed feedback. The delay in feedback increases the effective sample complexity of standard algorithms, but can be offset if we have access to partial feedback received before a pull is completed. We propose a general framework to model the relationship between partial and delayed feedback, and as a special case we introduce efficient algorithms for settings where the partial feedback are biased or unbiased estimators of the delayed feedback. Additionally, we propose a novel extension of the algorithms to the parallel MAB setting where an agent can control a batch of arms. Our experiments in real-world settings, involving policy search and hyperparameter optimization in computational sustainability domains for fast charging of batteries and wildlife corridor construction, demonstrate that exploiting the structure of partial feedback can lead to significant improvements over baselines in both sequential and parallel MAB.
We study the adversarial multi-armed bandit problem where partial observations are available and where, in addition to the loss incurred for each action, a emph{switching cost} is incurred for shifting to a new action. All previously known results incur a factor proportional to the independence number of the feedback graph. We give a new algorithm whose regret guarantee depends only on the domination number of the graph. We further supplement that result with a lower bound. Finally, we also give a new algorithm with improved policy regret bounds when partial counterfactual feedback is available.
We study a decentralized cooperative stochastic multi-armed bandit problem with $K$ arms on a network of $N$ agents. In our model, the reward distribution of each arm is the same for each agent and rewards are drawn independently across agents and time steps. In each round, each agent chooses an arm to play and subsequently sends a message to her neighbors. The goal is to minimize the overall regret of the entire network. We design a fully decentralized algorithm that uses an accelerated consensus procedure to compute (delayed) estimates of the average of rewards obtained by all the agents for each arm, and then uses an upper confidence bound (UCB) algorithm that accounts for the delay and error of the estimates. We analyze the regret of our algorithm and also provide a lower bound. The regret is bounded by the optimal centralized regret plus a natural and simple term depending on the spectral gap of the communication matrix. Our algorithm is simpler to analyze than those proposed in prior work and it achieves better regret bounds, while requiring less information about the underlying network. It also performs better empirically.
Bandit algorithms have various application in safety-critical systems, where it is important to respect the system constraints that rely on the bandits unknown parameters at every round. In this paper, we formulate a linear stochastic multi-armed bandit problem with safety constraints that depend (linearly) on an unknown parameter vector. As such, the learner is unable to identify all safe actions and must act conservatively in ensuring that her actions satisfy the safety constraint at all rounds (at least with high probability). For these bandits, we propose a new UCB-based algorithm called Safe-LUCB, which includes necessary modifications to respect safety constraints. The algorithm has two phases. During the pure exploration phase the learner chooses her actions at random from a restricted set of safe actions with the goal of learning a good approximation of the entire unknown safe set. Once this goal is achieved, the algorithm begins a safe exploration-exploitation phase where the learner gradually expands their estimate of the set of safe actions while controlling the growth of regret. We provide a general regret bound for the algorithm, as well as a problem dependent bound that is connected to the location of the optimal action within the safe set. We then propose a modified heuristic that exploits our problem dependent analysis to improve the regret.
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