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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.
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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.
We study decentralized stochastic linear bandits, where a network of $N$ agents acts cooperatively to efficiently solve a linear bandit-optimization problem over a $d$-dimensional space. For this problem, we propose DLUCB: a fully decentralized algorithm that minimizes the cumulative regret over the entire network. At each round of the algorithm each agent chooses its actions following an upper confidence bound (UCB) strategy and agents share information with their immediate neighbors through a carefully designed consensus procedure that repeats over cycles. Our analysis adjusts the duration of these communication cycles ensuring near-optimal regret performance $mathcal{O}(dlog{NT}sqrt{NT})$ at a communication rate of $mathcal{O}(dN^2)$ per round. The structure of the network affects the regret performance via a small additive term - coined the regret of delay - that depends on the spectral gap of the underlying graph. Notably, our results apply to arbitrary network topologies without a requirement for a dedicated agent acting as a server. In consideration of situations with high communication cost, we propose RC-DLUCB: a modification of DLUCB with rare communication among agents. The new algorithm trades off regret performance for a significantly reduced total communication cost of $mathcal{O}(d^3N^{2.5})$ over all $T$ rounds. Finally, we show that our ideas extend naturally to the emerging, albeit more challenging, setting of safe bandits. For the recently studied problem of linear bandits with unknown linear safety constraints, we propose the first safe decentralized algorithm. Our study contributes towards applying bandit techniques in safety-critical distributed systems that repeatedly deal with unknown stochastic environments. We present numerical simulations for various network topologies that corroborate our theoretical findings.
We propose a new online algorithm for minimizing the cumulative regret in stochastic linear bandits. The key idea is to build a perturbed history, which mixes the history of observed rewards with a pseudo-history of randomly generated i.i.d. pseudo-rewards. Our algorithm, perturbed-history exploration in a linear bandit (LinPHE), estimates a linear model from its perturbed history and pulls the arm with the highest value under that model. We prove a $tilde{O}(d sqrt{n})$ gap-free bound on the expected $n$-round regret of LinPHE, where $d$ is the number of features. Our analysis relies on novel concentration and anti-concentration bounds on the weighted sum of Bernoulli random variables. To show the generality of our design, we extend LinPHE to a logistic reward model. We evaluate both algorithms empirically and show that they are practical.
In the stochastic linear contextual bandit setting there exist several minimax procedures for exploration with policies that are reactive to the data being acquired. In practice, there can be a significant engineering overhead to deploy these algorithms, especially when the dataset is collected in a distributed fashion or when a human in the loop is needed to implement a different policy. Exploring with a single non-reactive policy is beneficial in such cases. Assuming some batch contexts are available, we design a single stochastic policy to collect a good dataset from which a near-optimal policy can be extracted. We present a theoretical analysis as well as numerical experiments on both synthetic and real-world datasets.
Modifying the reward-biased maximum likelihood method originally proposed in the adaptive control literature, we propose novel learning algorithms to handle the explore-exploit trade-off in linear bandits problems as well as generalized linear bandits problems. We develop novel index policies that we prove achieve order-optimality, and show that they achieve empirical performance competitive with the state-of-the-art benchmark methods in extensive experiments. The new policies achieve this with low computation time per pull for linear bandits, and thereby resulting in both favorable regret as well as computational efficiency.
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