Sepehr Elahi

Sepehr Elahi, Baran Atalar, Sevda Öğüt et al.

In federated multi-armed bandit problems, maximizing global reward while satisfying minimum privacy requirements to protect clients is the main goal. To formulate such problems, we consider a combinatorial contextual bandit setting with groups and changing action sets, where similar base arms arrive in groups and a set of base arms, called a super arm, must be chosen in each round to maximize super arm reward while satisfying the constraints of the rewards of groups from which base arms were chosen. To allow for greater flexibility, we let each base arm have two outcomes, modeled as the output of a two-output Gaussian process (GP), where one outcome is used to compute super arm reward and the other for group reward. We then propose a novel double-UCB GP-bandit algorithm, called Thresholded Combinatorial Gaussian Process Upper Confidence Bounds (TCGP-UCB), which balances between maximizing cumulative super arm reward and satisfying group reward constraints and can be tuned to prefer one over the other. We also define a new notion of regret that combines super arm regret with group reward constraint satisfaction and prove that TCGP-UCB incurs $\tilde{O}(\sqrt{λ^*(K)KT\overlineγ_{T}} )$ regret with high probability, where $\overlineγ_{T}$ is the maximum information gain associated with the set of base arm contexts that appeared in the first $T$ rounds and $K$ is the maximum super arm cardinality over all rounds. We lastly show in experiments using synthetic and real-world data and based on a federated learning setup as well as a content-recommendation one that our algorithm performs better then the current non-GP state-of-the-art combinatorial bandit algorithm, while satisfying group constraints.

Andi Nika, Sepehr Elahi, Cem Tekin

We consider a contextual bandit problem with a combinatorial action set and time-varying base arm availability. At the beginning of each round, the agent observes the set of available base arms and their contexts and then selects an action that is a feasible subset of the set of available base arms to maximize its cumulative reward in the long run. We assume that the mean outcomes of base arms are samples from a Gaussian Process (GP) indexed by the context set ${\cal X}$, and the expected reward is Lipschitz continuous in expected base arm outcomes. For this setup, we propose an algorithm called Optimistic Combinatorial Learning and Optimization with Kernel Upper Confidence Bounds (O'CLOK-UCB) and prove that it incurs $\tilde{O}(\sqrt{λ^*(K)KTγ_{KT}(\cup_{t\leq T}\mathcal{X}_t)} )$ regret with high probability, where $γ_{KT}(\cup_{t\leq T}\mathcal{X}_t)$ is the maximum information gain associated with the sets of base arm contexts $\mathcal{X}_t$ that appeared in the first $T$ rounds, $K$ is the maximum cardinality of any feasible action over all rounds, and $λ^*(K)$ is the maximum eigenvalue of all covariance matrices of selected actions up to time $T$, which is a function of $K$. To dramatically speed up the algorithm, we also propose a variant of O'CLOK-UCB that uses sparse GPs. Finally, we experimentally show that both algorithms exploit inter-base arm outcome correlation and vastly outperform the previous state-of-the-art UCB-based algorithms in realistic setups.

Andi Nika, Sepehr Elahi, Çağın Ararat et al.

We consider the problem of optimizing a vector-valued objective function $\boldsymbol{f}$ sampled from a Gaussian Process (GP) whose index set is a well-behaved, compact metric space $({\cal X},d)$ of designs. We assume that $\boldsymbol{f}$ is not known beforehand and that evaluating $\boldsymbol{f}$ at design $x$ results in a noisy observation of $\boldsymbol{f}(x)$. Since identifying the Pareto optimal designs via exhaustive search is infeasible when the cardinality of ${\cal X}$ is large, we propose an algorithm, called Adaptive $\boldsymbolε$-PAL, that exploits the smoothness of the GP-sampled function and the structure of $({\cal X},d)$ to learn fast. In essence, Adaptive $\boldsymbolε$-PAL employs a tree-based adaptive discretization technique to identify an $\boldsymbolε$-accurate Pareto set of designs in as few evaluations as possible. We provide both information-type and metric dimension-type bounds on the sample complexity of $\boldsymbolε$-accurate Pareto set identification. We also experimentally show that our algorithm outperforms other Pareto set identification methods.