Combinatorial Allocation Bandits with Nonlinear Arm Utility

We study bandit learning in matching markets, where players and arms constitute the two market sides, and the players' utilities are linear in the arm contexts. In each round, new arms arrive with observable contexts. Then, the algorithm matches them to players, aiming to minimize each player's regret against a stable matching benchmark. This contextual structure creates significant complexity: subtle context shifts can slightly alter one player's utility while completely reconfiguring the underlying benchmark, causing large regret spikes for others. We address this in two settings: stochastic contexts, drawn from a latent distribution, and adversarial contexts, which may be arbitrary. For the stochastic case, we introduce a novel minimum preference gap to capture learning difficulty and provide a fully adaptive algorithm with an instance-dependent poly-logarithmic regret upper bound. We also establish matching instance-independent regret upper and lower bounds under a mild distributional assumption. For the adversarial setting, we propose a tractable regret notion that remains valid under arbitrary contexts and achieves an instance-independent sublinear regret bound via an adaptive algorithm.

The Competing Bandits framework is a recently emerging area that integrates multi-armed bandits in online learning with stable matching in game theory. While conventional models assume that all players and arms are constantly available, in real-world problems, their availability can vary arbitrarily over time. In this paper, we formulate this setting as Sleeping Competing Bandits. To analyze this problem, we naturally extend the regret definition used in existing competing bandits and derive regret bounds for the proposed model. We propose an algorithm that simultaneously achieves an asymptotic regret bound of $\mathrm{O}\left(NK\log T_{i}/Δ^2\right)$ under reasonable assumptions, where $N$ is the number of players, $K$ is the number of arms, $T_{i}$ is the number of rounds of each player $p_i$, and $Δ$ is the minimum reward gap. We also provide a regret lower bound of $\mathrmΩ\left( N(K-N+1)\log T_{i}/Δ^2 \right)$ under the same assumptions. This implies that our algorithm is asymptotically optimal in the regime where the number of arms $K$ is relatively larger than the number of players $N$.

Combinatorial multi-armed bandits provide a fundamental online decision-making environment where a decision-maker interacts with an environment across $T$ time steps, each time selecting an action and learning the cost of that action. The goal is to minimize regret, defined as the loss compared to the optimal fixed action in hindsight under full-information. There has been substantial interest in leveraging what is known about offline algorithm design in this online setting. Offline greedy and linear optimization algorithms (both exact and approximate) have been shown to provide useful guarantees when deployed online. We investigate local search methods, a broad class of algorithms used widely in both theory and practice, which have thus far been under-explored in this context. We focus on problems where offline local search terminates in an approximately optimal solution and give a generic method for converting such an offline algorithm into an online stochastic combinatorial bandit algorithm with $O(\log^3 T)$ (approximate) regret. In contrast, existing offline-to-online frameworks yield regret (and approximate regret) which depend sub-linearly, but polynomially on $T$. We demonstrate the flexibility of our framework by applying it to three online stochastic combinatorial optimization problems: scheduling to minimize total completion time, finding a minimum cost base of a matroid and uncertain clustering.