预算约束动态Colonel Blotto博弈中的在线学习
Online Learning in Budget-Constrained Dynamic Colonel Blotto Games
AI总结：
本文研究预算约束动态Colonel Blotto博弈中的在线资源分配问题，将其转化为有向图路径规划，结合组合老虎机与带背包的老虎机算法求解，理论证明遗憾值具次线性上界并经仿真验证。
摘要：
本文在动态环境下使用Colonel Blotto game (CBG) 研究有限资源的策略分配，并采用在线学习方法分析该问题。在此模型中，一名参与者为学习者，其在有限时间范围内拥有有限的兵力可供分配，另一名参与者为对手。在每一轮中，学习者与对手进行一次单回合CBG，并根据过去的观测在战场间策略性地分配兵力。对手从某种学习者未知的固定分布中随机选择其分配动作。学习者的目标是最小化其遗憾值，即在遵循学习算法且不违反预算约束的情况下，最优混合策略的累积奖励与实际累积奖励之差。动态CBG中的学习在组合老虎机和带背包的老虎机框架下进行分析。我们首先将预算约束的动态CBG转化为有向图上的路径规划问题。随后，我们设计了一种高效算法，结合用于路径规划问题的特殊组合老虎机算法和带背包的老虎机算法来应对预算约束。理论分析表明，学习者的遗憾值被一个在时间范围内呈次线性且在其他参数上呈多项式的项所限制。最后，我们通过对多种场景进行仿真验证了理论结果。
英文摘要：
In this paper, we study the strategic allocation of limited resources using a Colonel Blotto game (CBG) under a dynamic setting and analyze the problem using an online learning approach. In this model, one of the players is a learner who has limited troops to allocate over a finite time horizon, and the other player is an adversary. In each round, the learner plays a one-shot Colonel Blotto game with the adversary and strategically determines the allocation of troops among battlefields based on past observations. The adversary chooses its allocation action randomly from some fixed distribution that is unknown to the learner. The learner's objective is to minimize its regret, which is the difference between the cumulative reward of the best mixed strategy and the realized cumulative reward by following a learning algorithm while not violating the budget constraint. The learning in dynamic CBG is analyzed under the framework of combinatorial bandits and bandits with knapsacks. We first convert the budget-constrained dynamic CBG to a path planning problem on a directed graph. We then devise an efficient algorithm that combines a special combinatorial bandit algorithm for path planning problem and a bandits with knapsack algorithm to cope with the budget constraint. The theoretical analysis shows that the learner's regret is bounded by a term sublinear in time horizon and polynomial in other parameters. Finally, we justify our theoretical results by carrying out simulations for various scenarios.
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