The Max $K$-Armed Bandit: A PAC Lower Bound and tighter Algorithms
This work addresses sample efficiency in multi-armed bandit problems for reinforcement learning and decision-making, but it is incremental as it builds on existing PAC frameworks.
The paper tackles the Max K-Armed Bandit problem by establishing a PAC lower bound on sample complexity for finding the best reward and proposes algorithms that achieve this bound up to logarithmic factors, with comparisons showing random arm selection can outperform when maximal rewards are similar.
We consider the Max $K$-Armed Bandit problem, where a learning agent is faced with several sources (arms) of items (rewards), and interested in finding the best item overall. At each time step the agent chooses an arm, and obtains a random real valued reward. The rewards of each arm are assumed to be i.i.d., with an unknown probability distribution that generally differs among the arms. Under the PAC framework, we provide lower bounds on the sample complexity of any $(ε,δ)$-correct algorithm, and propose algorithms that attain this bound up to logarithmic factors. We compare the performance of this multi-arm algorithms to the variant in which the arms are not distinguishable by the agent and are chosen randomly at each stage. Interestingly, when the maximal rewards of the arms happen to be similar, the latter approach may provide better performance.