A Frequency-Domain Analysis of the Multi-Armed Bandit Problem: A New Perspective on the Exploration-Exploitation Trade-off
Di Zhang
TL;DR
This paper reframes the stochastic multi-armed bandit problem from a time-domain regret focus to a frequency-domain perspective, modeling arm reward sequences as spectral components and learning as adaptive filtering. It establishes a formal Frequency-Domain Bandit Model and proves that the UCB exploration term acts as a time-varying high-pass gain on the uncertainty spectrum, with a decay proportional to the visit count. The authors derive finite-time dynamic bounds on the evolution of the policy spectrum and argue for an optimal exploration-rate decay, offering a fresh, physically intuitive interpretation of exploration-exploitation and guidance for adaptive parameter design. The framework opens avenues for new algorithms, such as frequency-domain adaptive variants, and suggests extensions to nonlinear filtering and non-stationary environments, marking a foundational step toward a unified spectral theory of decision-making.
Abstract
The stochastic multi-armed bandit (MAB) problem is one of the most fundamental models in sequential decision-making, with the core challenge being the trade-off between exploration and exploitation. Although algorithms such as Upper Confidence Bound (UCB) and Thompson Sampling, along with their regret theories, are well-established, existing analyses primarily operate from a time-domain and cumulative regret perspective, struggling to characterize the dynamic nature of the learning process. This paper proposes a novel frequency-domain analysis framework, reformulating the bandit process as a signal processing problem. Within this framework, the reward estimate of each arm is viewed as a spectral component, with its uncertainty corresponding to the component's frequency, and the bandit algorithm is interpreted as an adaptive filter. We construct a formal Frequency-Domain Bandit Model and prove the main theorem: the confidence bound term in the UCB algorithm is equivalent in the frequency domain to a time-varying gain applied to uncertain spectral components, a gain inversely proportional to the square root of the visit count. Based on this, we further derive finite-time dynamic bounds concerning the exploration rate decay. This theory not only provides a novel and intuitive physical interpretation for classical algorithms but also lays a rigorous theoretical foundation for designing next-generation algorithms with adaptive parameter adjustment.