Multi-armed Bandit Requiring Monotone Arm Sequences

TL;DR

This paper studies a continuum-armed bandit problem with a monotonicity constraint on the arm sequence, showing that such constraints increase regret from the standard optimal rate, with specific bounds depending on the objective function.

Contribution

It introduces algorithms for monotone arm sequences in continuum-armed bandits and establishes regret bounds, highlighting the impact of monotonicity on learning efficiency.

Findings

Regret is $O(T)$ for Lipschitz continuous objectives.

Regret is $ ilde O(T^{3/4})$ for unimodal or quasiconcave objectives.

Monotonicity constraints increase regret compared to unconstrained bandits.

Abstract

In many online learning or multi-armed bandit problems, the taken actions or pulled arms are ordinal and required to be monotone over time. Examples include dynamic pricing, in which the firms use markup pricing policies to please early adopters and deter strategic waiting, and clinical trials, in which the dose allocation usually follows the dose escalation principle to prevent dose limiting toxicities. We consider the continuum-armed bandit problem when the arm sequence is required to be monotone. We show that when the unknown objective function is Lipschitz continuous, the regret is $O(T)$. When in addition the objective function is unimodal or quasiconcave, the regret is $\tilde O(T^{3/4})$ under the proposed algorithm, which is also shown to be the optimal rate. This deviates from the optimal rate $\tilde O(T^{2/3})$ in the continuous-armed bandit literature and demonstrates the cost to the learning efficiency brought by the monotonicity requirement.