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On the Low-Complexity of Fair Learning for Combinatorial Multi-Armed Bandit

Xiaoyi Wu, Bo Ji, Bin Li

TL;DR

This work tackles CMAB with fairness constraints where the set of feasible super arms can grow exponentially, making traditional PO strategies computationally prohibitive. It introduces LCFL, a PC-inspired algorithm that randomly samples a fixed number $M$ of feasible subsets each round and selects the best among these (plus the prior choice), thereby achieving constant per-round comparisons while maintaining strong regret and fairness performance. Theoretical analysis provides a regret bound of the form $\text{Reg}(T)=\tilde{O}(\sqrt{T})$ with constants depending on $\eta$, $M$, and the sampling fraction $\alpha(M)=M/|\mathcal{S}|$, and establishes zero cumulative fairness violation after a time $t_0=O(\eta/\epsilon)$ under a suitable $\epsilon$. The results reveal a clear tradeoff: increasing $M$ improves both regret and fairness timing at higher computation cost, while increasing $\eta$ reduces regret but delays fairness satisfaction. Extensive simulations, including synthetic and motion-trace-based experiments, validate these findings and demonstrate the approach’s practicality in wireless networks with interference constraints.

Abstract

Combinatorial Multi-Armed Bandit with fairness constraints is a framework where multiple arms form a super arm and can be pulled in each round under uncertainty to maximize cumulative rewards while ensuring the minimum average reward required by each arm. The existing pessimistic-optimistic algorithm linearly combines virtual queue-lengths (tracking the fairness violations) and Upper Confidence Bound estimates as a weight for each arm and selects a super arm with the maximum total weight. The number of super arms could be exponential to the number of arms in many scenarios. In wireless networks, interference constraints can cause the number of super arms to grow exponentially with the number of arms. Evaluating all the feasible super arms to find the one with the maximum total weight can incur extremely high computational complexity in the pessimistic-optimistic algorithm. To avoid this, we develop a low-complexity fair learning algorithm based on the so-called pick-and-compare approach that involves randomly picking $M$ feasible super arms to evaluate. By setting $M$ to a constant, the number of comparison steps in the pessimistic-optimistic algorithm can be reduced to a constant, thereby significantly reducing the computational complexity. Our theoretical proof shows this low-complexity design incurs only a slight sacrifice in fairness and regret performance. Finally, we validate the theoretical result by extensive simulations.

On the Low-Complexity of Fair Learning for Combinatorial Multi-Armed Bandit

TL;DR

This work tackles CMAB with fairness constraints where the set of feasible super arms can grow exponentially, making traditional PO strategies computationally prohibitive. It introduces LCFL, a PC-inspired algorithm that randomly samples a fixed number of feasible subsets each round and selects the best among these (plus the prior choice), thereby achieving constant per-round comparisons while maintaining strong regret and fairness performance. Theoretical analysis provides a regret bound of the form with constants depending on , , and the sampling fraction , and establishes zero cumulative fairness violation after a time under a suitable . The results reveal a clear tradeoff: increasing improves both regret and fairness timing at higher computation cost, while increasing reduces regret but delays fairness satisfaction. Extensive simulations, including synthetic and motion-trace-based experiments, validate these findings and demonstrate the approach’s practicality in wireless networks with interference constraints.

Abstract

Combinatorial Multi-Armed Bandit with fairness constraints is a framework where multiple arms form a super arm and can be pulled in each round under uncertainty to maximize cumulative rewards while ensuring the minimum average reward required by each arm. The existing pessimistic-optimistic algorithm linearly combines virtual queue-lengths (tracking the fairness violations) and Upper Confidence Bound estimates as a weight for each arm and selects a super arm with the maximum total weight. The number of super arms could be exponential to the number of arms in many scenarios. In wireless networks, interference constraints can cause the number of super arms to grow exponentially with the number of arms. Evaluating all the feasible super arms to find the one with the maximum total weight can incur extremely high computational complexity in the pessimistic-optimistic algorithm. To avoid this, we develop a low-complexity fair learning algorithm based on the so-called pick-and-compare approach that involves randomly picking feasible super arms to evaluate. By setting to a constant, the number of comparison steps in the pessimistic-optimistic algorithm can be reduced to a constant, thereby significantly reducing the computational complexity. Our theoretical proof shows this low-complexity design incurs only a slight sacrifice in fairness and regret performance. Finally, we validate the theoretical result by extensive simulations.
Paper Structure (13 sections, 6 theorems, 77 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 13 sections, 6 theorems, 77 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. system model
  3. Algorithm Design
  4. Performance Analysis
  5. Simulations
  6. Synthetic Simulations
  7. Motion Trace-based Multi-user Interactive and Panoramic Scene Delivery Simulations
  8. Conclusion
  9. Proof of Lemma \ref{['lemma:exp_time_gap']}
  10. Proof of Proposition \ref{['prop:regret']}
  11. Proof of Lemma \ref{['lemma:large_prob_weight_gap']}
  12. Proof of Proposition \ref{['prop:zero_violation']}
  13. Proof of Lemma \ref{['lemma:drift']}

Key Result

Lemma 1

Denote $\mathcal{T}_0\triangleq\left\{t_{\tau}\right\}_{\tau=0,1,2, \ldots,\tau'}$ as the rounds when the subsets chosen at random include the one that has the maximum weight, i.e., $\mathbf{{S}^{\dagger}}(t_{\tau}) \in \mathcal{R}(t_{\tau})$. Here, $\mathbf{{S}^{\dagger}}(t)\in\operatornamewithlimi where $\alpha(M)\triangleq M/|\mathcal{S}|$ denotes the probability that the subsets chosen at rand

Figures (4)

  • Figure 1: Synthetic simulations: impact of parameter $\eta$.
  • Figure 2: Synthetic simulations: impact of parameter $M$.
  • Figure 3: Multi-user panoramic scene delivery: impact of parameter $\eta$.
  • Figure 4: Multi-user panoramic scene delivery: impact of $M$.

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Remark 1
  • Lemma 3
  • Proposition 2
  • Remark 2
  • Remark 3
  • Lemma 4