Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Gossiping Insert-Eliminate Algorithm for Multi-Agent Bandits

Ronshee Chawla, Abishek Sankararaman, Ayalvadi Ganesh, Sanjay Shakkottai

TL;DR

This work introduces GosInE, gossip-based insert-eliminate algorithms for decentralized multi-agent MABs where agents limit themselves to a small subset of arms and exchange only arm-IDs. The authors show that with a connected gossip network and a budget that grows at least logarithmically with time, each agent achieves a regret that scales like $O\left(\left(\sum_{j=2}^{\lceil K/N\rceil+2} \frac{1}{\Delta_j}\right) \alpha \ln T\right)$ up to a problem-dependent constant, effectively reducing regret by a factor of about $N$ compared to isolation. They provide a tight lower bound, analyze a network-conductance-driven second-order term, and demonstrate that higher connectivity and budget improve performance; asynchronous variants and initialization without agent IDs enhance practicality. Empirical results on synthetic and real data confirm that even modest collaboration yields substantial regret reductions, validating the fundamental benefit of minimal inter-agent learning.

Abstract

We consider a decentralized multi-agent Multi Armed Bandit (MAB) setup consisting of $N$ agents, solving the same MAB instance to minimize individual cumulative regret. In our model, agents collaborate by exchanging messages through pairwise gossip style communications on an arbitrary connected graph. We develop two novel algorithms, where each agent only plays from a subset of all the arms. Agents use the communication medium to recommend only arm-IDs (not samples), and thus update the set of arms from which they play. We establish that, if agents communicate $Ω(\log(T))$ times through any connected pairwise gossip mechanism, then every agent's regret is a factor of order $N$ smaller compared to the case of no collaborations. Furthermore, we show that the communication constraints only have a second order effect on the regret of our algorithm. We then analyze this second order term of the regret to derive bounds on the regret-communication tradeoffs. Finally, we empirically evaluate our algorithm and conclude that the insights are fundamental and not artifacts of our bounds. We also show a lower bound which gives that the regret scaling obtained by our algorithm cannot be improved even in the absence of any communication constraints. Our results thus demonstrate that even a minimal level of collaboration among agents greatly reduces regret for all agents.

The Gossiping Insert-Eliminate Algorithm for Multi-Agent Bandits

TL;DR

This work introduces GosInE, gossip-based insert-eliminate algorithms for decentralized multi-agent MABs where agents limit themselves to a small subset of arms and exchange only arm-IDs. The authors show that with a connected gossip network and a budget that grows at least logarithmically with time, each agent achieves a regret that scales like up to a problem-dependent constant, effectively reducing regret by a factor of about compared to isolation. They provide a tight lower bound, analyze a network-conductance-driven second-order term, and demonstrate that higher connectivity and budget improve performance; asynchronous variants and initialization without agent IDs enhance practicality. Empirical results on synthetic and real data confirm that even modest collaboration yields substantial regret reductions, validating the fundamental benefit of minimal inter-agent learning.

Abstract

We consider a decentralized multi-agent Multi Armed Bandit (MAB) setup consisting of agents, solving the same MAB instance to minimize individual cumulative regret. In our model, agents collaborate by exchanging messages through pairwise gossip style communications on an arbitrary connected graph. We develop two novel algorithms, where each agent only plays from a subset of all the arms. Agents use the communication medium to recommend only arm-IDs (not samples), and thus update the set of arms from which they play. We establish that, if agents communicate times through any connected pairwise gossip mechanism, then every agent's regret is a factor of order smaller compared to the case of no collaborations. Furthermore, we show that the communication constraints only have a second order effect on the regret of our algorithm. We then analyze this second order term of the regret to derive bounds on the regret-communication tradeoffs. Finally, we empirically evaluate our algorithm and conclude that the insights are fundamental and not artifacts of our bounds. We also show a lower bound which gives that the regret scaling obtained by our algorithm cannot be improved even in the absence of any communication constraints. Our results thus demonstrate that even a minimal level of collaboration among agents greatly reduces regret for all agents.

Paper Structure

This paper contains 33 sections, 26 theorems, 101 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Synchronous GosInE Algorithm
  4. Model Assumptions
  5. Regret Guarantee
  6. Discussion
  7. Proof Sketch
  8. Initialization without Agent IDs
  9. Asynchronous GosInE Algorithm
  10. Proof Sketch
  11. Lower Bound
  12. Insights
  13. Numerical Results
  14. Related Work
  15. Conclusions
  16. ...and 18 more sections

Key Result

Theorem 1

Suppose in a system of $N \geq 2$ agents connected by a communication matrix $P$ satisfying assumption (A.1) and $K \geq 2$ arms, each agent runs Algorithm algo:main-algo, with UCB parameter $\alpha > 3$ and communication budget $(B_t)_{t \in \mathbb{N}}$ and $\varepsilon > 0$ satisfying assumption where $(A_x)_{x \in \mathbb{N}}$ is given in Equation (eqn:A_x_defn) and $g((A_x)_{x \in \mathbb{N}

Figures (4)

  • Figure 1: $(N, K)$ are $(25, 75)$ and $(15, 50)$ respectively.
  • Figure 2: $(N, K)$ are $(25, 75)$ and $(15, 50)$ respectively.
  • Figure 3: $(N, K)$ as $(20, 70)$ and $(5, 20)$ and the graphs are complete and cycle respectively.
  • Figure 4: $(N, K)$ are $(10, 30)$ and $(15, 40)$ respectively.

Theorems & Definitions (47)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Corollary 6
  • Corollary 7
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 37 more