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Distributed Cooperative Decision-Making in Multiarmed Bandits: Frequentist and Bayesian Algorithms
Peter Landgren, Vaibhav Srivastava, Naomi Ehrich Leonard
TL;DR
Distributed MABs extend the explore–exploit problem to communicating agents, a setting motivated by prior research that primarily focused on single agents. The paper proposes cooperative frequentist and Bayesian algorithms using running consensus, and reports order-optimal group performance with graph-dependent variation in individual performance.
Problem
MAB research has primarily focused on single agents, while networked systems motivate distributed algorithms for multiple communicating agents facing the explore–exploit tradeoff.
Source: §I · p. 1Method
The paper proposes coop-UCB2 and coop-UCL, using running consensus to estimate arm means in cooperative distributed decision-making.
Source: §I · p. 1Results
The algorithms provide order-optimal group performance, and coop-UCB2 recovers the centralized-agent regret upper bound within a constant factor.
Source: B. The coop-UCB2 Algorithm · p. 4, §VII · p. 7Takeaways & Limitations
Nodal explore-exploit centrality characterizes how graph topology relates to individual performance, and its predicted node ordering matches numerical simulations.
Source: §I · p. 1
Abstract
from arXiv · showhide
We study distributed cooperative decision-making under the explore-exploit tradeoff in the multiarmed bandit (MAB) problem. We extend the state-of-the-art frequentist and Bayesian algorithms for single-agent MAB problems to cooperative distributed algorithms for multi-agent MAB problems in which agents communicate according to a fixed network graph. We rely on a running consensus algorithm for each agent's estimation of mean rewards from its own rewards and the estimated rewards of its neighbors. We prove the performance of these algorithms and show that they asymptotically recover the performance of a centralized agent. Further, we rigorously characterize the influence of the communication graph structure on the decision-making performance of the group.
I. INTRODUCTION
The introduction motivates distributed MABs through the explore–exploit tradeoff and the limited focus of prior work on single agents. It presents cooperative algorithms and graph-dependent performance analysis as a response.
I. INTRODUCTION
- Distributed MABs address the limited focus on single-agent policies while preserving the need to balance learning arm rewards and maximizing cumulative reward.The paper frames this balance as the explore–exploit tradeoff.Source: §I · p. 1
- The paper extends single-agent algorithms to cooperative distributed MABs, using running consensus for reward-mean estimation and designing an order-optimal group arm-selection heuristic.The agents communicate with neighbors in the graph.Source: §I · p. 1
- Nodal explore-exploit centrality characterizes graph-dependent performance, and its predicted node ordering matches simulation results.The paper also reports that priors informative about correlation structure markedly improve performance.Source: §I · pp. 1–2
II. COOPERATIVE MULTIARMED BANDIT PROBLEM
- The setting has multiple agents connected by an undirected, connected graph, with Gaussian arm rewards of known, shared variance.The proposed policies aim to sample each suboptimal arm within a constant factor of the stated lower bound.Source: §II · p. 2
III. COOPERATIVE ESTIMATION OF MEAN REWARDS
The cooperative estimator uses running consensus to estimate arm reward totals and sampling counts, then combines those estimates to obtain empirical mean rewards.
A. Cooperative Estimation Algorithm
- Two running consensus algorithms estimate each arm’s total reward and total number of samples across agents.These quantities are maintained per unit agent.Source: A. Cooperative Estimation Algorithm · p. 2
- Each agent forms an estimated empirical arm mean from its estimates of the arm’s total reward and sample count.The passage identifies these inputs as reward and count estimates per unit agent.Source: A. Cooperative Estimation Algorithm · p. 2
- The estimator updates reward-count and reward-total estimates using running consensus, with a row-stochastic matrix whose step size depends on maximum node degree.The reward update uses element-wise multiplication of reward and arm-selection indicators.Source: A. Cooperative Estimation Algorithm · p. 2
B. Analysis of the Cooperative Estimation Algorithm
The analysis characterizes estimator performance and supplies concentration bounds for the distributed estimates, with graph-topology-dependent constants measuring estimation performance.
B. Analysis of the Cooperative Estimation Algorithm
- The analysis identifies constants depending only on communication-graph topology as measures of distributed estimation performance.The constants are introduced alongside the estimator’s performance results.Source: B. Analysis of the Cooperative… · p. 3
- The cooperative-estimation performance proposition includes an inequality for the estimated arm-sample count.The supplied passage does not state the inequality’s explicit bound.Source: B. Analysis of the Cooperative… · p. 3
- The estimator analysis gives a concentration inequality for the distributed reward-total and sample-count estimates.The theorem concerns estimates obtained using the running-consensus updates.Source: B. Analysis of the Cooperative… · p. 3
IV. FREQUENTIST COOPERATIVE DECISION-MAKING
The frequentist section reviews coop-UCB and introduces coop-UCB2 to address its dependence on global graph information. The existing coop-UCB heuristic accounts for information received through communication and its propagation rate.
IV. FREQUENTIST COOPERATIVE DECISION-MAKING
- Coop-UCB2 is designed not to require agents to know the global graph structure, and the paper bounds group performance as a function of graph structure.The section also reviews the earlier coop-UCB algorithm.Source: §IV · p. 3
A. The coop-UCB Algorithm
- Coop-UCB’s decision heuristic incorporates additional information received through communication and the network’s rate of information propagation.The algorithm is analogous to single-agent UCB.Source: A. The coop-UCB Algorithm · p. 3
- The original coop-UCB requires each agent to know a graph-dependent quantity, ε_c^k, and can force high-ε_c^k agents to explore more while better-positioned agents exploit.The paper introduces coop-UCB2 to address these issues.Source: A. The coop-UCB Algorithm · p. 3
B. The coop-UCB2 Algorithm
coop-UCB2 uses cooperative estimates and an arm-selection heuristic for distributed agents. Its regret bound recovers the centralized-agent upper bound within a constant factor asymptotically, while graph position affects individual performance.
B. The coop-UCB2 Algorithm
- Each agent initializes by sampling every arm once, then updates cooperative mean estimates and selects the arm with the highest Qk heuristic.The heuristic requires knowledge of the total number of agents M, but not the global graph structure.Source: B. The coop-UCB2 Algorithm · p. 3
- As t grows, coop-UCB2 recovers the centralized-agent regret upper bound within a constant factor.The group bound’s logarithmic portion is independent of agent location, while its constant factor depends on location in the network.Source: B. The coop-UCB2 Algorithm · pp. 3–4
- Explore-exploit centrality, based on an agent’s graph position, predicts its contribution to group regret.The constant factor in the individual performance bound depends on the agent’s position through ϵk c.Source: B. The coop-UCB2 Algorithm · p. 4
V. BAYESIAN COOPERATIVE DECISION-MAKING
The paper extends coop-UCB2 to a Bayesian cooperative setting through coop-UCL. This setting accommodates correlated bandits and prior knowledge, and the algorithm’s performance is analyzed for an uninformative prior.
V. BAYESIAN COOPERATIVE DECISION-MAKING
- The section develops coop-UCL by extending coop-UCB2 to the Bayesian setting.Source: §V · p. 4
- The Bayesian setting models correlated bandits and incorporates prior knowledge about reward and correlation structure.Source: §V · p. 4
- The performance analysis of coop-UCL considers an uninformative prior.Source: §V · p. 4
A. The UCL Algorithm
UCL chooses arms using upper credible limits computed from posterior reward-mean distributions. For Gaussian rewards, the limit uses the posterior mean and standard deviation; the posterior is updated from a Gaussian prior.
A. The UCL Algorithm
- UCL selects the arm with the highest upper credible limit, computed at probability level 1−1/Kta.The limit is an upper bound that holds with probability 1−1/Kta.Source: A. The UCL Algorithm · p. 4
- For Gaussian rewards, the upper credible limit is expressed using νi(t), the posterior mean, and σi(t), the posterior standard deviation of mean reward.The expression also uses the standard Gaussian inverse cumulative distribution function Φ−1(·).Source: A. The UCL Algorithm · pp. 4–5
- The algorithm updates the posterior mean and covariance recursively from a multivariate Gaussian prior on arm rewards.The prior is specified by mean vector ν0 and covariance matrix Σ0; the update notation includes the selected-arm indicator φi(t) and empirical reward mean µi(t).Source: A. The UCL Algorithm · p. 5
B. The coop-UCL Algorithm
coop-UCL gives each agent an approximate Bayesian estimate using reward information from all agents, with the cooperative frequentist estimator supplying an input to the update. Agents then select arms by upper credible limit.
B. The coop-UCL Algorithm
- Each agent uses the approximate frequentist estimator from Section III in the Bayesian update to estimate a posterior conditioned on rewards obtained by all agents.Source: B. The coop-UCL Algorithm · p. 5
- Each agent’s cooperative approximate Bayesian estimation starts from its own multivariate Gaussian prior, specified by mean νk 0 and covariance Σk 0.Source: B. The coop-UCL Algorithm · p. 5
- At each time t, an agent chooses the option with the maximum (1−α(t))-upper-credible-limit.Source: B. The coop-UCL Algorithm · p. 5
C. Regret of the coop-UCL Algorithm
The section bounds suboptimal-arm selections under coop-UCL for Gaussian bandits with uninformative priors and obtains sub-logarithmic regret. Its proof uses inverse Gaussian CDF bounds.
C. Regret of the coop-UCL Algorithm
- The proof bounds the inverse standard-normal CDF and establishes positivity of an auxiliary minimum for 0 < ν ≤ 1.59.The section also specifies uninformative priors, under which the estimated mean and covariance simplify as described.Source: C. Regret of the coop-UCL Algorithm · p. 5
- Theorem 3 bounds how often all agents select a suboptimal arm under coop-UCL for Gaussian bandits with uninformative priors.The supplied theorem passage introduces the bound but does not include its expression.Source: C. Regret of the coop-UCL Algorithm · p. 5
- The analysis yields sub-logarithmic regret for coop-UCL, following the proof approach used for Theorem 2.For uninformative priors, the analysis relates coop-UCL to coop-UCB2.Source: C. Regret of the coop-UCL Algorithm · pp. 5–6
VI. NUMERICAL ILLUSTRATIONS
The simulations illustrate how graph position relates to regret across fixed and random networks. They also compare cooperative algorithms and show improved performance with informative priors.
Simulation setup
- The simulations use a 10-armed bandit with normally distributed arm means, selected to make results illustrative within the displayed time horizon.The passage says relevant conclusions hold across a wide variation of parameters.Source: §VI · p. 6
Example 1 (Regret on Fixed Graphs)
- In fixed-graph simulations, regret follows the predicted ordering: agent 3 is lowest, agents 1 and 2 are intermediate, and agent 4 is highest.The ordering matches the nodes’ explore-exploit centrality values.Source: Figure 1 · p. 6, §VI · p. 6
Example 2 (Regret on Random Graphs)
- In ER graphs, larger normalized explore-exploit centrality corresponds to sharply lower performance, attributed to less information at less-central nodes.The comparison is shown for 10 agents across 100 connected graphs.Source: §VI · pp. 6–7
Example 1 (Regret on Fixed Graphs)
- With the same run parameters, coop-UCL with informative priors markedly improves performance in the algorithm comparison.The illustrated priors are informative and have no correlation structure.Source: §VI · p. 7
VII. FINAL REMARKS
The paper presents frequentist coop-UCB2 and Bayesian coop-UCL, with performance bounds showing order-optimal group performance. It also relates individual-agent performance to graph topology and identifies communication schemes and communication constraints for future study.
VII. FINAL REMARKS
- Coop-UCB2 and coop-UCL need not know the graph structure, and the paper proves order-optimal performance for the group.The paper also studies individual performance as a function of graph topology using nodal explore-exploit centrality.Source: §VII · p. 7
- Future work includes other communication schemes, the tradeoff between communication frequency and performance, and noisy communications.Alternative schemes may offer better performance or better suit particular networked systems.Source: §VII · p. 7