Coordinating the Unknown Lipschitz Constant in Multiplayer Bandits

The paper studies cooperative multi-agent bandits in continuous (Lipschitz) action spaces when the Lipschitz constant is unknown. It considers three information structures: (A) unobserved actions with common rewards, (B) observed actions with independent rewards, and (C) unobserved actions with independent rewards. The central difficulty is that players cannot communicate once learning starts, so they must reach the same discretization from their own data.

The paper introduces a meta-algorithm, mECAB, that estimates an upper confidence bound on L by uniform exploration of a coarse grid, fixes a discretization, and runs an existing cooperative multiplayer MAB subroutine. The algorithm proceeds in three phases: initialization (players divide their action sets into bins and agree on an ordering), exploration (each player samples uniformly from each joint bin and computes empirical bin means), and exploitation (players run a multiplayer MAB subroutine on the induced discrete problem).

The paper proves regret guarantees for all three problems, each resting on a different agreement mechanism. For Problem A, common rewards make the exploration statistics shared as well once the schedule is fixed in advance, so all players obtain the same empirical bin means and therefore the same estimate of L. The regret bound is sup RT ≤ T(M d+1)/(M d+2) · [9L(M d/(M d+2)) + 5(2m)(M d/(M d+2)) sqrt(2/E ln(2T(M d+1))) + Em(M d) + 32 sqrt(T m̃(M d) log T + 1)].

For Problem B, actions are observable but rewards are not shared. The remedy is to exploit action observability to share reward information implicitly: after collecting E − 1 samples from a bin, a player devotes its final action in that bin to encoding its empirical mean, which the others decode and fold into their own estimate. This pooling improves the effective sample size from E to M E, sharpening the estimate. The regret bound is sup RT ≤ T(M d+1)/(M d+2) × [9L(M d/(M d+2)) + 5(2m)(M d/(M d+2)) sqrt(2/(M E) ln(2T(M d+1))) + Em(M d) + 32 sqrt(T m̃(M d) log T + 1)].

For Problem C, neither common rewards nor observable actions are available. The paper restores agreement by quantizing the estimate using a dithered quantization: players agree in advance on a dither U ∼ Unif[0, 1) and set L̂ i = X i + U, where X i is the raw estimate. The paper shows that "a deterministic rounding will not do: if Lm sits near a rounding boundary, two players whose estimates straddle it round differently however many samples they collect, and the failure probability approaches 1/2 regardless of E." Randomizing the offset makes the distance from Lm to the nearest boundary uniform rather than instance-dependent. The probability that the M players do not all obtain the same value of L̂ is at most 17 sqrt(m ln(A)/E), where A = 4M(2m)(M d). The regret bound is sup RT ≤ T(M d+1)/(M d+2) × [9L(M d/(M d+2)) + 5(2m)(M d/(M d+2)) sqrt(2/E ln(2T(M d+1))) + Em(M d) + C sqrt(log T) T sqrt(m̃(M d)) + 17 T sqrt(m ln(A)/E)]. If E ≥ m2 T(2/(M d+2)) ln A, the last term is at most 17 T(M d+1)/(M d+2), so Problem C matches Problems A and B up to constants.

The paper also reports simulations with M = 2 players, d = 1, T = 10 5 rounds, comparing Lipschitz-adaptive (Est-L) with non-adaptive (No-L) discretization for L = 1 and L = 1000. The results show that when L is small, the two rules produce comparable resolutions; when L is large, Est-L overtakes No-L despite the initial linear exploration cost. The information structure modulates the gain: pooling in Problem B makes L̂ more accurate and flattens the later slope relative to Problems A and C, while Problem C, with the weakest feedback, is the most variable.

The paper concludes that common rewards and observable actions each deliver that agreement for free, and when neither is present a dithered quantization delivers it at no cost in the leading order of the regret. Natural next steps are adversarial rewards and structural assumptions beyond Lipschitz continuity.

Improvement 1: Uncertainty-Aware Multi-Agent Coordination via Adaptive Discretization

The improved AI system can dynamically adjust its action-space granularity based on an online-estimated Lipschitz constant, without requiring prior knowledge of environment smoothness. In multi-agent reinforcement learning (MARL) or federated optimization, agents can use the mECAB three-phase protocol (coarse exploration → Lipschitz estimation → refined exploitation) to autonomously agree on a shared discretization level, even when communication is disabled. This enables robust coordination in tasks like robotic swarm exploration or distributed resource allocation where the reward landscape’s smoothness is unknown and varies across tasks.

Improvement 2: Communication-Free Information Sharing via Action Encoding

The improved AI system can encode reward statistics into action choices during exploration, allowing agents to implicitly share information without explicit messages. This is useful in privacy-sensitive or bandwidth-limited multi-agent systems (e.g., autonomous vehicles, sensor networks) where direct communication is costly or forbidden. The system can pool empirical estimates across agents (increasing effective sample size by factor M) to achieve tighter confidence bounds and faster convergence, even when rewards are independent and actions are observable.

Improvement 3: Randomized Quantization for Consensus Under Asymmetric Feedback

The improved AI system can use dithered quantization (adding a shared random dither before rounding) to guarantee agreement on model parameters or policy updates across agents, even when they receive heterogeneous observations (e.g., different reward signals, partial observability). This prevents divergent discretizations that cause coordination failures. The system can apply this to decentralized federated learning, where clients with non-IID data must agree on global model updates without a central server, ensuring identical quantization outcomes with high probability and bounded regret.

Improvement 4: Regret-Optimal Exploration-Exploitation Trade-off in Continuous Action Spaces

The improved AI system can automatically balance exploration cost (linear in coarse-grid size) against exploitation gain (sublinear in time) by selecting the exploration horizon E based on the problem horizon T and action dimension d. This yields near-optimal regret scaling (T((M d+1)/(M d+2))) for cooperative multi-agent bandits, which can be transferred to hyperparameter tuning, A/B testing, or multi-agent Bayesian optimization where the action space is continuous and the Lipschitz constant is unknown. The system can outperform non-adaptive methods, especially when the true Lipschitz constant is large, by avoiding over-refinement early on.

Improvement 5: Robustness to Feedback Structure Variations

The improved AI system can automatically detect which information structure it operates under (common rewards, observable actions, or neither) and switch between the corresponding agreement mechanisms (shared statistics, action-encoding, or dithering). This makes it deployable in heterogeneous multi-agent environments (e.g., mixed human-AI teams, cross-organization systems) where feedback modalities differ across agents or change over time, without manual reconfiguration. The system maintains sublinear regret in all cases, with only constant-factor differences, ensuring graceful degradation under weakest feedback (Problem C).

Motivated by decentralized applications, we study cooperative multi-agent bandits in continuous (Lipschitz) action spaces when the Lipschitz constant is unknown. We consider three information structures: (A) unobserved actions with common rewards, (B) observed actions with independent rewards, and (C) unobserved actions with independent rewards. In each case we design and analyze an algorithm that estimates the Lipschitz constant, chooses a discretization of the joint action space, and applies a cooperative bandit method to the induced discrete problem. Players never communicate once learning starts, so the central difficulty is that they must reach the same discretization from their own data. We prove regret guarantees showing that common rewards and observable actions each supply this agreement for free, and that in their absence agreement can still be bought, through a dithered quantization of the estimate, at no cost in the leading order of the regret.