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Distributed Clustering of Linear Bandits in Peer to Peer Networks
Nathan Korda, Balazs Szorenyi, Shuai Li
TL;DR
The paper addresses collaborative linear bandits in peer-to-peer networks where communication is limited. It introduces gossip-based distributed confidence-ball algorithms for shared and clustered bandit problems, proving near-optimal regret guarantees and evaluating them on real-world datasets.
Problem
The paper studies how agents can collaboratively solve linear bandits and share information without overloading limited communication channels, including when agents belong to unknown clusters with different parameters.
Method
The paper develops DCB for a shared bandit problem and DCCB for clustered bandits, using gossip-based information sharing and online cluster discovery.
Results
Near-optimal regret improvements are proved for DCB and DCCB, with DCB asymptotically optimal up to log factors relative to perfect information sharing; experiments evaluate DCCB against CLUB and bandit benchmarks on three real-world datasets.
Takeaways & Limitations
Gossip-based peer-to-peer sharing can support distributed linear-bandit learning while keeping communication costs low and preserving strong regret performance in shared and clustered settings.
Takeaways & Limitations
Both DCB and DCCB are synchronous algorithms, although more powerful gossip-based distributed-computation methods could alleviate this restriction.
Abstract
from arXiv · showhide
We provide two distributed confidence ball algorithms for solving linear bandit problems in peer to peer networks with limited communication capabilities. For the first, we assume that all the peers are solving the same linear bandit problem, and prove that our algorithm achieves the optimal asymptotic regret rate of any centralised algorithm that can instantly communicate information between the peers. For the second, we assume that there are clusters of peers solving the same bandit problem within each cluster, and we prove that our algorithm discovers these clusters, while achieving the optimal asymptotic regret rate within each one. Through experiments on several real-world datasets, we demonstrate the performance of proposed algorithms compared to the state-of-the-art.
1. Introduction
The paper develops gossip-based distributed confidence-ball algorithms for collaborative linear bandits, targeting lower communication costs while preserving regret performance. It first handles a shared bandit problem, then extends the approach to unknown clusters of agents with differing parameters.
- Motivation: Collaborative linear bandits can improve network and individual performance, but communicating every observation would overload peer-to-peer channels.The proposed setting targets limited-resource systems where agents exchange information directly and compute locally.
- Motivation: Gossip sharing diffuses information across the network through limited exchanges, providing a communication-efficient and robust alternative to instant global sharing.Each agent shares with one randomly selected peer per round; a delay of typically O(log t) helps remove the resulting information-mixing bias.
- Shared bandit problem: DCB is a gossip-based distributed variant of the generic Confidence Ball algorithm for agents solving the same underlying linear bandit problem.Unlike the earlier epoch-based distributed approach, DCB uses a simpler structure, although its buffers grow logarithmically with time.
- Clustered bandits: The clustered setting allows agents’ parameter vectors to vary, requiring the unknown groups and their coefficient vectors to be learned online before information sharing can accelerate learning.This model is motivated by recommendation problems with massive numbers of users.
- Clustered bandits: DCCB distributes a centralized clustering approach by starting with one cluster and pruning edges when evidence indicates that agents have different bandit parameters.The paper reports near-optimal regret improvements for DCB and DCCB, with low communication cost, and evaluates them on real-world benchmark data.
2. Linear Bandits and the DCB Algorithm
The DCB algorithm adapts confidence-ball linear bandits to peer-to-peer gossip sharing by delaying and buffering information, mitigating imperfect mixing while reducing communication. Its regret is asymptotically near-optimal relative to perfect information sharing, with substantially lower communication costs than instant sharing.
- Information delay: A delay τ(t) ensures buffers contain only older information, reducing bias from duplicated or incompletely mixed data.The analysis attributes the remaining performance loss to delayed information use and incomplete network mixing.
- DCB algorithm: DCB combines OFUL-style confidence balls with gossip sharing, using each agent’s active covariance matrix and b-vector to choose actions.Agents maintain buffers of covariance matrices and b-vectors, update them through peer exchanges, and calculate confidence balls from delayed active statistics.
- Regret guarantees: After an initial burn-in period, DCB improves regret over CB-NoSharing by an almost-order term while remaining asymptotically optimal up to log factors.The comparison is made against independent confidence-ball learners and against algorithms with perfect information sharing.
- Communication cost: The DCB communication cost is at most O(log^2(|V|t)d^2|V|) bits per round, versus d|V|^2 per round for instant sharing.The reduction is significant when log(|V|t)d ≪ |V|.
- Communication cost: An epoch-based variant reduces gossip communication to O(d^2|V|) per round and is more efficient over any horizon when d ≪ |V|.The growing-buffer approach is preferable until horizons of order O(exp(|V|)).
3. Clustering and the DCCB Algorithm
DCCB extends distributed confidence-ball learning with gossip-based clustering, pruning links when agents' parameter estimates are sufficiently separated. With high probability, agents identify their true clusters after finite time and then share only within-cluster information, achieving a cluster-level regret bound.
- DCCB Algorithm: Each agent maintains local estimators, covariance statistics, buffers, and a neighbor set that is initialized to the full network.The algorithm updates local confidence-ball quantities while gossiping with randomly contacted neighbors.
- DCCB Algorithm: DCCB augments DCB with a pruning protocol that removes links between agents whose parameter estimates exceed a threshold.Agents compare local estimates and prune one another when the separation condition is met.
- Clustering guarantee: Under the context-distribution assumption, agents identify their true cluster with high probability after finite time.The assumption requires bounded context vectors and a full-rank expected covariance matrix with minimum eigenvalue λ > 0.
- Clustering guarantee: After pruning, buffer and active-matrix resets prevent pre-clustering information from outside the true cluster from introducing bias.Once clusters are identified, each cluster shares only information gathered within that cluster.
- Regret guarantee: Theorem 6 bounds each cluster's cumulative regret with probability 1 − δ, with a constant C(γ, |V|, λ, δ) representing the time needed to identify the clustering.The analysis reduces to the multi-agent case after correct cluster identification and separately bounds the identification period.
4. Experiments and Discussion
The paper evaluates DCCB on three real-world recommendation datasets against centralized and distributed baselines, then discusses its relation to prior distributed and clustered bandit methods. The discussion identifies synchronous operation and confidence-ball-specific analysis as important scope boundaries.
- Experiments: DCCB is evaluated on LastFM, Delicious, and MovieLens against CLUB, CB-NoSharing, and CB-InstSharing.The experiments follow the dataset-construction principles of earlier clustered-bandit work.
- Related work: Prior work spans distributed multi-armed bandits, no-communication settings, communication-aware methods, and clustered bandit algorithms.The paper positions DCCB among methods including distributed ε-greedy learning, CLUB-related clustering, and item-user clustering.
- Related work: A related contextual-bandit approach achieves at best Ω(T^2/3) regret under a different setting where agents request actions rather than share information across a limited channel.The paper attributes the setting difference to worse regret bounds.
- Discussion: The analysis is tailored to confidence-ball algorithms, although several proof components could support analogous guarantees for distributed Thompson-sampling methods.This is presented as a possible reuse of the analysis rather than as a result established by the paper.
- Discussion: Both DCB and DCCB are synchronous, while stronger gossip-computation techniques could apply when agents identify only a sub-network of the true clustering.The paper presents this as a direction for addressing the synchronous setting.
A.1. Pseudocode for the generic CB algorithm and the DCB algorithm
The generic confidence-ball algorithm initializes its statistics, constructs confidence sets, selects actions, and updates from rewards. DCB adds gossip sharing and delayed buffer updates to distribute this process across agents.
- Confidence Ball: The confidence-ball procedure initializes A_0 = I and b_0 = 0 before iterating over time.These quantities provide the starting covariance and reward statistics.
- Confidence Ball: At each round, the algorithm receives the action set, constructs a confidence ball from its current statistics, and chooses an action.The action then produces a reward used in the next update.
- Distributed Confidence Ball: The distributed algorithm initializes per-agent active matrices, b-vectors, and information buffers before communication begins.The buffers support delayed use of information received through gossip.
- Distributed Confidence Ball: DCB uses a delay function based on logarithmic time and network size to manage gossip-shared observations.The protocol draws a random permutation of agents for the communication process.
- Distributed Confidence Ball: Agents share information and update their buffers after acting, then use the delayed information in subsequent rounds.The pseudocode separates action selection from sharing and buffer updates.
A.2. More on Communication Complexity
The communication analysis compares full instant sharing, gossip-based DCB, epoch-based variants, and DCCB. Gossip reduces per-round communication while retaining near-optimal regret, and DCCB adds clustering with the same asymptotic communication order.
- Communication costs: Full instant sharing costs O(d|V|^2) bits per round, whereas DCB requires at most O(log^2(|V|t)d^2|V|) bits per agent per round.The reduction is significant when log(|V|t)d ≪ |V|.
- Communication costs: Epoch-based gossip reduces communication to O(d^2|V|) per round and is more efficient over any time horizon when d ≪ |V|.The paper contrasts this with growing buffers, which become an issue only after O(exp(|V|)) rounds.
- Protocol alternatives: Tracking which agents received each observation would require at least quadratic communication, while the proposed approach remains linear in |V|.The comparison motivates diffusion and delay rather than explicit observation tracking.
- Protocol alternatives: Round-robin communication creates a delay at least linear in |V|, unlike gossip's logarithmic dependence, and worsens the additive regret constant by a factor of |V|.The paper also notes that round-robin lacks gossip's generalisability and robustness.
- DCCB communication: DCCB adds at most O(d|V|) for sharing estimated parameters, so its per-round communication remains O(log^2(|V|t)d^2|V|).The theoretical summary states that DCB and DCCB retain near-optimal regret while reducing communication dependence on network size.
A.3. Proofs of Intermediary Results for DCB
The intermediary proofs establish weighted confidence estimates by applying self-normalized linear-bandit arguments to transformed observations, then control determinant and eigenvalue terms through matrix inequalities.
- Weighted confidence analysis: The analysis transforms observations and subGaussian noise by multiplying each sample by the square root of its positive weight.The transformed vectors are used to formulate the weighted maximum-likelihood estimate.
- Weighted confidence analysis: The weighted estimator is bounded using Cauchy–Schwarz together with a self-normalized concentration result holding with probability 1 − δ.The proof sets x using the inverse regularized design matrix to obtain the confidence bound.
- Matrix control: The determinant term is controlled by repeatedly applying the rank-one determinant identity across agents and time indices.The proof enumerates V × {1, . . . , τ(t)} and updates the matrix sequentially.
- Matrix control: The remaining inequalities use the Sherman–Morrison lemma, Cauchy–Schwarz, and the smallest-eigenvalue bound for positive definite matrices.These steps produce the required uniform control over vectors indexed by agents and prior rounds.
A.4. Proof of Theorem 6
The proof shows that clustering is identified after a finite high-probability time, after which information sharing and weight behavior permit the standard DCB regret analysis within each true cluster.
- Finite-time clustering: After clustering, agents share information only with members of their own cluster, while pre-clustering cross-cluster sharing is bounded.This separates the finite discovery phase from the subsequent within-cluster analysis.
- Finite-time clustering: With probability 1 − δ, there is a constant C(|V |, γ, λ, δ) after which each agent’s local neighbor set equals its cluster.The event combines the required clustering conditions and guarantees the correct neighbor set from time C onward.
- Post-clustering weights: The post-clustering weights satisfy the independence and structural properties required to reuse the proof of Theorem 1.Lemma 7 establishes these properties for agents inside and outside the fixed agent’s cluster.
- Post-clustering weights: The influence of earlier imperfect information sharing remains controlled after time C, allowing the bias from pre-clustering data to be bounded.The proof explicitly decomposes information gathered before complete cluster discovery and controls its contribution.
- Regret bound: The weights’ variance decreases geometrically after clustering, with the bound |U k| 2^−max{t′,C}.This property supplies the remaining ingredient for applying the DCB analysis.
- Regret bound: The final regret bound combines the post-C DCB analysis with a finite pre-C cost upper bounded by |U k|C(|V |, γ, λ, δ)∥θ∥.The proof then selects constants, sums simple regret, and combines the result with the clustering event.
A.5. Proofs of Intermediary Results for DCCB
The intermediary DCCB proofs analyze the pruning protocol’s information resets and show that, once an agent identifies its cluster, subsequent sharing is restricted to that cluster.
- Pruning protocol: When pruning cuts an edge, both affected agents reset their buffers to local information scaled by their current neighbor-set sizes.The scaling is retained for analytical convenience because its practical effect is washed out in confidence bounds and local estimates.
- Cluster isolation: On the event E, each agent reaches a time before C when its local neighbor set becomes its local cluster.After this reset, the agent shares only with agents that have likewise identified the same cluster.
- Cluster isolation: Once an agent’s neighbor set equals U k, the sharing protocol ensures that the relevant within-cluster weight properties hold by construction.The proof uses this restriction to establish the statements corresponding to parts (ii) and (iii) of Lemma 7.
- Influence bound: The proof of the influence bound follows the earlier lemma through time C and adds inequalities based on positive-definite matrix properties.The determinant and eigenvalue arguments are used to complete the post-clustering bound.
- Influence bound: The remaining inequality steps rely on standard positive-definite matrix bounds and parallel the argument used for the preceding lemma.This completes the two parts of the intermediary result.