Decentralized Collaborative Learning of Personalized Models over Networks

Paul Vanhaesebrouck; Aurélien Bellet; Marc Tommasi

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Decentralized Collaborative Learning of Personalized Models over Networks

Paul Vanhaesebrouck, Aurélien Bellet, Marc Tommasi

arXiv:1610.05202v2Oct 17, 2016cs.LGcs.AIcs.DCeess.SYstat.ML

TL;DR

The paper asks how agents with personalized objectives can improve locally trained models by communicating with similar neighbors without centralized coordination. It proposes two asynchronous decentralized gossip methods—confidence-weighted model propagation and ADMM-based collaborative learning—and reports that both improve substantially over solitary models in the presented classification experiment, while collaborative learning consistently outperforms model propagation across training sizes.

Problem
Agents need to improve locally trained personalized models using information from similar neighbors, while avoiding centralized collection, aggregation, and coordination.
Method
The paper proposes asynchronous fully decentralized gossip algorithms: model propagation smooths pretrained models using confidence values, while collaborative learning jointly learns and propagates models through decentralized ADMM.
Results
Both model propagation and collaborative learning substantially improve over solitary models in the classification experiment, with collaborative learning consistently outperforming model propagation across training sizes.
Takeaways & Limitations
Personalized decentralized collaboration can improve local models, while jointly interweaving learning and propagation is more effective than post-training propagation in the reported training-size analysis.
Takeaways & Limitations
The methods assume a given similarity graph and agents with only local network views; the paper identifies formal analysis of graph similarity and generalization as future work.

Abstract

from arXiv · show

We consider a set of learning agents in a collaborative peer-to-peer network, where each agent learns a personalized model according to its own learning objective. The question addressed in this paper is: how can agents improve upon their locally trained model by communicating with other agents that have similar objectives? We introduce and analyze two asynchronous gossip algorithms running in a fully decentralized manner. Our first approach, inspired from label propagation, aims to smooth pre-trained local models over the network while accounting for the confidence that each agent has in its initial model. In our second approach, agents jointly learn and propagate their model by making iterative updates based on both their local dataset and the behavior of their neighbors. To optimize this challenging objective, our decentralized algorithm is based on ADMM.

1 Introduction

The paper addresses personalized collaborative learning in fully decentralized peer-to-peer networks, proposing two asynchronous gossip algorithms that avoid centralized coordination. One smooths pretrained local models using confidence values, while the other jointly learns and propagates models via decentralized ADMM.

Centralized learning requires users to surrender personal data and can impose high bandwidth and device battery costs.
The proposed setting lets agents with local datasets learn personalized models through a fully decentralized peer-to-peer network.Agents communicate only with direct neighbors whose objectives are considered similar.
The paper proposes two asynchronous, fully decentralized gossip algorithms in which agents communicate with one neighbor at a time.This communication pattern is intended to support deployment in large peer-to-peer networks.
Model propagation first trains local models independently, then regularizes their parameters to be smooth over the network while accounting for confidence differences.The paper states that the resulting asynchronous gossip algorithm converges to the optimal solution.
Collaborative learning interweaves learning and propagation by trading off network smoothness against accuracy on local datasets.Its asynchronous gossip algorithm is based on a decentralized version of ADMM.

2 Preliminaries

The paper formulates decentralized learning for agents with heterogeneous local data and personalized objectives, connected by a similarity-weighted graph. Unlike consensus methods that produce one global model, the goal is to improve each agent’s solitary model using information from neighbors.

Each agent learns a model that minimizes expected loss under its own unknown data distribution, using locally sampled training examples.The local training-set size may vary widely, including agents with no data.
The objective is to improve solitary local models by leveraging information from other users in the network.
The network is a weighted connected graph whose edge weights represent similarity between agents’ learning objectives.Agents know their neighbors and associated weights but not the global topology or total number of participants.
Existing decentralized averaging and optimization methods solve consensus problems that produce a common global solution for all agents.That formulation is unsuitable when agents have personalized objectives.
The setting resembles multi-task learning but allows widely varying data sizes, does not require all tasks to be positively related, and targets fully decentralized operation.The cited multi-task-learning approaches are described as centralized, except for a synchronous method relying on a central server.

3 Model Propagation

Model propagation smooths locally trained models across a similarity-weighted network while weighting deviations by confidence, then realizes the solution through asynchronous gossip updates that converge to the optimum.

3.1 Problem Formulation: Each agent first trains a solitary model locally, then adapts it so neighboring models become smoother while retaining confidence-weighted proximity to their original values.High-confidence models are constrained from large deviations; low-confidence models can be determined largely by neighboring models.
3.1 Problem Formulation: The optimization penalizes differences between neighboring models according to graph weights and penalizes deviations from solitary models according to confidence.The parameter µ controls the trade-off between network smoothing and fidelity to initial models.
3.1 Problem Formulation: The formulation generalizes label propagation by allowing unequal confidence values, which cannot generally be represented through different solitary models or a different graph.Equal confidence recovers the standard label-propagation setting.
3.1 Problem Formulation: Although the closed-form solution requires global network and solitary-model knowledge, its iterative form converges regardless of initialization and decomposes into neighborhood-based updates.The resulting process is decentralized but synchronous when every agent contacts all neighbors at each iteration.
3.2 Asynchronous Gossip Algorithm: The asynchronous gossip algorithm uses local Poisson clocks, one-neighbor communication, pairwise updates, and maintained possibly outdated neighbor-model knowledge.Each waking agent selects a neighbor, exchanges model information, and updates both agents’ states while other network variables remain unchanged.
3.2 Asynchronous Gossip Algorithm: The algorithm converges to a state in which every node has its optimal model and the optimal models of its neighbors.The convergence result holds for arbitrary initial values and extends to agents with clocks ticking at different rates.

4 Collaborative Learning

Collaborative learning jointly adapts personalized models using local data and neighboring agents’ behavior, rather than only propagating pre-trained models. The objective combines graph smoothness with local-data loss and is optimized through an asynchronous decentralized ADMM algorithm.

Problem formulation: Agents iteratively update personalized models using both their local datasets and their neighbors’ behavior.This approach is computationally more costly than propagating pre-trained models alone.
Problem formulation: The collaborative objective accounts for each personal model’s loss on its local dataset, rather than only its distance from a solitary model.The optimization problem minimizes this collaborative-learning objective over all agents’ model parameters.
Problem formulation: Graph smoothness favors neighboring models that vary smoothly, while the local-loss term preserves flexibility when different parameters yield similar predictions.The local loss also incorporates confidence through its sum over each agent’s dataset.
Problem reformulation: The personalized optimization problem is reformulated as partial consensus, requiring neighboring agents to agree on each other’s personalized models.Local copies and edge-based secondary variables decouple the objectives for decentralized optimization.
Asynchronous gossip algorithm: Asynchronous decentralized ADMM alternates updates of primal, secondary, and dual variables when one agent wakes and selects a neighbor.All other network variables remain unchanged during each interaction, and the method converges almost surely at O(1/t) for convex objectives.

5 Experiments

Experiments evaluate accuracy, convergence, and communication for mean estimation and linear classification. Confidence-aware propagation helps with unbalanced data, while collaborative learning improves accuracy and asynchronous methods approach synchronous performance.

The experiments evaluate two synthetic collaborative tasks—mean estimation and linear classification—for accuracy, convergence rate, and communication.
Collaborative Mean Estimation: For ε = 1, confidence-aware model propagation wins about 0.85 of instances, while its L2 error remains constant as dataset imbalance increases.
Collaborative Mean Estimation: Asynchronous model propagation achieves an accuracy/communication trade-off almost as good as synchronous propagation without synchronization.
Collaborative Linear Classification: Collaborative learning consistently outperforms model propagation and significantly improves over solitary models, while global consensus performs poorly.
Collaborative Linear Classification: Collaborative learning yields similar accuracy across training-set sizes and corrects initial unbalancedness, whereas confidence can limit model propagation gains for data-rich agents.
Collaborative Linear Classification: Asynchronous collaborative learning performs as well as its synchronous counterpart, while asynchronous model propagation converges an order of magnitude faster.
The decentralized algorithms experimentally require a number of iterations that scales favorably with network size.

6 Conclusion

The paper proposes and evaluates two asynchronous peer-to-peer algorithms for decentralized collaborative learning of personalized models, while identifying several directions for extending and formalizing the framework.

The work proposes, analyzes, and evaluates two asynchronous peer-to-peer algorithms for decentralized collaborative learning of personalized models.
Future work includes formally analyzing how similarity graphs affect generalization and developing methods to estimate graph weights.
Other proposed directions include privacy-preserving algorithms, time-evolving networks, and sequential data arrival.

Appendix A Proof of Proposition 1

The appendix proves the closed-form model-propagation solution by expressing its objective in matrix form and showing the relevant matrix is invertible.

The proposition gives a closed-form solution for the model-propagation objective using the stochastic similarity matrix and solitary-model matrix.
The formulation uses α ∈ (0, 1) and defines μ as the ratio of 1 − α to α.
The proof writes the objective with the graph Laplacian and uses convex quadratic minimization by setting the derivative to zero.
Because the stochastic matrix has eigenvalues in [−1, 1] and the spectral radius is below one, the matrix required for the solution is invertible.

Appendix B Convergence of the Iterative Form (5)

The supplied appendix passages introduce an equation rewrite and a condition below one, but do not provide enough context to summarize the convergence argument further.

The appendix rewrites the equation and invokes a condition that is less than one.

Appendix C Proof of Theorem 1

The proof models asynchronous model propagation as a random iterative process and establishes convergence by showing the expected update matrix has spectral radius below one.

Equivalent iterative process: The model propagation algorithm is represented as a random iterative process over the horizontally stacked agent models.The equivalent process uses communication and update matrices to encode pairwise interactions and node updates.
Expected dynamics: The global recursion uses a block-diagonal matrix M and a stacked optimal solution to characterize the expected dynamics.M repeats (αI + ᾱC)^-1αP on its diagonal, while the stacked solution repeats Θsol n times.
Spectral argument: Convergence follows after proving that the spectral radius of the expected matrix Ā is less than one.The proof analyzes the absence of −1 and +1 eigenvalues and uses this to establish ρ(Ā) < 1.
Conclusion: The expected trajectory is obtained by unrolling the recursion after establishing the spectral-radius condition.The proof then invokes the resulting matrix properties to complete Theorem 1.

Appendix D Synchronous Decentralized ADMM Algorithm for Collaborative Learning

The synchronous decentralized ADMM procedure alternates primal, secondary, and dual updates using information exchanged among neighboring agents. Although it converges for convex objectives, global synchronization can be costly in practice.

ADMM updates: Each agent first updates its primal variables in the synchronous decentralized ADMM procedure.The procedure starts with local primal updates before communication-dependent secondary and dual updates.
ADMM updates: Agents then use values received from neighbors to update secondary variables on incident edges.The construction preserves the constraint Z(t + 1) ∈ C^E.
ADMM updates: Agents update their dual variables for each relevant edge after the secondary-variable step.This completes the three-stage synchronous ADMM iteration.
Convergence and limitation: Synchronous ADMM converges at O(1/t) for convex objectives and faster linearly for strongly convex objectives, but requires costly global synchronization.The synchronization requirement is identified as a practical drawback in decentralized networks.

Appendix E Additional Experimental Results

The experiments represent personalized target models geometrically and connect agents with similar model directions. Across networks of 100 to 1000 agents, the iterations needed to reach 90% of optimal-model accuracy scale linearly with network size.

Target models: Target models are represented as points in R2, with links indicating small angular differences after projection onto the unit circle.Small angular differences correspond to small Euclidean distances on the projected circle.
Scalability: The scalability experiment varies the number of agents from 100 to 1000 using a k-nearest-neighbor graph based on angle similarity.Each agent connects to the k agents with the largest angle similarity.
Scalability: The number of iterations needed to reach 90% of the optimal models’ accuracy scales linearly with the number of agents.The experiment measures convergence to 90% of the accuracy of the optimal set of models.
Scalability: Because parallelizable asynchronous gossip iterations also scale roughly linearly with network size, the authors expect favorable scaling to very large networks.This is the authors’ stated expectation based on the observed iteration trend and gossip structure.