### Abstract

Given an undirected graph G = (N, ϵ) of agents N = {1, ⋯, N} connected with edges in ϵ, we study how to compute an optimal decision on which there is consensus among agents and that minimizes the sum of agent-specific private convex composite functions {Φ_{i}}_{i∈N}, where Φ_{i} ≙ ξ_{i} + f_{i} belongs to agent-i. Assuming only agents connected by an edge can communicate, we propose a distributed proximal gradient algorithm (DPGA) for consensus optimization over both unweighted and weighted static (undirected) communication networks. In one iteration, each agent-i computes the prox map of ξ_{i} and gradient of f_{i}, and this is followed by local communication with neighboring agents. We also study its stochastic gradient variant, SDPGA, which can only access to noisy estimates of ∇f_{i} at each agent-i. This computational model abstracts a number of applications in distributed sensing, machine learning and statistical inference. We show ergodic convergence in both suboptimality error and consensus violation for the DPGA and SDPGA with rates O(1/t) and O(1/√t), respectively.

Original language | English (US) |
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Pages (from-to) | 5-20 |

Number of pages | 16 |

Journal | IEEE Transactions on Automatic Control |

Volume | 63 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2018 |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering

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## Cite this

*IEEE Transactions on Automatic Control*,

*63*(1), 5-20. https://doi.org/10.1109/TAC.2017.2713046