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Distributed and Inexact Proximal Gradient Method for Online Convex Optimization

Nicola Bastianello, Emiliano Dall'Anese

Abstract

This paper develops and analyzes an online distributed proximal-gradient method (DPGM) for time-varying composite convex optimization problems. Each node of the network features a local cost that includes a smooth strongly convex function and a non-smooth convex function, both changing over time. By coordinating through a connected communication network, the nodes collaboratively track the trajectory of the minimizers without exchanging their local cost functions. The DPGM is implemented in an online fashion, that is, in a setting where only a limited number of steps are implemented before the function changes. Moreover, the algorithm is analyzed in an inexact scenario, that is, with a source of additive noise, that can represent e.g. communication noise or quantization. It is shown that the tracking error of the online inexact DPGM is upper-bounded by a convergent linear system, guaranteeing convergence within a neighborhood of the optimal solution.

Distributed and Inexact Proximal Gradient Method for Online Convex Optimization

Abstract

This paper develops and analyzes an online distributed proximal-gradient method (DPGM) for time-varying composite convex optimization problems. Each node of the network features a local cost that includes a smooth strongly convex function and a non-smooth convex function, both changing over time. By coordinating through a connected communication network, the nodes collaboratively track the trajectory of the minimizers without exchanging their local cost functions. The DPGM is implemented in an online fashion, that is, in a setting where only a limited number of steps are implemented before the function changes. Moreover, the algorithm is analyzed in an inexact scenario, that is, with a source of additive noise, that can represent e.g. communication noise or quantization. It is shown that the tracking error of the online inexact DPGM is upper-bounded by a convergent linear system, guaranteeing convergence within a neighborhood of the optimal solution.

Paper Structure

This paper contains 17 sections, 9 theorems, 56 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction and Motivation
  2. Inexact Composite Optimization
  3. Convergence analysis
  4. Online Composite Optimization
  5. Convergence analysis
  6. Numerical Results
  7. Simulation setup
  8. Results
  9. Proofs of Lemmas in Section \ref{['sec:time-invariant']}
  10. Proof of Lemma \ref{['lem:expectation_norm']}
  11. Proof of Lemma \ref{['lem:properties-gamma']}
  12. Proof of Proposition \ref{['pr:time-invariant-convergence']}
  13. Auxiliary results
  14. Convergence analysis
  15. Proofs of section \ref{['sec:time-varying']}
  16. ...and 2 more sections

Key Result

Lemma 1

Let $\mathbold{e}$ be a random vector with finite mean $\bm{\mu}$ and finite covariance matrix $\bm{\Sigma}$. Then, one has that

Figures (2)

  • Figure 1: A qualitative illustration of the distributed, inexact and time-varying framework considered in the paper. Each node is observing time-varying data (e.g. measurements with a sensor) which imply that the problem is time-varying. Moreover, communication errors and the limited resources available at each node introduce inexactness in the algorithm's updates.
  • Figure 1: Comparison in terms of cumulative tracking error of DPGM (proposed in this paper), PG-EXTRA shi_proximal_2015, and NIDS li_decentralized_2019 for a time-varying sparse linear regression problem, without and with state errors.

Theorems & Definitions (16)

  • Remark 1
  • Lemma 1: Expectation of error norm
  • Lemma 2: Relaxed problem
  • Proposition 1: Time-invariant convergence
  • Proposition 2: Time-varying convergence
  • Corollary 1: Asymptotic error bound
  • Remark 2: Sources of inexactness
  • Lemma 3: Implicit update
  • proof
  • Lemma 4: Bounded subgradients
  • ...and 6 more