[Paper Review] A Linearly Convergent Proximal Gradient Algorithm for Decentralized Optimization

Sulaiman A. Alghunaim, Kun Yuan+1 more|arXiv (Cornell University)|May 20, 2019

Distributed Control Multi-Agent Systems54 references33 citations

TL;DR

The paper introduces a proximal gradient decentralized algorithm with a common non-smooth regularizer and proves its global linear convergence under strong convexity of the smooth part. It also provides a concise analysis that extends to EXTRA when R=0 and offers rate/step-size insights.

ABSTRACT

Decentralized optimization is a powerful paradigm that finds applications in engineering and learning design. This work studies decentralized composite optimization problems with non-smooth regularization terms. Most existing gradient-based proximal decentralized methods are known to converge to the optimal solution with sublinear rates, and it remains unclear whether this family of methods can achieve global linear convergence. To tackle this problem, this work assumes the non-smooth regularization term is common across all networked agents, which is the case for many machine learning problems. Under this condition, we design a proximal gradient decentralized algorithm whose fixed point coincides with the desired minimizer. We then provide a concise proof that establishes its linear convergence. In the absence of the non-smooth term, our analysis technique covers the well known EXTRA algorithm and provides useful bounds on the convergence rate and step-size.

Motivation & Objective

• Motivate decentralized composite optimization with a common non-smooth regularizer across agents.
• Design a proximal gradient decentralized algorithm whose fixed point matches the global minimizer.
• Prove global linear convergence under strong convexity of the aggregate smooth part.
• Show compatibility with known EXTRA when the non-smooth term is absent and derive convergence insights.

Proposed method

• Formulate the decentralized problem as a constrained optimization with a consensus constraint using a network matrix A.
• Derive a saddle-point reformulation and a proximal primal-dual diffusion (P2D2) algorithm with updates (10a)-(10c).
• Switch to a decentralized implementation that updates a single auxiliary vector Z and uses the proximal operator for R as in (14).
• Provide a concrete per-agent procedure (P2D2) (15) that only requires local communication with neighbors.
• Establish fixed-point existence and uniqueness for the global solution (Lemma 1).
• Develop a linear-convergence proof framework that leverages an augmented cost with a B-structured penalty and descent inequalities (Lemmas 2–4).

Experimental results

Research questions

• RQ1 Can a decentralized proximal gradient method with a common non-smooth regularizer achieve linear convergence to the global minimizer?
• RQ2 Under what conditions on the smooth part and step-sizes does linear convergence hold in the decentralized setting?
• RQ3 How does the proposed method relate to and specialize to existing algorithms like EXTRA when the non-smooth term is absent?
• RQ4 What bounds on convergence rate and step-sizes are implied by the analysis?

Key findings

• The proximal primal-dual diffusion (P2D2) algorithm converges linearly to the global minimizer under Assumption 1 and appropriate step-size conditions.
• Existence and uniqueness of a fixed point corresponding to the global solution are established (Lemma 1).
• When R=0, the method reduces to a form related to EXTRA, with the analysis providing rate-related bounds and step-size guidance.
• An augmented cost framework with a B-penalty yields restricted strong convexity, enabling linear convergence guarantees (Lemma 2).
• Descent and error bounds are derived to relate the iterates’ errors to contraction factors, leading to the main linear convergence result (Lemmas 3–4).