A General Analysis of the Convergence of ADMM (1502.02009v3)

Abstract: We provide a new proof of the linear convergence of the alternating direction method of multipliers (ADMM) when one of the objective terms is strongly convex. Our proof is based on a framework for analyzing optimization algorithms introduced in Lessard et al. (2014), reducing algorithm convergence to verifying the stability of a dynamical system. This approach generalizes a number of existing results and obviates any assumptions about specific choices of algorithm parameters. On a numerical example, we demonstrate that minimizing the derived bound on the convergence rate provides a practical approach to selecting algorithm parameters for particular ADMM instances. We complement our upper bound by constructing a nearly-matching lower bound on the worst-case rate of convergence.

• The paper establishes linear convergence of ADMM under strong convexity using a novel dynamical systems proof framework.
• It introduces numerical parameter tuning by minimizing an upper-bound on the convergence rate through semidefinite programming.
• It extends the analysis to over-relaxed ADMM, revealing a broader range of parameters that guarantee algorithm convergence.

Analyzing Convergence Characteristics of the Alternating Direction Method of Multipliers (ADMM)

The paper "A General Analysis of the Convergence of ADMM" provides a rigorous examination of convergence properties associated with the Alternating Direction Method of Multipliers (ADMM), particularly under conditions where one of the objective terms is strongly convex. The paper introduces a new proof framework that leverages a dynamical systems perspective to analyze optimization algorithms, thereby extending results obtained by employing integral quadratic constraints, as detailed in earlier works by Lessard et al. (2014).

Main Contributions

1. Linear Convergence Under Strong Convexity: The authors establish a new proof demonstrating linear convergence rates for ADMM when one of the participating functions in the optimization problem is strongly convex. The significantly streamlined argument uses a dynamical system stability lens, effectively sidestepping previous constraints that correlated convergence properties with specific choices of parameters.
2. Numerical Parameter Tuning: Another crucial aspect of this research is its practical implication of optimizing ADMM parameters. By minimizing the derived upper-bound on the convergence rate, the authors suggest a numerical method for tuning these parameters, an approach validated through numerical experiments.
3. Bound Construction: A central technical innovation in this paper is constructing both upper and lower bounds for the convergence rate. The derived semidefinite programs provide tight rate estimates, detailing a methodology how changing algorithm parameters dynamically alters the program, eliminating the need for repetitive proofs with each parameter adjustment.
4. Extensions to Over-Relaxed ADMM: The paper goes further to extend the analysis to the over-relaxed variant of ADMM. The authors reveal that a wider set of the over-relaxation parameter values leads to algorithm convergence compared to traditional beliefs.

Theoretical and Practical Implications

On the theoretical front, this research provides a generalized analysis framework for ADMM convergence, encapsulating various existing results while maintaining enough flexibility to accommodate new algorithm variants introduced by parameter perturbations. The dynamical systems approach opens pathways for automated rate analysis utilizing semidefinite programming, streamlining calculations that traditionally demanded intricate, manual derivations.

Practically, this adaptive parameter selection method using semidefinite programs can substantially improve convergence speeds in applications involving distributed computing settings, model fitting, and resource allocation, which are rife with ADMM implementations. As a result, the numerical procedures offered for selecting optimal parameters stand poised to enhance the effectiveness of ADMM in real-world scenarios by affording practitioners a systematic method for tuning implementations.

Potential for Future Research

The insights from this paper pave the way for expanded exploration into other operator splitting methods beyond ADMM, such as Douglas-Rachford splitting or forward-backward splitting, appealing to robust control concepts from dynamical systems theory. Future work could involve exploring the interplay of these mathematical principles to develop comprehensive frameworks that unify analyses across various convex and non-convex optimization problems.

In summary, the presented analysis not only reinforces the understanding of ADMM convergence under strong convexity but also evolves it into a flexible, numerical toolkit applicable for advanced optimization tasks, indicative of its potential to catalyze further innovations in optimization theory and practice.