Towards Parameter-free Distributed Optimization: a Port-Hamiltonian Approach

TL;DR

This work tackles parameter-free distributed consensus optimization by casting the problem in a port-Hamiltonian systems (PHS) framework, deriving continuous-time dynamics that leverage passivity to ensure convergence to the optimizer of $\sum_i f_i(\bm{\theta})$. It then introduces a parameter-free discrete-time discretization, the Mixed Implicit Discretization (MID), built on discrete gradients to preserve the equilibrium and stability for any $\tau>0$. MID additionally enables fully distributed computation by avoiding dependence on neighbors’ future states, with LMIs providing tractable conditions for convergence. Numerical experiments show that MID achieves faster convergence than conventional discretizations and existing discrete-time methods, especially at larger step sizes, highlighting the practical impact of parameter-free, energy-based design in distributed optimization.

Abstract

This paper introduces a novel distributed optimization technique for networked systems, which removes the dependency on specific parameter choices, notably the learning rate. Traditional parameter selection strategies in distributed optimization often lead to conservative performance, characterized by slow convergence or even divergence if parameters are not properly chosen. In this work, we propose a systems theory tool based on the port-Hamiltonian formalism to design algorithms for consensus optimization programs. Moreover, we propose the Mixed Implicit Discretization (MID), which transforms the continuous-time port-Hamiltonian system into a discrete time one, maintaining the same convergence properties regardless of the step size parameter. The consensus optimization algorithm enhances the convergence speed without worrying about the relationship between parameters and stability. Numerical experiments demonstrate the method's superior performance in convergence speed, outperforming other methods, especially in scenarios where conventional methods fail due to step size parameter limitations.

Figures (4)

• Figure 1: Comparison of the number of iterations $K_B$ required to reach $\|\mathbf{q}[k]-(\mathds{1}\otimes\mathbf{I}_m)\bm{\theta}^*\|\leq B$ with $B=10^{-6}$. The proposed MID allows the reduction of the number of iterations by increasing $\tau$, with an optimal value around $\tau=4$. In contrast, increasing $\tau$ for Euler discretization does not feature such great reduction and leads to divergence at around $\tau=0.77$.
• Figure 2: Comparison of MID, Euler discretization as well as DTGT Notarstefano2019, CTGT guido2023, DIGing nedic2017achieving and COOR kia2015distributed for a quadratic optimization problem. The fastest values of $\tau$ were found to be approximately $3.78, 0.71, 0.58, 0.65, 0.05, 0.078$. The proposal MID has a clear advantage in terms of speed of convergence.
• Figure 3: Comparison of MID, Euler discretization as well as DTGT Notarstefano2019, CTGT guido2023, DIGing nedic2017achieving and COOR kia2015distributed for a quadratic optimization problem with $\tau=10$.
• Figure 4: Comparison of MID for a quadratic optimization problem with $\tau=1, 10, 100, 1000$.

Theorems & Definitions (24)

• Proposition 1
• proof
• Corollary 1
• proof
• Example 1: Centralized optimization
• Example 2: Unconstrained distributed optimization
• Remark 1
• Definition 1
• Theorem 1
• proof