Affine-coupled Distributed Optimization via Distributed Proximal Jacobian ADMM with Quantized Communication

Abstract: This paper investigates distributed resource allocation optimization over directed graphs with limited communication bandwidth. We develop a novel distributed algorithm that integrates the centralized Proximal Jacobian Alternating Direction Method of Multipliers (PJ-ADMM) with a finite-level quantized consensus scheme, enabling nodes to cooperatively solve the optimization in a distributed fashion. Under the assumption of convex objective functions, we establish that the proposed algorithm achieves sublinear convergence to a neighborhood of the optimal solution, with the convergence accuracy explicitly bounded by the quantization level. Numerical experiments validate that the algorithm achieves competitive performance compared to existing approaches while exhibiting communication efficiency.

• The paper presents QDPJ-ADMM, a decentralized algorithm that applies proximal Jacobian updates with a quantized consensus layer to solve affine-coupled convex optimization problems.
• It achieves sublinear convergence to a neighborhood of the optimum with explicit bandwidth–accuracy tradeoffs influenced by network size and quantization granularity.
• Simulations on large-scale digraphs validate its efficiency, showing reduced cumulative bandwidth usage and precise convergence under stringent quantization constraints.

Affine-Coupled Distributed Optimization via Distributed Proximal Jacobian ADMM with Quantized Communication

Problem Context and Prior Art

Distributed optimization underpins scalable machine learning, control, and coordination in networked systems, particularly where communication resources are limited. The standard Alternating Direction Method of Multipliers (ADMM) and its extensions have matured as mainstream approaches for solving affine-coupled convex optimization problems, such as resource allocation, across networked agents. Nevertheless, canonical PJ-ADMM (Proximal Jacobian ADMM) schemes rely predominantly on centralized coordination and full-precision, real-valued communication, rendering them unsuitable in bandwidth-constrained or privacy-preserving settings.

Efforts to decentralize ADMM include dual consensus ADMM, variable-splitting, tracking-based, and proximal regularization variants. Yet, for resource allocation problems entailing general affine coupling, no prior methods achieve simultaneously: (i) full decentralization, (ii) compatibility with directed graphs, and (iii) quantized finite-level inter-agent communications. Moreover, consensus-based quantized approaches, while previously explored for consensus constraints, do not directly extend to affine-coupling due to fundamental symmetry limitations in quantized averaging protocols.

Problem Formulation

The addressed problem is an $N$-agent affine-coupled convex minimization under quantized communication: $$\min_{x_i\in\mathbb{R}^{n_i}} \sum_{i=1}^N f_i(x_i)\quad\text{s.t.}\quad \sum_{i=1}^N A_i x_i = b$$ where each local $f_i$ is closed, proper, and convex, and only quantized values can be exchanged across the communication digraph.

Algorithmic Contributions

The paper develops the Quantized Distributed Proximal Jacobian ADMM (QDPJ-ADMM) algorithm. The design is based on a two-layer architecture:

• Outer optimization layer: Each agent performs primal-dual updates utilizing only local variables and local estimates of the global affine constraint violation (primal residual). The local subproblems incorporate a diagonal or block-diagonal proximal term, as in multi-block PJ-ADMM.
• Inner quantized consensus layer: Each agent updates its local estimate $\hat{d}_i^{[k]}$ of the coupled residual by executing a quantized average consensus protocol over the digraph. This protocol admits finite alphabet communication and is robust to quantization.

Notably, inter-agent messaging is strictly quantized, and all coordination is achieved locally—no central aggregator or coordinator is required, in contrast to previous centralized or partially-centralized hybrid schemes.

Theoretical Analysis

Formal convergence results are established under standard convexity and connectivity assumptions.

• For convex $f_i$, strong connectivity, and proper parameter selection, the iterates of QDPJ-ADMM achieve sublinear convergence to a neighborhood of the global optimum. Specifically,

$$\mathcal{L}\left(\bar{x}^{[K]}, \lambda^*\right)-\mathcal{L}(x^*,\lambda^*)\leq \frac{C}{K}+\mathcal{O}(\Delta)$$

where $\Delta$ is the quantization level, characterizing both the accuracy of consensus in the dual consensus step as well as the radius of the neighborhood around the exact optimum. Crucially, the convergence rate constants are polynomially dependent on network size and quantization granularity, and the ultimate solution accuracy degrades gracefully with coarsening quantization.

• The theoretical treatment leverages the error properties of the underlying quantized consensus protocol, combining them with a telescoping analysis of saddle-point gap, standard in multi-block augmented Lagrangian methods.
• The per-iteration communication complexity scales with the graph degree and logarithmically with $1/\Delta$. This offers an explicit bandwidth–accuracy tradeoff.

Numerical Results

Simulations on large random digraphs (100 agents, 100-dimensional variables, high-dimensional linear constraints) validate the theoretical guarantees. The method demonstrates:

• Accurate convergence to the optimal set, with solution precision matching the finest available quantization.
• Communication efficiency over the entire optimization trajectory, with substantially lower cumulative bandwidth usage compared to baseline (centralized) PJ-ADMM.
• Lower accumulative quantization errors than approaches requiring repeated quantization of both primal and dual iterates.

Smaller quantization step sizes $\Delta$ yield higher accuracy, at the cost of increased communication volume per exchanged message. The experiments highlight QDPJ-ADMM's scalability and tight communication–accuracy tradeoffs in high-dimensional, bandwidth-constrained environments.

Implications and Outlook

This work rigorously closes a significant gap in the literature on distributed resource allocation by delivering:

• The first fully decentralized, quantized communication-compatible PJ-ADMM for affine-coupled convex problems on digraphs.
• Explicit convergence-rate and network-induced error characterizations, facilitating system-level design for bandwidth-limited systems.
• A method directly applicable to practical scenarios in distributed control, federated learning, and networked resource allocation with stringent communication constraints.

From a theoretical viewpoint, the quantized consensus layer opens avenues for embedding advanced coding, quantization adaptation (e.g., variable-rate quantization), and robustness against link failures. Practically, the work lays the algorithmic foundation for high-throughput, energy-efficient, and privacy-conscious distributed optimization in IoT, edge learning, and cooperative control.

Potential future developments include:

• Tightening variable convergence analysis and constraint satisfaction error bounds, particularly with inexact or adaptive quantization,
• Extension to time-varying or open networks with dynamic agent participation,
• Incorporating inexact or stochastic subproblem solvers and asynchrony.

Conclusion

QDPJ-ADMM provides a principled, fully-distributed framework for solving affine-coupled convex resource allocation problems under quantized communication constraints. It achieves strong theoretical guarantees, a tunable accuracy–bandwidth tradeoff, and favorable empirical performance versus existing distributed and centralized PJ-ADMM schemes. This result is a critical enabler for scalable deployment of networked optimization protocols in bandwidth-limited settings (2604.14861).