A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates (1704.07807v2)

Abstract: This paper proposes a novel proximal-gradient algorithm for a decentralized optimization problem with a composite objective containing smooth and non-smooth terms. Specifically, the smooth and nonsmooth terms are dealt with by gradient and proximal updates, respectively. The proposed algorithm is closely related to a previous algorithm, PG-EXTRA \cite{shi2015proximal}, but has a few advantages. First of all, agents use uncoordinated step-sizes, and the stable upper bounds on step-sizes are independent of network topologies. The step-sizes depend on local objective functions, and they can be as large as those of the gradient descent. Secondly, for the special case without non-smooth terms, linear convergence can be achieved under the strong convexity assumption. The dependence of the convergence rate on the objective functions and the network are separated, and the convergence rate of the new algorithm is as good as one of the two convergence rates that match the typical rates for the general gradient descent and the consensus averaging. We provide numerical experiments to demonstrate the efficacy of the introduced algorithm and validate our theoretical discoveries.

• The paper proposes NIDS, a decentralized proximal-gradient method that enables each agent to use uncoordinated, network-independent step-sizes based solely on local objective properties.
• It achieves linear convergence for strongly convex cases and o(1/k) sublinear rates for general convex problems by clearly separating network influence from objective characteristics.
• Numerical validations confirm NIDS's superior performance in decentralized environments, enhancing scalability and robustness in applications like sensor networks and distributed machine learning.

Overview of Decentralized Proximal-Gradient Methods

The paper presents a novel algorithm for decentralized optimization, named NIDS (Network Independent Decentralized Proximal-Gradient). The algorithm addresses a composite optimization problem with a focus on both smooth and nonsmooth components by leveraging gradient and proximal updates, respectively. A notable highlight of the NIDS algorithm is the independence of its step-sizes from network topologies, a distinguishing feature when compared to previous works like PG-EXTRA.

Key Features and Contributions

This work introduces several important enhancements and theoretical insights in decentralized optimization:

1. Uncoordinated Step-Sizes: NIDS distinguishes itself by enabling each agent within the network to choose its own step-size independently, based on the local properties of its objective function. The requirement for coordination across the network, a constraint in traditional methods, is thereby lifted.
2. Network-Independent Step-Sizes: The algorithm features step-sizes that are not constrained by the network topology. The upper bounds for these step-sizes simply rely on the properties of the local objective functions, and they can approach those typically utilized in centralized gradient descent methods.
3. Separated Convergence Rates: For cases with smooth objective functions and under strong convexity assumptions, NIDS achieves linear convergence rates. These rates distinctively separate the influence of the network topology from the functional properties, each adhering to traditional convergence bounds in gradient descent and consensus averaging literature.
4. Sublinear Convergence for the General Convex Case: The paper establishes an $o(1/k)$ convergence rate for NIDS under general convexity. This is slightly improved over existing methods like PG-EXTRA, further emphasizing NIDS's computational efficiency.

Numerical Validation and Implications

Numerical experiments further affirm the efficacy of NIDS. Particularly relevant is the deployment in applications necessitating decentralized computing architectures, such as those found in sensor networks or distributed machine learning scenarios. The experiments conducted span both strongly convex cases and scenarios incorporating nonsmooth terms, showcasing NIDS's applicability and superior performance relative to competing algorithms like DIGing and PG-EXTRA.

Theoretical and Practical Implications

The introduction of a step-size strategy decoupled from network topology provides a significant practical advantage in dynamic network situations or networks where global parameter coordination is challenging or infeasible. Specifically, this design choice can enhance scalability and robustness in real-world applications involving fluctuating connectivity or heterogeneous processing units.

On a theoretical level, the clear separation of convergence dependencies offers substantial insight into potential areas of performance bottleneck—whether they originate from network connectivity or the intrinsic properties of the objective functions. This clarity allows for targeted optimization, whether through network design improvements or function conditioning strategies.

Conclusion and Future Work

The paper presents substantial advancements in the field of decentralized optimization. The approach not only broadens the possible application domains for proximal-gradient techniques by addressing their typical limitations in decentralized settings but also sets a strong foundation for future exploration, such as integrating advanced acceleration strategies like Nesterov’s method or expanding applicability to non-stationary networks. This work promises to invigorate research and developments in decentralized optimization and its manifold applications.