Distributed Stochastic Gradient Tracking Methods (1805.11454v5)

Abstract: In this paper, we study the problem of distributed multi-agent optimization over a network, where each agent possesses a local cost function that is smooth and strongly convex. The global objective is to find a common solution that minimizes the average of all cost functions. Assuming agents only have access to unbiased estimates of the gradients of their local cost functions, we consider a distributed stochastic gradient tracking method (DSGT) and a gossip-like stochastic gradient tracking method (GSGT). We show that, in expectation, the iterates generated by each agent are attracted to a neighborhood of the optimal solution, where they accumulate exponentially fast (under a constant stepsize choice). Under DSGT, the limiting (expected) error bounds on the distance of the iterates from the optimal solution decrease with the network size $n$, which is a comparable performance to a centralized stochastic gradient algorithm. Moreover, we show that when the network is well-connected, GSGT incurs lower communication cost than DSGT while maintaining a similar computational cost. Numerical example further demonstrates the effectiveness of the proposed methods.

Analysis of Distributed Stochastic Gradient Tracking Methods

The academic paper "Distributed Stochastic Gradient Tracking Methods" by Shi Pu and Angelia Nedić explores efficiently solving convex optimization problems in distributed network settings. The authors propose two distributed algorithms: a Distributed Stochastic Gradient Tracking method (DSGT) and a Gossip-like Stochastic Gradient Tracking method (GSGT). These methods address the challenges presented by distributed environments, where each network node (agent) only has access to local noisy gradients of a global objective function to be minimized.

For context, distributed convex optimization frequently arises in machine learning, decentralized network control, and resource management in wireless networks. The paper specifically focuses on scenarios where agents must collaboratively minimize a global cost function, represented as the average of their local strongly convex and smooth cost functions.

Distributed Stochastic Gradient Tracking Method (DSGT)

The DSGT method revolves around maintaining an auxiliary variable at each agent to track the average stochastic gradients of the global cost function. The paper demonstrates that with a chosen constant step size, the algorithm's iterative sequence converges to a neighborhood of the optimal solution at an exponential rate. Key results here include:

• Limiting (expected) error bounds diminish as the network size increases, achieving a performance gradient comparable to centralized algorithms—a promising trait for distributed settings.
• Though the auxiliary variables and tracking increase communication costs twice compared to standard subgradient methods, the decrease in error bounds justifies such costs in scenarios demanding high accuracy.

Gossip-like Stochastic Gradient Tracking Method (GSGT)

The GSGT method offers a communication-efficient alternative. It randomizes node activation and pairwise communication to reduce overall network communication burdens. This protocol performs favorably particularly in well-connected networks and offers:

• Lower communication costs compared to DSGT (especially when node connectivity is high) while maintaining comparable computational effort.
• Ultimately, despite a slower convergence rate compared to DSGT, GSGT's reduced communication burden makes it appealing for large-scale network applications where inter-node communication can become a bottleneck.

Theoretical Contributions and Assumptions

Both algorithms were rigorously analyzed under standard assumptions, such as strong convexity and Lipschitz continuity of gradient functions. The paper's theoretical contributions expand upon existing frameworks by:

1. Providing error bounds and convergence rates illustrating performance dependency on network topology.
2. Proving that DSGT under diminishing step sizes converges with an optimal $\mathcal{O}(1/k)$ rate.

Additionally, the authors presented comprehensive bounds on spectral norms to facilitate optimal weight selection in communication matrices, directly influencing convergence properties.

Implications and Future Directions

The research broadens existing knowledge on decentralized optimization algorithms, demonstrating that distributed stochastic methods can match centralized approaches' accuracy under correct conditions. Notably, it motivates exploring further network models beyond fixed and connected graphs, incorporating dynamics of time-variance and directed edges. Future investigations might delve into adaptive schemes, automatic optimization of communication protocols, or tackling heterogeneous and adversarial network conditions.

Upon reflecting on the observations from extensive numerical validations, they reinforced the theoretical findings and offered insights into practical settings and parameter tuning strategies.

In summary, the comprehensive theoretical analysis and practical implications of DSGT and GSGT offer promising pathways toward scalable distributed optimization strategies fundamental to advancements in decentralized AI applications.