Distributed optimization over time-varying directed graphs (1303.2289v2)

Abstract: We consider distributed optimization by a collection of nodes, each having access to its own convex function, whose collective goal is to minimize the sum of the functions. The communications between nodes are described by a time-varying sequence of directed graphs, which is uniformly strongly connected. For such communications, assuming that every node knows its out-degree, we develop a broadcast-based algorithm, termed the subgradient-push, which steers every node to an optimal value under a standard assumption of subgradient boundedness. The subgradient-push requires no knowledge of either the number of agents or the graph sequence to implement. Our analysis shows that the subgradient-push algorithm converges at a rate of $O(\ln(t)/\sqrt{t})$, where the constant depends on the initial values at the nodes, the subgradient norms, and, more interestingly, on both the consensus speed and the imbalances of influence among the nodes.

• The paper introduces the subgradient-push algorithm that combines subgradient methods with the push-sum protocol to optimize distributed convex functions.
• It rigorously demonstrates a convergence rate of O(ln t/√t) by extending the push-sum protocol to manage perturbations from subgradients.
• The work underscores the algorithm's scalability and practical relevance for sensor networks, robotics, and distributed machine learning.

Distributed Optimization over Time-Varying Directed Graphs

This paper, "Distributed optimization over time-varying directed graphs," by Angelia Nedić and Alex Olshevsky, addresses the problem of performing distributed convex optimization via a network of nodes, each possessing its own convex function. The primary aim is to minimize the sum of these functions. The research is particularly relevant for applications in sensor networks, distributed control systems, and large-scale machine learning problems where centralized data collection and processing are infeasible.

Key Contributions

The paper makes the following noteworthy contributions:

1. Algorithm Development: The authors introduce a novel algorithm named Subgradient-Push, which combines subgradient methods with the push-sum protocol to handle optimization over time-varying, directed communication graphs. The subgradient-push is designed to achieve consensus among the nodes while ensuring convergence to an optimal solution.
2. Theoretical Analysis: The convergence analysis of the subgradient-push algorithm is rigorously performed. Specifically, the authors show that the algorithm converges to an optimal value with a rate of $O\left(\frac{\ln t}{\sqrt{t}}\right)$. The proof hinges on extending the push-sum protocol to handle perturbations introduced by subgradients and leveraging results from the theory of non-stationary Markov chains.
3. Scalability Results: The constants in the convergence rates are shown to depend on initial conditions, subgradient norms, the speed of information diffusion in the network, and imbalances of influence among nodes. This dependency provides insights into the scalability of the algorithm for large and complex networks.

Detailed Results

Convergence Rate

A critical result established in the paper is that the convergence of the subgradient-push algorithm is logarithmic in time adjusted by a square root term: $O\left(\frac{\ln t}{\sqrt{t}}\right).$ Specifically, it was shown that: $$F(\widetilde{z}_i(t+1)) - F(z^*) \leq \frac{n}{2}\frac{\|\bar{z}(0) - z^*\|^2 }{\sqrt{t+1}} + O\left(\frac{1 + \ln t}{\sqrt{t+1}}\right),$$ where $\widetilde{z}_i(t+1)$ is a convex combination of the values at node $i$ and $F(z^*)$ is the optimal value. This result indicates that the objective function $F(\widetilde{z}_i(t+1))$ converges to the optimal value $F(z^*)$ at the stated rate.

Algorithm Simplicity and Practicality

The subgradient-push algorithm requires each node to know its out-degree but does not require knowledge of the entire graph structure or the number of nodes. This decentralization is practical for dynamic and large-scale networks where such global information is either unavailable or costly to obtain.

Implications and Future Directions

The practical and theoretical implications of this work are significant for decentralized computation in distributed systems. The proposed method is particularly applicable to problems in robotics, sensor networks, and distributed machine learning, where nodes must operate based on local information and intermittent communication.

Theoretical Implications

• Non-asymptotic Performance: Understanding the constants involved in the convergence rate helps in determining the non-asymptotic performance of distributed algorithms in finite time.
• Information Diffusion Measurement: The research underlines how the rate of information diffusion within the network influences convergence rates, potentially guiding the design of more efficient network structures.

Practical Implications

• Scalability: The subgradient-push algorithm's reliance on local information and robustness to time-varying directed graphs suggests excellent scalability for very large networks, which is crucial for modern distributed systems.
• Implementation Feasibility: The broadcast-based nature of the subgradient-push allows for straightforward implementation using existing networking infrastructure.

Future Research Directions

The paper opens up several avenues for future research:

• Graph Sequences: Investigating how various graph sequences and their properties (like cut size and diameter) affect the constants $\delta$ and $\lambda$ in convergence metrics.
• Algorithm Extensions: Developing algorithms that can handle even more general conditions, such as asynchronous updates or stochastic subgradients.
• Experimental Validation: While the paper includes theoretical proof and simulations, further real-world experimental validation in various application domains would be beneficial.

This work lays a robust foundation for solving distributed optimization problems in complex, real-world networks and provides a scalable, efficient algorithm suitable for modern computational demands.