Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs (1406.2075v2)

Abstract: We investigate the convergence rate of the recently proposed subgradient-push method for distributed optimization over time-varying directed graphs. The subgradient-push method can be implemented in a distributed way without requiring knowledge of either the number of agents or the graph sequence; each node is only required to know its out-degree at each time. Our main result is a convergence rate of $O \left((\ln t)/t \right)$ for strongly convex functions with Lipschitz gradients even if only stochastic gradient samples are available; this is asymptotically faster than the $O \left((\ln t)/\sqrt{t} \right)$ rate previously known for (general) convex functions.

• The paper establishes that the subgradient-push algorithm achieves a convergence rate of O((log t)/t) for strongly convex functions.
• The method demonstrates robustness to noisy gradient estimates, reflecting realistic conditions in distributed networks.
• The approach scales to large, dynamic networks by relying only on minimal topological data, extending consensus protocols.

Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs

The paper "Stochastic Gradient-Push for Strongly Convex Functions on Time-Varying Directed Graphs" by Angelia Nedic and Alex Olshevsky explores a distributed optimization technique that is particularly pertinent to networks represented by time-varying directed graphs. This work offers significant advancements in understanding the behavior of distributed optimization algorithms under the constraints of such dynamic network environments.

The researchers propose and analyze the "subgradient-push" method—a decentralized algorithm designed to minimize a sum of convex functions distributed across a network. Each node in the network has knowledge of only its local function and can communicate with a subset of neighboring nodes. The network is modeled as a directed graph where the topology changes over time, a common scenario in wireless communication systems with mobile nodes or intermittent connections.

Key Contributions

1. Improved Convergence Rate: A primary contribution of this paper is the establishment of a convergence rate of $O((\log t)/t)$ for strongly convex functions. This rate represents a substantial improvement over the previously known rate for general convex functions, providing a more efficient pathway to optimality.
2. Robustness to Noisy Gradients: The algorithm is robust to noise in gradient estimates, a critical feature since it assumes only stochastic subgradient samples are available at the nodes. This assumption reflects real-world conditions where precise gradient calculations are often infeasible.
3. Scalability & Flexibility: The subgradient-push algorithm requires minimal information from the network's topology, making it scalable to large networks without the need for centralized control. Each node only needs to be aware of its out-degree at each timestep, enabling the algorithm's application to large-scale, dynamically changing networks.
4. Relation to Consensus Protocols: The algorithm builds upon and extends the push-sum protocol for computing averages on directed graphs, highlighting similarities in structure and providing theoretical convergence guarantees even in decentralized environments with directional communication.

Implications and Future Directions

The implications of the work are multifold. Theoretically, it bridges the gap between deterministic consensus methods and stochastic optimization, offering a viable solution to distributed graph optimization challenges. Practically, it provides a foundational technique for applications in sensor networks, distributed machine learning tasks, and any domain requiring decentralized decision making over unstable networks.

The paper suggests several avenues for future research. These include developing consensus algorithms with faster convergence rates that scale polynomially with network size, and exploring optimization scenarios with other types of convexity assumptions or noise models. Another promising direction is quantifying trade-offs between convergence speed, communication cost, and computational overhead in different network scenarios, which could lead to more efficient distributed systems design.

In conclusion, Nedic and Olshevsky's work on the subgradient-push algorithm advances the state of distributed optimization by enhancing our understanding of how strongly convex problems can be effectively solved in complex network settings. Their results provide both a technical framework and a motivation for ongoing research into distributed algorithms that must operate under constraints of limited information and communication.