A Sharp Estimate on the Transient Time of Distributed Stochastic Gradient Descent

Area of Science:

• Optimization Theory
• Distributed Systems
• Machine Learning

Background:

• Decentralized optimization problems involve minimizing average cost functions across networks.
• Agents often rely on noisy gradient information for decision-making.
• Distributed stochastic gradient descent (DSGD) is a key method for such scenarios.

Purpose of the Study:

• To perform a non-asymptotic convergence analysis of DSGD.
• To characterize the transient time for DSGD to reach its asymptotic convergence rate.
• To establish the sharpness of the theoretical results through a constructed optimization problem.

Main Methods:

• Non-asymptotic convergence analysis of DSGD.
• Theoretical analysis for strongly convex and smooth objective functions.
• Construction of a challenging optimization problem to validate theoretical bounds.

Main Results:

• DSGD achieves an optimal network-independent convergence rate in expectation, comparable to centralized stochastic gradient descent (SGD).
• The study quantifies the transient time required for DSGD to approach its asymptotic convergence rate.
• A "hard" optimization problem demonstrates the sharpness of the derived convergence bounds.

Conclusions:

• DSGD is an effective method for distributed optimization with noisy gradients.
• The characterization of transient time provides crucial insights into DSGD's practical performance.
• Theoretical results are validated by numerical experiments, confirming their tightness.