[Paper Review] A Communication-efficient Algorithm with Linear Convergence for Federated Minimax Learning

Zhenyu Sun, Ermin Wei|arXiv (Cornell University)|Jun 2, 2022

Stochastic Gradient Optimization Techniques4 citations

TL;DR

This paper proposes FedGDA-GT, a communication-efficient federated minimax learning algorithm that achieves linear convergence with constant stepsize by leveraging gradient tracking. It matches the centralized GDA's time complexity of O(log(1/ε)) communication rounds and outperforms Local SGDA in both convergence speed and accuracy.

ABSTRACT

In this paper, we study a large-scale multi-agent minimax optimization problem, which models many interesting applications in statistical learning and game theory, including Generative Adversarial Networks (GANs). The overall objective is a sum of agents' private local objective functions. We first analyze an important special case, empirical minimax problem, where the overall objective approximates a true population minimax risk by statistical samples. We provide generalization bounds for learning with this objective through Rademacher complexity analysis. Then, we focus on the federated setting, where agents can perform local computation and communicate with a central server. Most existing federated minimax algorithms either require communication per iteration or lack performance guarantees with the exception of Local Stochastic Gradient Descent Ascent (SGDA), a multiple-local-update descent ascent algorithm which guarantees convergence under a diminishing stepsize. By analyzing Local SGDA under the ideal condition of no gradient noise, we show that generally it cannot guarantee exact convergence with constant stepsizes and thus suffers from slow rates of convergence. To tackle this issue, we propose FedGDA-GT, an improved Federated (Fed) Gradient Descent Ascent (GDA) method based on Gradient Tracking (GT). When local objectives are Lipschitz smooth and strongly-convex-strongly-concave, we prove that FedGDA-GT converges linearly with a constant stepsize to global $ε$-approximation solution with $\mathcal{O}(\log (1/ε))$ rounds of communication, which matches the time complexity of centralized GDA method. Finally, we numerically show that FedGDA-GT outperforms Local SGDA.

Motivation & Objective

• Address the trade-off between communication efficiency and convergence accuracy in federated minimax learning.
• Overcome the limitations of Local SGDA, which fails to converge exactly with constant stepsizes due to gradient noise.
• Develop a method that ensures linear convergence with constant stepsize while minimizing communication rounds.
• Provide generalization bounds for distributed minimax learning using Rademacher complexity analysis.
• Demonstrate superior empirical performance compared to existing methods like Local SGDA.

Proposed method

• Propose FedGDA-GT, a federated gradient descent ascent algorithm enhanced with gradient tracking (GT) to reduce communication overhead.
• Integrate gradient tracking to maintain accurate gradient estimates across local updates, enabling constant stepsize convergence.
• Use local multiple-step updates with periodic global aggregation to reduce communication frequency.
• Apply Rademacher complexity analysis to derive generalization bounds for the empirical minimax objective.
• Prove linear convergence to an ε-approximate solution in O(log(1/ε)) communication rounds under strongly-convex-strongly-concave and Lipschitz smooth assumptions.
• Leverage theoretical analysis of gradient tracking to ensure convergence stability without diminishing stepsize.

Experimental results

Research questions

• RQ1 Can a federated minimax algorithm achieve linear convergence with constant stepsize while minimizing communication rounds?
• RQ2 Why does Local SGDA fail to converge exactly with constant stepsizes despite multiple local updates?
• RQ3 How does gradient tracking improve convergence in federated minimax settings compared to standard local update schemes?
• RQ4 What generalization bounds can be established for distributed empirical minimax learning under statistical sampling?
• RQ5 Can the proposed method match the time complexity of centralized GDA in terms of communication efficiency?

Key findings

• FedGDA-GT achieves linear convergence to an ε-approximate solution in O(log(1/ε)) communication rounds with constant stepsize.
• The algorithm matches the time complexity of centralized GDA, which is optimal for such problems.
• FedGDA-GT outperforms Local SGDA in numerical experiments, demonstrating faster convergence and higher accuracy.
• Generalization bounds for distributed minimax learning are derived with the same order O(1/√N) as centralized cases, where N is the total number of samples.
• Theoretical analysis confirms that Local SGDA cannot achieve exact convergence with constant stepsizes due to persistent gradient noise.
• Rademacher complexity analysis provides a generalization bound that scales with the metric entropy of the hypothesis space and sample size.