[Paper Review] On the Suboptimality of Negative Momentum for Minimax Optimization

Guodong Zhang, Yuanhao Wang|arXiv (Cornell University)|Aug 17, 2020

Advanced Bandit Algorithms Research49 references4 citations

TL;DR

This paper analyzes negative momentum in smooth, strongly-convex strongly-concave minimax optimization by linking it to Chebyshev polynomials via a variational inequality framework. It shows that while negative momentum accelerates local convergence with complexity 𝒪(κ¹.⁵), it is suboptimal compared to the 𝒪(κ) complexity of Extra-Gradient and Optimistic Gradient Descent Ascent, establishing the first explicit convergence rate for negative momentum in this setting.

ABSTRACT

Smooth game optimization has recently attracted great interest in machine learning as it generalizes the single-objective optimization paradigm. However, game dynamics is more complex due to the interaction between different players and is therefore fundamentally different from minimization, posing new challenges for algorithm design. Notably, it has been shown that negative momentum is preferred due to its ability to reduce oscillation in game dynamics. Nevertheless, the convergence rate of negative momentum was only established in simple bilinear games. In this paper, we extend the analysis to smooth and strongly-convex strongly-concave minimax games by taking the variational inequality formulation. By connecting momentum method with Chebyshev polynomials, we show that negative momentum accelerates convergence of game dynamics locally, though with a suboptimal rate. To the best of our knowledge, this is the \emph{first work} that provides an explicit convergence rate for negative momentum in this setting.

Motivation & Objective

• To analyze the convergence behavior of negative momentum in smooth, strongly-convex strongly-concave minimax problems beyond the bilinear case.
• To determine whether negative momentum achieves optimal convergence rates in this setting, given its strong performance in bilinear games.
• To formally connect negative momentum dynamics to polynomial approximation and Chebyshev polynomials for theoretical analysis.
• To establish a local convergence rate for negative momentum and compare it with the known optimal rates of EG and OGDA.
• To clarify the fundamental difference between negative momentum and OGDA, despite their similar rates in bilinear games.

Proposed method

• Formulate the minimax problem as a variational inequality using the Jacobian of the game's gradient field.
• Model first-order methods as discrete dynamical systems and linearize the dynamics around the equilibrium for local analysis.
• Establish a correspondence between momentum methods and polynomial approximation, particularly linking Polyak’s momentum to Chebyshev polynomials.
• Derive the optimal momentum parameter and asymptotic convergence rate by analyzing the extremal properties of Chebyshev polynomials.
• Use spectral analysis of the linearized system to bound the convergence rate in terms of the condition number κ.
• Compare the derived rate of negative momentum with the known lower bound Ω(κ) for first-order methods and the 𝒪(κ) rate of EG and OGDA.

Experimental results

Research questions

• RQ1 Does negative momentum achieve optimal convergence in smooth, strongly-convex strongly-concave minimax games?
• RQ2 What is the explicit local convergence rate of negative momentum in this setting?
• RQ3 How does the convergence rate of negative momentum compare to that of Extra-Gradient and Optimistic Gradient Descent Ascent?
• RQ4 Why does negative momentum perform well in bilinear games but remain suboptimal in more general settings?
• RQ5 Can the dynamics of negative momentum be formally analyzed via polynomial approximation and Chebyshev polynomials?

Key findings

• Negative momentum achieves a local convergence rate of 𝒪(κ¹.⁵) in smooth, strongly-convex strongly-concave minimax games.
• This rate is suboptimal compared to the Ω(κ) lower bound for first-order methods, which is matched by Extra-Gradient and Optimistic Gradient Descent Ascent.
• The analysis establishes the first explicit convergence rate for negative momentum in this general setting, extending beyond prior results limited to bilinear games.
• The connection between negative momentum and Chebyshev polynomials enables precise derivation of the convergence rate and optimal parameter tuning.
• Despite matching the rate of OGDA and EG in bilinear games, negative momentum fails to achieve the optimal rate in the broader strongly-convex strongly-concave class.
• Numerical simulations confirm that negative momentum outperforms GDA but underperforms OGDA, validating the theoretical rate gap.