Research map / Optimization / Nonconvex and smooth optimization
Continuous-Time Analysis of Heavy Ball Momentum in Min-Max Games
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Continuous-time (first-order ODE) models with O(h^3) local error for simultaneous and alternating heavy ball momentum in min-max games, showing smaller momentum aids local convergence and implicit gradient regularization, opposite to minimization.
Problem
Heavy ball momentum in min-max games is poorly understood compared with minimization; it is unclear how momentum parameter and update scheme (simultaneous vs alternating) affect local convergence and implicit regularization of min-max learning dynamics.
Approach
Derives first-order continuous-time models for simultaneous and alternating HB whose local error is O(h^3), with the momentum and step size decoupled, and uses the Jacobians of these models to show smaller (even negative) momentum widens the step-size range for local convergence while alternating updates converge faster; it also shows smaller momentum and alternating updates steer trajectories to
Key results
- Smaller momentum beta widens the range of step sizes giving local convergence for simultaneous HB, since the bound 2(1-beta)^2/(1+beta) is decreasing in beta.
- Alternating updates reduce the largest eigenvalue real part by h|sigma|^2/(1-beta)^2 relative to simultaneous updates for small h and alpha.
- In GAN training on CIFAR-10 with Adam-style HB, smaller momentum and alternating updates yield lower average gradient slopes, and lower slopes correlate with lower FID.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use smaller momentum (lower β) | Sim-HB · Min-max games with dominant player interactions | range of step sizes permitting local convergence | Helped |
| use excessively negative momentum | Sim-HB · Min-max games with dominant player interactions | local convergence rate | Hurt |
| use positive momentum with a sufficiently small step size | Sim-HB · Min-max games with dominant player interactions | optimal local convergence rate | Helped |
| use alternating rather than simultaneous updates | HB · Min-max games with small step size and weak potential component | local convergence rate | Helped |
| use smaller momentum (lower β) | Sim-HB · 2D test functions 1 and 2 | average slope | Helped |
| use smaller momentum (lower β) | Alt-HB · 2D test functions 1 and 2 | average slope | Helped |
| use alternating rather than simultaneous updates | HB · 2D test functions 1 and 2 | average slope | Helped |
| use smaller momentum (lower β) | Alt-HB with Adam · CIFAR-10 GAN training | cumulative average slope | Helped |
| use alternating rather than simultaneous updates | HB with Adam · CIFAR-10 GAN training | cumulative average slope | Helped |
| achieve lower average slopes | HB with Adam · CIFAR-10 GAN training | FID | Helped |
Surprising findings
- Unlike minimization, negative momentum enlarges the step-size range for local convergence in min-max games.
- The implicit regularization effect is opposite to minimization: smaller rather than larger momentum leads to shallower gradient regions.
Limitations
- Analysis is in continuous time and does not account for stochastic gradients.
- Coefficients of the O(h*alpha) term require perturbed eigenvectors and are left open; superiority of alternating updates requires h and alpha small.
Experimental setup
Datasets: CIFAR-10
Benchmarks: CIFAR-10 (WGAN training, FID) 2D test functions (trajectory average slope)
Metrics: eigenvalue real part (local convergence) average slope of gradient norm FID
Baselines: Simultaneous heavy ball Alternating heavy ball Gradient descent-ascent (beta=0) O(h^2) continuous-time model of GDA
Builds on: Heavy ball momentum (Polyak) Implicit gradient regularization via continuous-time HB models Gradient descent-ascent continuous-time models
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