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Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems
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Extends mean-field Langevin dynamics to symmetric distributional minimax problems via MFL-AG (weighted-averaged gradient, single loop) and MFL-ABR (anchored best response, double loop), with convergence guarantees and a uniform-in-time propagation of chaos result.
Problem
Global convergence and non-asymptotic discretization error control for mean-field Langevin-type dynamics solving entropy-regularized minimax optimization over probability distributions (mixed Nash equilibrium).
Approach
A symmetric single-loop MFL-AG in which each player's drift is the weighted historical average of past gradients (dual-averaging style), giving average-iterate NI convergence; plus MFL-ABR, a double-loop best-response scheme with Langevin inner loops yielding linear outer-loop convergence.
Key results
- MFL-AG average-iterate NI error bound of order (M^2/alpha^2 + M^2/alpha^2) * 4(r+1)^2/(r lambda^2 t), optimized at beta_t = t.
- MFL-ABR outer-loop NI error decays linearly in k, giving an epsilon-MNE in O((1/eps) log(1/eps)) outer iterations.
- In a 1D bilinear sigmoid experiment, MFL-AG and MFL-ABR outperform MFL-DA in 3-point NI error, with MFL-AG reaching zero error after a while.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use MFL-AG instead of ordinary descent-ascent dynamics (MFL-DA) | Mean-field Langevin dynamics · Simulated bilinear objective | 3-point NI error | Helped |
Surprising findings
- MFL-DA converges rapidly to a different distribution than MFL-AG and MFL-ABR, yet its early output is better in the NI comparison before the averaged methods dominate.
Limitations
- Experiments are limited to a 1D bilinear toy problem with no neural or Markov game experiments.
- MFL-ABR lacks a particle-level discrete analysis of the outer loop.
- Discretization constant C_1 depends on eta with roughly eta^{-1/alpha} order, and the bound has curse-of-dimensionality dependence via LSI constants.
Experimental setup
Benchmarks: bilinear sigmoid toy minimax problem
Metrics: Nikaido-Isoda error W1 distance between iterates
Baselines: MFL-DA
Builds on: mean-field Langevin dynamics dual averaging fictitious play mean-field best response flow
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