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Monotone Inclusion Approach to Weakly Monotone Discrete-Time Finite-Horizon Mean-Field Games

NeurIPS 2026 · Optimization › Nonconvex and smooth optimization · approach: Equilibrium learning in zero-sum games · 0 citations (OpenAlex)

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Casts finite-horizon discrete-time mean-field equilibrium computation under weak monotonicity as a constrained monotone inclusion over occupation measures and solves it with anchored projected gradient steps, giving an O(1/sqrt(T)) last-iterate tangential residual rate without regularization.

Problem

Computing mean-field equilibria in discrete-time finite-horizon mean-field games that are only weakly monotone and Lipschitz in the state measure, where no contraction holds and uniqueness is not guaranteed.

Approach

Reformulate MFE as a variational inequality over the polytope of occupation flows (valid when transitions do not depend on the state measure), so that the problem becomes a Lipschitz monotone inclusion 0 in F(rho)+N_K(rho); then apply anchored projected gradient steps with step size 1/(L sqrt(k+gamma)) and anchor weight gamma/(k+gamma), with no regularization.

Key results

• Tangential residual after T iterations is bounded by 25*gamma*sqrt(2H)*|S||A|*L_r/sqrt(T-1+gamma), i.e., O(1/sqrt(T)) without regularization or uniqueness.
• A two-state-plus-absorbing example (H=2, |S|=3, |A|=2) is constructed showing a matching Omega(1/sqrt(T)) lower bound for the iteration.
• Every accumulation point of the iterates induces an MFE, and the iterates converge to the projection of the anchor onto the MFE occupation-flow set.

Surprising findings

• The lower bound shows the O(1/sqrt(T)) rate is tight for this algorithm under the stated assumptions.

Limitations

• Requires the transition kernel to be independent of the state measure (Assumption on P), which is the main limitation the authors acknowledge.
• Requires oracle access to the reward and transition kernel; no sample-based or model-free version is given.
• Convergence in the main theorem for the iterates is stated for the tangential residual; the full last-iterate convergence proof is deferred to the appendix.

Experimental setup

Metrics: tangential residual convergence rate in T

Baselines: KL-regularized policy-update methods for weakly monotone MFGs contraction-based MFG algorithms (e.g., exponential convergence) O(1/T) oracle-based method

Builds on: anchoring (Halpern iteration) projected gradient descent weak monotonicity MFG framework

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