Pith. sign in

REVIEW 2 major objections 6 minor 40 references

Kullback-Leibler Mirror-Prox for Measure-Valued Variational Inequalities and Mean-Field Equilibria

T0 review · 2 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that two explicit exponential reweightings per step, run on a fine enough mesh for enough iterations, compute static mean-field equilibria to any prescribed accuracy; a KL penalty selects among multiple equilibria.

desk verdict A rigorous and useful KL Mirror-Prox method for static MFGs; the core ergodic and mesh guarantees hold up, but Section 5's algorithmic selection step is asserted rather than proved. read the letter →

arxiv 2608.10293 v1 pith:SEDJNKXT submitted 2026-08-10 math.OC

Erhan Bayraktar , Ibrahim Ekren , Lu Vy , Ziqing Zhang This is my paper · ORCID
classification math.OC MSC 91A1649J4065K15
keywords meanfieldgamesvariationalinequalitiesextragradientmethodrelativeentropyLasry-Lionsmonotonicitymirror-proxequilibriumselectionmeshrefinement
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes that static mean field equilibria—population distributions in which every agent's chosen state is a best response to that same distribution—can be computed by a two-step reweighting scheme with fully quantitative guarantees. The method, KL Mirror-Prox, replaces the Euclidean projections of the classical extragradient algorithm with relative-entropy proximal steps, so each half-iteration is an explicit exponential reweighting of the current measure: no projection, line search, PDE solve, or best-response computation is needed. Under Lasry–Lions monotonicity, a $W_1$-Lipschitz condition on the measure variable, and spatial Lipschitz continuity of the cost, the averaged iterate satisfies a Minty residual bound of order $\log(1/m_h)/(\lambda K) + (2L_m+L_x)h$, which the paper converts into a certificate: for any accuracy $\delta>0$, explicit choices of mesh $h$ and iterations $K$ produce a $\delta$-approximate mean field equilibrium. Under strong monotonicity the same machinery yields $W_1$-metric convergence rates for the last, best, and averaged iterates, and a KL-type Tikhonov penalty is shown to select, in the vanishing-penalty limit, the equilibrium of minimal relative entropy with respect to a chosen reference measure.

What carries the argument

The machinery is the KL three-point identity for the relative-entropy Bregman geometry: for a cost $g$ and its exponential tilt $\zeta(dx)\propto e^{-\lambda g(x)}\rho(dx)$, the identity $\lambda\langle g,\zeta-\eta\rangle = D_{\rm KL}(\eta\|\rho)-D_{\rm KL}(\eta\|\zeta)-D_{\rm KL}(\zeta\|\rho)$ holds exactly for every competing measure $\eta$. Applied to the predictor and corrector half-steps, the two identities add up so that intermediate KL terms cancel and the only leftover is the predictor–corrector mismatch $\langle J(\nu_k,\cdot)-J(\mu_k,\cdot),\nu_k-\mu_{k+1}\rangle$, bounded by $\lambda L_m D_\Omega$ times the KL dissipation; summing over iterations telescopes the relative-entropy terms into the ergodic Minty residual. A measurable mesh projection $Q_h$ with $\sup_x|Q_h(x)-x|\le h$ carries these finite-mesh estimates to a general compact state space at the cost of the $(2L_m+L_x)h$ term, and the paper's approximate MFE/VI/MVI implications (with the square-root conversion $\varepsilon\mapsto 2\sqrt{2L_m D_\Omega\varepsilon}$) complete the chain from residual to certified $\delta$-approximate equilibrium.

What would settle it

On the arctangent example (6.1) with $M=1001$ mesh points, compute $\sup_\eta \langle J(\eta,\cdot),\bar\nu_{K,h}-\eta\rangle$ and check that it stays below $\log M/(\lambda K)+5/(M-1)$: a violation at the predicted $K$ would falsify the Theorem 3.2 bound. A second test: run the fixed-stepsize method on a monotone static game whose cost is continuous but only Hölder in the measure, e.g., $J(\mu,x)=\int |x-y|^{1/2}\,\mu(dy)$ on $\Omega=[-1,1]$; this operator has no finite $L_m$, so the paper's step-size rule offers no guarantee, and observing divergence or the need for a line search would confirm that the Lipschitz assumption is truly load-bearing.

Watch

Extended reading notes

Core claim

The paper's central claim is that a static mean field game, written as the variational inequality $\langle J(\mu^*,\cdot),\eta-\mu^*\rangle\ge 0$ for all $\eta\in\mathcal P(\Omega)$, is solved by iterating two explicit exponential tilts—the predictor $\nu_k\propto \mu_k e^{-\lambda J(\mu_k,\cdot)}$ and the corrector $\mu_{k+1}\propto \mu_k e^{-\lambda J(\nu_k,\cdot)}$—and that the error of the iterates is explicitly quantifiable from the algorithm's constants. Under the standard Lasry–Lions monotonicity condition $\langle J(\mu,\cdot)-J(\nu,\cdot),\mu-\nu\rangle\ge 0$ and a stepsize $\lambda<1/(L_m D_\Omega)$, the KL three-point identity telescopes over iterations into an $O(1/K)$ Minty residual bound on any finite mesh; projecting an arbitrary comparison measure onto the mesh and using the measure- and spatial-Lipschitz constants $L_m,L_x$ upgrades this to a uniform bound $\varepsilon_{K,h}=\log(1/m_h)/(\lambda K)+(2L_m+L_x)h$ on the full compact state space. The approximate-equivalence results then convert this residual into an equilibrium statement: an $\varepsilon$-Minty solution is a $\delta$-VI solution with $\delta=2\sqrt{2L_m D_\Omega\,\varepsilon}$, so for every $\delta>0$ the choices $h\le \varepsilon^*_\delta/(2(2L_m+L_x))$ and $K\ge 2\log(1/m_h)/(\lambda\varepsilon^*_\delta)$ certify that the averaged iterate is a $\delta$-MFE of the original problem. Under strong Lasry–Lions monotonicity the same one-step estimate yields sublinear and then geometric last-iterate $W_1$-convergence on a fixed mesh, and a separate discretization bound $W_1(\mu^*_h,\mu^*)\le \sqrt{L_x/\alpha}\,h$ splits the total metric error into optimization plus approximation terms. Finally, adding $\varepsilon\,D_{\rm KL}(\cdot\|\rho)$ as a Tikhonov penalty and letting $\varepsilon\downarrow 0$ selects the equilibrium of minimal relative entropy with respect to the reference measure $\rho$, whenever that minimizer is unique and finite.

Load-bearing premise

The load-bearing premise is that the cost operator changes at a bounded, known rate as the population measure varies—$W_1$-Lipschitz continuity with a known constant $L_m$—together with the fixed stepsize restriction $\lambda<1/(L_m D_\Omega)$; if the operator is only continuous in the measure, or if $L_m$ is unknown, every quantitative guarantee in the paper collapses and stepsize selection would need a different mechanism.

Editorial extensions

If this is right

  • Any prescribed accuracy is reachable: for each $\delta>0$ the paper specifies the mesh size and the iteration count explicitly, so a user can compute a $\delta$-approximate mean field equilibrium without tuning, line search, or a posteriori selection.
  • The method is projection-free and needs only cost evaluations: each half-step is one evaluation of $J$ plus an exponential reweighting, so it applies to potential and nonpotential costs, to costs that are neither convex nor differentiable in the state variable, and to problems with no PDE structure.
  • Genuinely non-Dirac equilibria are in scope: because the analysis never imposes displacement monotonicity or spatial convexity, equilibria supported on several states (such as the $\frac12\delta_{-1}+\frac12\delta_1$ double-well example) are computed directly rather than excluded by assumption.
  • Under strong monotonicity the metric error separates into an optimization term and a mesh term, yielding, under the paper's scaling assumptions, explicit iteration counts of order $d^2\varepsilon^{-2}|\log\varepsilon|$ for $\varepsilon$-accuracy in $W_1$, while storage on a quasi-uniform grid still scales like $\varepsilon^{-d}$.
  • The KL Tikhonov penalty turns equilibrium selection into a modeling choice: the vanishing-penalty limit is the equilibrium minimizing $D_{\rm KL}(\cdot\|\rho)$, so the reference measure $\rho$ explicitly controls which equilibrium is returned when several exist.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the updates are explicit tilts, the algorithm is an oracle method: I would expect the same iteration bounds to survive if $J$ were evaluated by Monte Carlo or particle samples, with the sampling error folded into the $W_1$-Lipschitz mismatch term; the paper does not analyze this stochastic variant, so this is a conjecture to test.
  • The uniform Minty bound acts as a run-time certificate: on a finite mesh one can evaluate the gap $\langle J(\eta,\cdot),\bar\nu_{K,h}-\eta\rangle$ at adversarial test measures such as point masses and verify the $\delta$-MFE claim during the run, a cheap a posteriori check the paper leaves implicit.
  • The explicit dependence on $\log(1/m_h)$ makes the initialization a computational lever: for a quasi-uniform mesh with uniform initialization ($D_{0,h}=O(d|\log h|)$), the minimal atom $m_h$ rather than the mesh resolution can dominate the iteration count, so designing exploration measures with controlled small masses may matter as much as refining the grid.
  • For non-unique equilibria the reference measure $\rho$ is a selection dial; taking $\rho$ uniform selects the maximum-entropy equilibrium (minimal relative entropy to uniform), which is plausibly the equilibrium reached by vanishing-noise or learning dynamics—an identification that would connect this paper's selection mechanism to the common-noise selection literature.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. This paper develops a KL-divergence version of Korpelevich's extragradient method for static mean-field equilibria, formulated as a variational inequality on the space of probability measures over a compact state space. On finite meshes X_h, each half-step reduces to an explicit exponential reweighting, and Proposition 3.1 gives a consistency result between discrete and continuous iterates. Under Lasry–Lions monotonicity and W1-Lipschitz continuity of the cost operator (Assumptions 3.2–3.4), the paper proves an O(1/K) ergodic Minty-residual bound (Lemma 3.3), a joint mesh/iteration bound (Theorem 3.2), and, via the approximate-MFE conversions in Propositions 3.2–3.5, explicit choices of h and K that certify a δ-MFE of the original compact-state problem. Under strong Lasry–Lions monotonicity (Assumption 4.1), it derives last-, best-, and averaged-iterate W1 rates (Theorems 4.1–4.2). Section 5 adds a KL-type Tikhonov regularization intended to select the entropy-minimizing equilibrium: Theorem 5.1 proves this selection for exact solutions of the regularized VI, and Remark 5.1 sketches an algorithmic construction via KL Mirror-Prox. The appendix contains the full proofs of the results in Sections 3 and 4.

Significance. If the central claims hold, this is a valuable contribution to computational static mean-field games. The quantitative analysis is genuinely joint in the mesh size h and the iteration count K, and it provides explicit, finite computation certificates without requiring PDE structure, potentiality, or convexity of the cost in the state variable. The framework covers non-Dirac equilibria and nonpotential operators. The proofs in Sections 3 and 4 are written out in full with explicit constants, the numerical examples satisfy the stated assumptions, and accompanying code is provided. I also note that the central quantitative claim, Theorem 3.2 together with Propositions 3.2 and 3.5, is internally consistent: the mesh-projection argument, the Minty-residual bound, and the conversion to a δ-MFE all check out under the stated assumptions. The main caveat is the equilibrium-selection part: Theorem 5.1 is nonconstructive, and the algorithmic claim in Remark 5.1 is only a sketch. Once that gap is resolved, the paper would be a strong addition to the literature.

major comments (2)
  1. [Section 5, Remark 5.1 (residual estimates)] Remark 5.1 claims that the KL Mirror-Prox algorithm applied to the regularized operator F^{ε_h}_h can produce a sequence satisfying the approximate discrete regularized VI with r_h/ε_h→0. The load-bearing step is the assertion that boundedness of J together with 0<λ_hε_h<1 implies a uniform positive lower bound a_h>0 on every coordinate of every predictor and corrector along the whole trajectory. No proof is given; the statement is not immediate because the corrector update contains the factor (ν_k/ρ_h)^{-λ_hε_h}, whose logarithm can in principle become unbounded if coordinates collapse. Furthermore, the proposed choice of λ_h from the 'trajectory-local step-size condition' depends on \tilde L_h, which depends on a_h, which in turn depends on λ_h; the sketch does not demonstrate that a positive λ_h satisfying both conditions exists. Since Theorem 5.1 concerns exact solutions of the regularized VI, this remark is the only argument making the entropy-selection mechanism algorithmic. Please either provide a complete proof of the uniform lower bound and of the consistency of the step-size choice, or withdraw the algorithmic claim and present Theorem 5.1 as a nonconstructive selection result.
  2. [Section 5, Remark 5.1 (residual estimates)] The residual conversion in the same remark is also only sketched. Even assuming a uniform lower bound a_h, the proof must show that the local W1-to-L∞ Lipschitz constant \tilde L_h on the set of probability vectors with coordinates at least a_h/2 is compatible with the chosen λ_h, that all iterates and the comparison mixtures ρ_{h,t} stay in the local Lipschitz regime for the full trajectory, and that the constants in s_{h,K} and r_{h,K} have the correct dependence on ε_h so that r_{h,K_h}/ε_h→0 is achievable. The remark asserts s_{h,K}=O(K^{-1}) and r_{h,K}≤2√(2\tilde L_hD_Ωs_{h,K}) without tracking this dependence. A rigorous argument, or an equivalent lemma, is needed before the approximate-solution extension of Theorem 5.1 can be used.
minor comments (6)
  1. [Section 5, Remark 5.2] The hypomonotone discussion is presented in two sentences and is explicitly conditional; please mark it as informal or provide a proof, since it currently reads as an unproved assertion about a modification of the main theorem.
  2. [Notation] The symbol ∥μ−ν∥_1 is defined as ∫|dμ−dν|, which is twice total variation, but the paper later uses this norm together with W1 and Pinsker-type inequalities. Adding an explicit reminder in Lemma 3.2 or in the notation paragraph would prevent confusion with the standard L1 norm.
  3. [Proof of Theorem 3.1] The sentence 'Applying Lemma 3.3 to P(X_h)' is formally unusual because Lemma 3.3 is stated for a general compact Ω; a one-line remark that X_h is compact and the same proof applies would remove ambiguity.
  4. [Section 6] The claimed equilibria μ*=δ_{−1/2} for Example (6.1) and μ*=1/2δ_{−1}+1/2δ_1 for Example (6.3) are stated without derivation; for a numerical section intended to illustrate the theory, adding the one-line verification would improve reproducibility.
  5. [Sections 3–4] The paper does not discuss how L_m, L_x, and m_h are estimated in practice if they are not known a priori; since the step-size and mesh/iteration prescriptions depend on these constants, a short remark on this practical issue would be useful.
  6. [Remark 4.1] The asymptotic statement 'p_k ∼ 1/(λk)' is asserted from the recurrence p_{k+1}=p_k−λp_k^2+O(p_k^3); including a short asymptotic argument would make the impossibility statement self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence and approximation guarantees are derived from stated Lipschitz/monotonicity assumptions via standard inequalities, with no fitted constants and no load-bearing self-citations.

full rationale

The paper's derivation chain is self-contained. Theorem 3.2 and the δ-MFE conversion use only the stated assumptions (Lasry–Lions monotonicity, W1-Lipschitz continuity with constant Lm, spatial Lipschitz continuity with Lx), the KL three-point identity, Pinsker's inequality, Kantorovich–Rubinstein duality, and the finite-mesh projection estimates; no constant is fitted to force the target guarantee, and the step-size condition λ < 1/(Lm D_Omega) is an explicit assumption rather than a calibrated value. The approximate-MFE translation (Propositions 3.2 and 3.5) is a definitional equivalence plus a standard Minty-to-VI inequality, not an input disguised as an output. Section 4's metric rates similarly follow from strong monotonicity and finite-alphabet reverse Pinsker inequalities, with mesh-error separated from optimization error. Section 5's equilibrium-selection theorem is independent of the algorithmic convergence results and relies on the regularized VI, explicit Gibbs structure, and balance conditions; no self-citations by the authors appear in the reference list, and the numerical experiments select models that satisfy the stated assumptions rather than fitting parameters to the theory. The only notable weakness is Remark 5.1's asserted uniform lower bound a_h for the trajectories of the entropy-regularized operator, which is not proved; that is an incomplete correctness argument, not a circular reduction, because the claimed regularized-VI residual is not equal to the input assumptions by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper's guarantees rest on four explicit structural assumptions (monotonicity, W1-Lipschitz, spatial Lipschitz, strong monotonicity when rates are claimed) plus standard inequalities. No hidden fitted parameters or invented entities. The algorithmic hyperparameters lambda, epsilon_h and the initialization are design choices, clearly exposed in the theorems.

free parameters (3)
  • Stepsize lambda = 0 < lambda < 1/(Lm D_Omega); lambda = 0.49 and 0.24 in Section 6
    Algorithmic hyperparameter prescribed by theory, not fitted to measured data.
  • Tikhonov penalty epsilon_h = epsilon_h = sqrt(h) suggested in Remark 5.1
    Chosen to satisfy balance conditions h/epsilon_h -> 0 and epsilon_h log(1/rho_min,h) -> 0; no data fitting.
  • Initial distribution mu0 and mesh masses m_h = Uniform or perturbed uniform in Section 6
    Full-support initialization required for exploration; affects convergence via log(1/m_h).
assumptions (6)
  • domain assumption Assumption 3.2: Lasry-Lions monotonicity of J
    Central monotonicity; used in Lemma 3.3 and Proposition 3.4 to relate VI and MVI.
  • domain assumption Assumption 3.3: W1-Lipschitz continuity of J in the measure variable
    Controls the extragradient mismatch; required for step-size restriction and mesh consistency.
  • domain assumption Assumption 3.4: spatial Lipschitz continuity of J in x
    Used for the quantitative finite-mesh Minty bound (Theorem 3.2) and mesh-VI comparison (Proposition 4.1).
  • domain assumption Assumption 4.1: strong Lasry-Lions monotonicity with W1
    Gives metric convergence rates, uniqueness, and the Fejer-type descent estimates.
  • domain assumption Continuity along mixtures in Proposition 3.3
    Bridges MVI to VI; follows from Assumption 3.3 in the paper's setting.
  • standard math Standard background results: Pinsker's inequality, compactness of P(Omega), Kantorovich-Rubinstein duality, reverse Pinsker
    Imported lemmas used in proofs without modification; cited to Csiszar-Talata and Sason-Verdu.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Kullback-Leibler Mirror-Prox for Measure-Valued Variational Inequalities and Mean-Field Equilibria." pith.science (2026). https://pith.science/paper/SEDJNKXT

@misc{pith2026260810293,
  author       = {Pith},
  title        = {Pith review of: Kullback-Leibler Mirror-Prox for Measure-Valued Variational Inequalities and Mean-Field Equilibria},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SEDJNKXT}},
  note         = {Machine review of arXiv:2608.10293}
}
read the original abstract

We study the computation of static mean-field equilibria on a compact state space by formulating the equilibrium condition as a variational inequality over probability measures. We propose an entropic variant of Korpelevich's extragradient algorithm---the Kullback--Leibler Mirror-Prox method---in which Euclidean projections are replaced by relative-entropy proximal steps. Each half-step is therefore an explicit exponential reweighting of the current measure, implemented on a finite state-space discretization. Under Lasry--Lions monotonicity and continuity assumptions, we prove convergence of mesh-refined ergodic averages and obtain finite-iteration Minty-residual and approximate-equilibrium bounds that jointly quantify iteration and discretization errors. Under strong monotonicity, we derive metric convergence rates for the last, best, and averaged iterates. We also develop a KL-type Tikhonov regularization that selects the equilibrium minimizing relative entropy with respect to a reference measure. The framework applies to potential and nonpotential cost operators and does not require differentiability or convexity of the cost in the individual state.

Figures

Figures reproduced from arXiv: 2608.10293 by the authors.

Figure 1
Figure 1. Relations among MFE, VI, and Minty VI in the exact and approximate settings. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Minty gap (left), variational inequality gap (right), and the common upper bound from [PITH_FULL_IMAGE:figures/full_fig_p023_2.png] view at source ↗
Figure 3
Figure 3. Averaged predictor measure ¯νK,h for Example (6.1) at selected iterations with M = 8. The dashed line marks the equilibrium x ∗ = −1/2. Moreover, by (6.2), ⟨J(η, ·), ν¯K,h − η⟩ = ⟨J(¯νK,h, ·), ν¯K,h − η⟩. Thus the Minty gap and the VI gap coincide for this particular choice of J. Consequently, sup η∈P(Ω) [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Evolution of the averaged predictor measure ¯ν [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Minty gap (left) and variational inequality gap (right), together with their respective upper [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Last iterates µk,h for Example (6.6) at selected iterations, using M = 8 and λ = 0.49. The crosses mark the finite-mesh equilibrium µ ∗ h [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]
Figure 7
Figure 7. Figure 7: Observed last-iterate error and the geometric upper bound from Theorem [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 33 canonical work pages

  1. [1]

    Bregman-projected mirror methods for regularized stationary mean-field games

    Hussain Al Abdulaziz, Yuri Ashrafyan, Yeva Gevorgyan, and Diogo Gomes. Bregman-projected mirror methods for regularized stationary mean-field games.arXiv preprint arXiv:2606.19611, 2026

    work page Pith review arXiv 2026

  2. [2]

    Mean field games: convergence of a finite difference method.SIAM Journal on Numerical Analysis, 51(5):2585–2612, 2013

    Yves Achdou, Fabio Camilli, and Italo Capuzzo-Dolcetta. Mean field games: convergence of a finite difference method.SIAM Journal on Numerical Analysis, 51(5):2585–2612, 2013

    work page 2013

  3. [3]

    Mean field games: numerical methods.SIAM Journal on Numerical Analysis, 48(3):1136–1162, 2010

    Yves Achdou and Italo Capuzzo-Dolcetta. Mean field games: numerical methods.SIAM Journal on Numerical Analysis, 48(3):1136–1162, 2010

    work page 2010

  4. [4]

    Mean field games and applications: Numerical aspects.Mean Field Games: Cetraro, Italy 2019, pages 249–307, 2020

    Yves Achdou and Mathieu Lauri` ere. Mean field games and applications: Numerical aspects.Mean Field Games: Cetraro, Italy 2019, pages 249–307, 2020

    work page 2019

  5. [5]

    CEMRACS 2017: numerical probabilistic approach to MFG.ESAIM: Proceedings and Surveys, 65:84–113, 2019

    Andrea Angiuli, Christy V Graves, Houzhi Li, Jean-Fran¸ cois Chassagneux, Fran¸ cois Delarue, and Ren´ e Carmona. CEMRACS 2017: numerical probabilistic approach to MFG.ESAIM: Proceedings and Surveys, 65:84–113, 2019

    work page 2017

  6. [6]

    Approxi- mate variational inequalities and equilibria.Computational Management Science, 20(1):43, 2023

    Giancarlo Bigi, Lorenzo Lampariello, Simone Sagratella, and Valerio Giuseppe Sasso. Approxi- mate variational inequalities and equilibria.Computational Management Science, 20(1):43, 2023

    work page 2023

  7. [7]

    Proximal methods for stationary mean field games with local couplings.SIAM Journal on Control and Optimization, 56(2):801–836, 2018

    Luis M Briceno-Arias, Dante Kalise, and Francisco J Silva. Proximal methods for stationary mean field games with local couplings.SIAM Journal on Control and Optimization, 56(2):801–836, 2018

    work page 2018

  8. [8]

    A policy iteration method for mean field games.ESAIM: Control, Optimisation and Calculus of Variations, 27:85, 2021

    Simone Cacace, Fabio Camilli, and Alessandro Goffi. A policy iteration method for mean field games.ESAIM: Control, Optimisation and Calculus of Variations, 27:85, 2021

    work page 2021

Show all 40 references
  1. [9]

    Learning in mean field games: the fictitious play

    Pierre Cardaliaguet and Saeed Hadikhanloo. Learning in mean field games: the fictitious play. ESAIM: Control, Optimisation and Calculus of Variations, 23(2):569–591, 2017

    2017

  2. [10]

    Selection by vanishing common noise for potential finite state mean field games.Communications in Partial Differential Equations, 47(1):89–168, 2022

    Alekos Cecchin and Fran¸ cois Delarue. Selection by vanishing common noise for potential finite state mean field games.Communications in Partial Differential Equations, 47(1):89–168, 2022

    2022

  3. [11]

    Context tree estimation for not necessarily finite memory processes, via BIC and MDL.IEEE Transactions on Information Theory, 52(3):1007–1016, 2006

    Imre Csisz´ ar and Zsolt Talata. Context tree estimation for not necessarily finite memory processes, via BIC and MDL.IEEE Transactions on Information Theory, 52(3):1007–1016, 2006

    2006

  4. [12]

    Approximately solving mean field games via entropy-regularized deep reinforcement learning

    Kai Cui and Heinz Koeppl. Approximately solving mean field games via entropy-regularized deep reinforcement learning. InInternational Conference on Artificial Intelligence and Statistics, pages 1909–1917. PMLR, 2021

    1909

  5. [13]

    Learning mean-field games.Advances in Neural Information Processing Systems, 32, 2019

    Xin Guo, Anran Hu, Renyuan Xu, and Junzi Zhang. Learning mean-field games.Advances in Neural Information Processing Systems, 32, 2019

    2019

  6. [14]

    Entropy regularization for mean field games with learning.Mathematics of Operations Research, 47(4):3239–3260, 2022

    Xin Guo, Renyuan Xu, and Thaleia Zariphopoulou. Entropy regularization for mean field games with learning.Mathematics of Operations Research, 47(4):3239–3260, 2022

    2022

  7. [15]

    Learning in anonymous nonatomic games with applications to first-order mean field games.arXiv preprint arXiv:1704.00378, 2017

    Saeed Hadikhanloo. Learning in anonymous nonatomic games with applications to first-order mean field games.arXiv preprint arXiv:1704.00378, 2017

    2017 arXiv

  8. [16]

    Learning in nonatomic games, part i: Finite action spaces and population games.Journal of Dynamics and Games, 9(4):433–460, 2022

    Saeed Hadikhanloo, Rida Laraki, Panayotis Mertikopoulos, and Sylvain Sorin. Learning in nonatomic games, part i: Finite action spaces and population games.Journal of Dynamics and Games, 9(4):433–460, 2022

    2022

  9. [17]

    Finite mean field games: fictitious play and convergence to a first order continuous mean field game.Journal de Math´ ematiques Pures et Appliqu´ ees, 132:369–397, 2019

    Saeed Hadikhanloo and Francisco J Silva. Finite mean field games: fictitious play and convergence to a first order continuous mean field game.Journal de Math´ ematiques Pures et Appliqu´ ees, 132:369–397, 2019

    2019

  10. [18]

    Minyi Huang, Roland P Malham´ e, and Peter E Caines. Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle.Com- munications in Information and Systems, 6(3):221–252, 2006

    2006

  11. [19]

    Evolution of mixed strategies in monotone games.SIAM Journal on Optimization, 33(4):2750–2771, 2023

    Ryan Hynd. Evolution of mixed strategies in monotone games.SIAM Journal on Optimization, 33(4):2750–2771, 2023. 27

    2023

  12. [20]

    Last iterate convergence in monotone mean field games.Advances in Neural Information Processing Systems, 38:18086–18119, 2025

    Noboru Isobe, Kenshi Abe, and Kaito Ariu. Last iterate convergence in monotone mean field games.Advances in Neural Information Processing Systems, 38:18086–18119, 2025

    2025

  13. [21]

    The extragradient method for finding saddle points and other problems

    Galina M Korpelevich. The extragradient method for finding saddle points and other problems. Matecon, 12:747–756, 1976

    1976

  14. [22]

    Regularized iterative stochastic approxi- mation methods for stochastic variational inequality problems.IEEE Transactions on Automatic Control, 58(3):594–609, 2012

    Jayash Koshal, Angelia Nedic, and Uday V Shanbhag. Regularized iterative stochastic approxi- mation methods for stochastic variational inequality problems.IEEE Transactions on Automatic Control, 58(3):594–609, 2012

    2012

  15. [23]

    Mean field games.Japanese Journal of Mathematics, 2(1):229–260, 2007

    Jean-Michel Lasry and Pierre-Louis Lions. Mean field games.Japanese Journal of Mathematics, 2(1):229–260, 2007

    2007

  16. [24]

    Numerical methods for mean field games and mean field type control.arXiv preprint arXiv:2106.06231, 2021

    Mathieu Lauriere. Numerical methods for mean field games and mean field type control.arXiv preprint arXiv:2106.06231, 2021

    2021 arXiv

  17. [25]

    Scalable deep reinforce- ment learning algorithms for mean field games

    Mathieu Lauriere, Sarah Perrin, Sertan Girgin, Paul Muller, Ayush Jain, Theophile Cabannes, Georgios Piliouras, Julien P´ erolat, Romuald Elie, Olivier Pietquin, et al. Scalable deep reinforce- ment learning algorithms for mean field games. InInternational Conference on Machin...

    2022

  18. [26]

    Learning in mean field games: A survey.arXiv preprint arXiv:2205.12944, 2022

    Mathieu Lauri` ere, Sarah Perrin, Julien P´ erolat, Sertan Girgin, Paul Muller, Romuald ´Elie, Matthieu Geist, and Olivier Pietquin. Learning in mean field games: A survey.arXiv preprint arXiv:2205.12944, 2022

    2022 arXiv

  19. [27]

    Generalized conditional gradient and learning in potential mean field games.Applied Mathematics & Optimization, 88(3):89, 2023

    Pierre Lavigne and Laurent Pfeiffer. Generalized conditional gradient and learning in potential mean field games.Applied Mathematics & Optimization, 88(3):89, 2023

    2023

  20. [28]

    Computational methods for first-order nonlocal mean field games with applications.SIAM Journal on Numerical Analysis, 59(5):2639–2668, 2021

    Siting Liu, Matthew Jacobs, Wuchen Li, Levon Nurbekyan, and Stanley J Osher. Computational methods for first-order nonlocal mean field games with applications.SIAM Journal on Numerical Analysis, 59(5):2639–2668, 2021

    2021

  21. [29]

    Solving continuous mean field games: Deep reinforcement learning for non-stationary dynamics.Advances in Neural Information Processing Systems, 38:104325–104354, 2025

    Lorenzo Magnino, Kai Shao, Zida Wu, Jiacheng Shen, and Mathieu Lauriere. Solving continuous mean field games: Deep reinforcement learning for non-stationary dynamics.Advances in Neural Information Processing Systems, 38:104325–104354, 2025

    2025

  22. [30]

    Extragradient methods for mean field games of controls and mean field type FBSDEs.arXiv preprint arXiv:2602.14621, 2026

    Charles Meynard. Extragradient methods for mean field games of controls and mean field type FBSDEs.arXiv preprint arXiv:2602.14621, 2026

    2026 arXiv

  23. [31]

    Prox-method with rate of convergenceO(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems

    Arkadi Nemirovski. Prox-method with rate of convergenceO(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM Journal on Optimization, 15(1):229–251, 2004

    2004

  24. [32]

    Scaling mean field games by online mirror descent

    Julien P´ erolat, Sarah Perrin, Romuald Elie, Mathieu Lauri` ere, Georgios Piliouras, Matthieu Geist, Karl Tuyls, and Olivier Pietquin. Scaling mean field games by online mirror descent. InProceedings of the 21st International Conference on Autonomous Agents and Multiagent Sys...

    2022

  25. [33]

    Fictitious play for mean field games: Continuous time analysis and applications.Ad- vances in Neural Information Processing Systems, 33:13199–13213, 2020

    Sarah Perrin, Julien P´ erolat, Mathieu Lauri` ere, Matthieu Geist, Romuald Elie, and Olivier Pietquin. Fictitious play for mean field games: Continuous time analysis and applications.Ad- vances in Neural Information Processing Systems, 33:13199–13213, 2020

    2020

  26. [34]

    Upper bounds on the relative entropy and R´ enyi divergence as a function of total variation distance for finite alphabets

    Igal Sason and Sergio Verd´ u. Upper bounds on the relative entropy and R´ enyi divergence as a function of total variation distance for finite alphabets. In2015 IEEE Information Theory Workshop-Fall (ITW), pages 214–218. IEEE, 2015

    2015

  27. [35]

    Particle system approximation of Nash equilibria in large games

    Ludovic Tangpi and Nizar Touzi. Particle system approximation of Nash equilibria in large games. arXiv preprint arXiv:2510.19211, 2025

    2025 arXiv

  28. [36]

    A variational inequality approach to independent learning in static mean-field games.ACM/IMS Journal of Data Science, 2(2):1–43, 2025

    Batuhan Yardim, Semih Cayci, and Niao He. A variational inequality approach to independent learning in static mean-field games.ACM/IMS Journal of Data Science, 2(2):1–43, 2025. 28 A Proofs of Subsection 3.1 A.1 Proof of Lemma 3.1 Proof.Since Ω is compact andg∈C(Ω), the functio...

    2025

  29. [37]

    For everyµ∈ P(X)with full support satisfying ∥µ−µ ∗∥1 ≤ m∗ 2 , one has DKL(µ∗∥µ)≤2∥µ−µ ∗∥1

  30. [38]

    Proof.Let µ= nX i=1 piδxi ∈ P(X) withp i >0 for alli

    If, in addition,µ ∗ has full support onX, namely m∗ = min 1≤i≤n p∗ i >0, then, for everyµ∈ P(X)with full support satisfying ∥µ−µ ∗∥1 ≤ m∗ 2 , one has the sharper quadratic estimate DKL(µ∗∥µ)≤ 1 m∗ ∥µ−µ ∗∥2 1. Proof.Let µ= nX i=1 piδxi ∈ P(X) withp i >0 for alli. Using logu≤u−1...

  31. [39]

    The composite Tikhonov VI: J(µ ε h,·), ηh −µ ε h +ε Rh(ηh)−R h(µε h) ≥0,∀η h ∈ P(Xh).(F.1)

  32. [40]

    Proof.We first prove that (F.1) implies (F.2)

    The Euler, or first-order, VI: J(µ ε h,·) +ε δRh δµ (µε h,·), ηh −µ ε h ≥0,∀η h ∈ P(Xh).(F.2) Here the first variation is understood modulo additive constants, since ⟨c, ηh −µ ε h⟩= 0∀c∈R,∀η h ∈ P(Xh). Proof.We first prove that (F.1) implies (F.2). Fixη h ∈ P(Xh) and set ηh,t ...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.