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arxiv: 2605.19392 · v1 · pith:OZ2OVOZZnew · submitted 2026-05-19 · 💻 cs.LG

Understanding Dynamics of Adam in Zero-Sum Games: An ODE Approach

Yi Feng , Weiming Ou , Xiao Wang This is my paper

Pith reviewed 2026-05-20 07:47 UTC · model grok-4.3

classification 💻 cs.LG
keywords Adamzero-sum gamesODE analysisGAN trainingmomentum parameterslocal convergenceimplicit regularization
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The pith

In zero-sum games the first- and second-order momentum terms of Adam-DA reverse the convergence roles they play in minimization.

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

The paper derives ordinary differential equations that serve as continuous-time limits of the discrete Adam-DA updates used for zero-sum games. These ODEs make it possible to analyze local convergence and implicit gradient regularization in a tractable way. The central result is that the first-order momentum parameter slows convergence while the second-order momentum parameter accelerates it, exactly opposite to the well-known effects in ordinary minimization. The predictions are checked by running GAN training on several architectures and datasets. If the ODE approximation holds, the same reversed momentum behavior should appear in any zero-sum setting where Adam-DA is applied.

Core claim

By taking the continuous-time limit of the Adam-DA iterates, the authors obtain a system of ODEs whose equilibria and stability properties can be studied directly. Analysis of these ODEs shows that raising the first-order momentum coefficient destabilizes the saddle while raising the second-order coefficient stabilizes it; the signs of these effects are reversed relative to the standard minimization case. The same ODEs also reveal an implicit regularization term whose form depends on the momentum parameters in the opposite manner from gradient descent.

What carries the argument

The system of ordinary differential equations obtained as the continuous-time limit of the Adam-DA discrete updates.

If this is right

  • Local convergence of Adam-DA to a saddle can be read off from the eigenvalues of the linearized ODE.
  • The implicit regularization induced by Adam-DA in games takes the opposite functional form from the regularization induced in minimization.
  • Tuning guidelines for Adam-DA in GANs should invert the usual momentum recommendations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same ODE construction could be applied to other adaptive optimizers such as RMSProp-DA to check whether the momentum reversal is specific to Adam or generic.
  • If the reversal persists in non-convex zero-sum problems, it may explain why small first-order momentum values are often preferred in practice for GAN training.
  • The ODE view suggests a possible continuous-time schedule for the momentum coefficients that could improve stability without changing the discrete algorithm.

Load-bearing premise

The discrete Adam-DA steps with typical learning rates and momentum values stay close enough to their continuous ODE trajectories that local stability and regularization results carry over.

What would settle it

Run Adam-DA on a simple bilinear zero-sum game with known saddle and measure whether increasing the first-order momentum visibly slows convergence or increasing the second-order momentum visibly speeds it up; a reversal of either trend would contradict the claim.

Figures

Figures reproduced from arXiv: 2605.19392 by Weiming Ou, Xiao Wang, Yi Feng.

Figure 1
Figure 1. Figure 1: Trajectories of Adam-DA, Continuous Adam-DA, and SignGDA-flow on three test functions from (Compagnoni et al., 2024b). Continuous Adam-DA closely approximates Adam-DA. Especially in 1(b) and 1(c), where SignGDA-flow either diverges or approaches to a different equilibrium, while the trajectories of the other two methods remain similar. More details are provided in Appendix B.1. between ODEs and algorithms.… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical experiments on quadratic test functions for the local convergence of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The ℓ1 norm of gradients of Adam-DA with varying β and ρ during GANs training. Datasets: CIFAR-10 and STL-10. Architectures: ResNet and CNN. As shown in 3(a), 3(c), 3(e), and 3(g), smaller β values result in smaller gradient norms. According to 3(b), 3(d), 3(f), and 3(h), larger ρ values also lead to smaller gradient norms. Both findings support the thesis. scapes in terms of ℓ1 norm, i.e., regions with lo… view at source ↗
Figure 4
Figure 4. Figure 4: Inception Score for the corresponding experimental settings in Figure [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Figures 5(a), 5(b), and 5(c) show the distances of two continuous-time models between Adam, with results averaged over 30 random initial conditions. In the following [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Additional experiments with different parameters. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effect of ϵ. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Self-Attention GAN Experiments on CelebA, Evaluated by FID. In [PITH_FULL_IMAGE:figures/full_fig_p038_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: IIn this figure, we reproduce one set of the experimental results from Section 5 of the submission on CNN GANs trained on the CIFAR-10 dataset. We evaluate performance using FID and include a comparison with the optimistic adaptive method. The conclusion is the same as that in [PITH_FULL_IMAGE:figures/full_fig_p039_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 2D sweep over (β, ρ) jointly. Each figure represents the final cumulative average gradient norms on 25 GANs training. Each figure shows the final cumulative average gradient norms over 25 GAN training runs. We observe that the upper-left corner of each figure exhibits smaller gradient norms than the lower-right corner, indicating that smaller β and larger ρ guide the optimization trajectories toward flatt… view at source ↗
Figure 11
Figure 11. Figure 11: Sample images generated by the models trained in [PITH_FULL_IMAGE:figures/full_fig_p041_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sample images for different β. Architecture: ResNet. Data Set: CIFAR-10. 42 [PITH_FULL_IMAGE:figures/full_fig_p042_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Sample images for different ρ. Architecture: ResNet. Data Set: CIFAR-10. (a) β = −0.3, ρ = 0.9 (b) β = −0.2, ρ = 0.9 (c) β = 0.0, ρ = 0.9 (d) β = 0.2, ρ = 0.9 (e) β = 0.3, ρ = 0.9 (f) β = 0.5, ρ = 0.9 [PITH_FULL_IMAGE:figures/full_fig_p043_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Sample images for different β. Architecture: CNN. Data Set: STL-10. 43 [PITH_FULL_IMAGE:figures/full_fig_p043_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Sample images for different ρ. Architecture: CNN. Data Set: STL-10. 44 [PITH_FULL_IMAGE:figures/full_fig_p044_15.png] view at source ↗
read the original abstract

The remarkable success of the Adam in training neural networks has naturally led to the widespread use of its descent-ascent counterpart, Adam-DA, for solving zero-sum games. Despite its popularity in practice, a rigorous theoretical understanding of Adam-DA still lags behind. In this paper, we derive ordinary differential equations (ODEs) that serve as continuous-time limits of the Adam-DA. These ODEs closely approximate the discrete-time dynamics of Adam-DA, providing a tractable analytical framework for understanding its behavior in zero-sum games. Using this ODE approach, we investigate two fundamental aspects of Adam-DA: local convergence and implicit gradient regularization. Our analysis reveals that the roles of the first- and second-order momentum parameters in zero-sum games are exactly the opposite of their well-documented effects in minimization problems. We validate these predictions through GAN experiments across multiple architectures and datasets, demonstrating the practical implications of this reversed momentum effect.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper derives ordinary differential equation (ODE) limits for the discrete Adam-DA algorithm applied to zero-sum games. These ODEs are used to analyze local convergence and implicit gradient regularization, leading to the claim that the first-order momentum parameter (β1) and second-order momentum parameter (β2) play exactly opposite roles compared to their effects in standard minimization problems. The analysis is validated through qualitative GAN experiments on multiple architectures and datasets.

Significance. If the ODE approximation is faithful at practical step sizes and momentum values, the work supplies a useful continuous-time framework for understanding momentum in non-monotone settings and could inform hyperparameter selection for GAN training. The explicit reversal result, if rigorously supported, distinguishes this contribution from prior ODE analyses of Adam in convex or minimization settings.

major comments (2)
  1. [§3] §3 (ODE derivation): The continuous-time limit is obtained via standard Euler discretization and momentum rescaling, but the manuscript provides no explicit error bounds, timescale-separation conditions, or verification that the approximation remains valid for β2 ≈ 0.999 and learning rates ≈ 10^{-3} when the underlying vector field is non-monotone and oscillatory. This assumption is load-bearing for transferring local convergence and regularization conclusions from the ODE to the discrete Adam-DA updates.
  2. [§5] §5 (Experiments): The GAN results are presented without quantitative metrics (e.g., FID scores, convergence rates), ablation controls on β1/β2, or direct comparisons to Adam in minimization tasks that would demonstrate the claimed reversal. This weakens the empirical support for the central theoretical prediction.
minor comments (2)
  1. [§3] The notation for the rescaled momentum terms in the ODE should be aligned more explicitly with the discrete Adam-DA update equations to improve readability.
  2. [Introduction] Add a brief discussion of how the derived ODEs relate to existing continuous-time analyses of Adam in minimization (e.g., prior works on momentum in convex optimization) for clearer positioning.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed and insightful comments on our manuscript. We have carefully considered each point and outline our responses and planned revisions below.

read point-by-point responses

  1. Referee: [§3] §3 (ODE derivation): The continuous-time limit is obtained via standard Euler discretization and momentum rescaling, but the manuscript provides no explicit error bounds, timescale-separation conditions, or verification that the approximation remains valid for β2 ≈ 0.999 and learning rates ≈ 10^{-3} when the underlying vector field is non-monotone and oscillatory. This assumption is load-bearing for transferring local convergence and regularization conclusions from the ODE to the discrete Adam-DA updates.

    Authors: We thank the referee for this observation. The ODE limit is derived using the standard Euler method with appropriate rescaling of the momentum terms, as is common in the literature on continuous-time analyses of adaptive optimizers. We acknowledge that the manuscript does not provide explicit error bounds or detailed timescale separation conditions, particularly for the non-monotone case. Deriving such bounds rigorously for oscillatory dynamics would require substantial additional analysis. In the revised manuscript, we will include a new subsection in §3 discussing the assumptions underlying the approximation and provide numerical verification by comparing discrete trajectories with the ODE solutions for β2 close to 1 and small learning rates in the context of our GAN experiments. This will offer practical evidence for the validity of the limit in the relevant parameter regime. revision: partial

  2. Referee: [§5] §5 (Experiments): The GAN results are presented without quantitative metrics (e.g., FID scores, convergence rates), ablation controls on β1/β2, or direct comparisons to Adam in minimization tasks that would demonstrate the claimed reversal. This weakens the empirical support for the central theoretical prediction.

    Authors: We agree that incorporating quantitative metrics and ablations would strengthen the empirical section. In the revision, we will augment §5 with FID scores and other relevant quantitative measures for the GAN experiments. We will also add ablation studies on the effects of varying β1 and β2, as well as direct comparisons to the behavior of Adam in standard minimization settings. These changes will better substantiate the claimed reversal of roles for the momentum parameters. revision: yes

standing simulated objections not resolved
  • Deriving explicit error bounds and timescale-separation conditions for the ODE approximation in non-monotone and oscillatory settings.

Circularity Check

0 steps flagged

No circularity: standard ODE limit derivation with independent analysis

full rationale

The paper derives ODEs as continuous-time limits of discrete Adam-DA updates via standard Euler discretization and momentum rescaling techniques. Local convergence and implicit regularization properties are then analyzed directly on the resulting ODE system in the zero-sum setting, yielding the reversed-momentum observation as a consequence of the vector field structure. This chain is self-contained, does not reduce any prediction to a fitted input or prior self-citation by construction, and is externally validated via GAN experiments. No load-bearing step equates to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of taking continuous-time limits of the discrete Adam-DA updates and on the transfer of local stability properties from the resulting ODEs to the original algorithm. No new entities are postulated.

axioms (1)
  • domain assumption Discrete Adam-DA updates admit a continuous-time ODE limit that approximates their trajectory for small learning rates.
    This is the foundational modeling step that converts the discrete optimizer into an analyzable dynamical system.

pith-pipeline@v0.9.0 · 5680 in / 1354 out tokens · 33080 ms · 2026-05-20T07:47:16.248366+00:00 · methodology

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