受控极小极大：通过控制理论稳定对抗优化

# Controlled-Minimax: Stabilizing Adversarial Optimization via Control Theory
Source: [https://arxiv.org/html/2610.12915](https://arxiv.org/html/2610.12915)
Tianjiao LuoJianfei ChenAffiliation:Department of Computer Science and Technology, Tsinghua University, Beijing, ChinaJun ZhuAffiliation:Department of Computer Science and Technology, Tsinghua University, Beijing, ChinaCorrespondence to:[dcszj@mail\.tsinghua\.edu\.cn](mailto:[email protected])

###### Abstract

Minimax optimization is fundamental to adversarial learning yet suffers from severe stability challenges, often manifesting as persistent limit cycles or divergence\. This paper investigates these instabilities through the lens of continuous\-time dynamical systems\. By modeling training dynamics as a second\-order Ordinary Differential Equation \(ODE\), we rigorously identify the root cause of non\-convergence: the problematic eigenstructure of the loss function’s Hessian near equilibrium points\. To address this, we introduce Controlled\-Minimax, a novel stabilization framework that reformulates the optimization trajectory as a controllable physical system\. Unlike heuristic regularization techniques, our approach derives a specific feedback controller that provides theoretical guarantees of local exponential stability, regardless of the underlying Hessian spectrum\. Consequently, our method stabilizes general minimax problems where traditional gradient descent\-ascent methods fail\. Extensive experiments on pathological benchmarks including the Dirac\-GAN, 2D Gaussian mixtures, and Swiss Roll datasets, as well as high\-resolution image synthesis with StyleGAN2\-ada, demonstrate that Controlled\-Minimax effectively suppresses oscillatory behavior while ensuring fast and robust convergence\.

###### Keywords:

Machine Learning, ICML

## 1Introduction

Minimax optimization is a cornerstone of modern machine learning, transcending traditional single\-objective minimization tasks to address problems characterized by competing interests\. This paradigm frames learning as a competition between two agents: a minimizing player controlling parametersθ\\mathbf\{\\theta\}and a maximizing player controllingϕ\\mathbf\{\\phi\}, coupled through a shared objective functionL⁡\(θ,ϕ\)L\(\\mathbf\{\\theta\},\\mathbf\{\\phi\}\)\. The goal is to find a saddle point\(θ∗,ϕ∗\)\(\\mathbf\{\\theta\}^\{\*\},\\mathbf\{\\phi\}^\{\*\}\)that solves the problemminθ⁡maxϕ⁡L⁡\(θ,ϕ\)\\min\_\{\\mathbf\{\\theta\}\}\\max\_\{\\mathbf\{\\phi\}\}L\(\\mathbf\{\\theta\},\\mathbf\{\\phi\}\)\. This formulation has proven to be incredibly powerful and versatile, underpinning the success of Generative Adversarial Networks \(GANs\) in synthesizing high\-fidelity data\([Goodfellow et al\., 2020](https://arxiv.org/html/2610.12915#bib.bib8)\), enhancing model robustness through adversarial training\([Madry et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib19)\), and enabling complex decision\-making in multi\-agent reinforcement learning\([Busoniu et al\., 2008](https://arxiv.org/html/2610.12915#bib.bib3)\)\.

However, efficiently solving these minimax problems in the context of deep learning remains a formidable challenge\. Unlike classical convex\-concave settings, the optimization landscapes parameterized by deep neural networks are highly non\-convex and non\-concave, leading to complex local geometries\. Standard approaches, most notably Gradient Descent\-Ascent \(GDA\)\([Bottou, 2010](https://arxiv.org/html/2610.12915#bib.bib2)\), often fail to converge to a stable Nash equilibrium in such regimes\. Instead of simply descending a potential surface, the optimization dynamics frequently exhibit rotational behaviors caused by the interaction between the two players\. Consequently, training trajectories are prone to persistent oscillations, limit cycles, or even divergence, which severely hampers the stability and generalization capability of the resulting models\.

To address the inherent instability of minimax optimization, existing literature has predominantly explored two directions: refining the optimization dynamics and imposing structural regularizations\. The former category attempts to correct the rotational behavior of GDA by incorporating lookahead mechanisms\([Korpelevich, 1976](https://arxiv.org/html/2610.12915#bib.bib14);[Mertikopoulos et al\., 2018](https://arxiv.org/html/2610.12915#bib.bib20);[Gidel et al\., 2019](https://arxiv.org/html/2610.12915#bib.bib7)\), multi\-step updates\([Goodfellow et al\., 2020](https://arxiv.org/html/2610.12915#bib.bib8)\), or timescale separation between the players\([Heusel et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib10)\)\. The latter seeks to smooth the optimization landscape by enforcing Lipschitz constraints or adding gradient penalties\([Gulrajani et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib9);[Mescheder et al\., 2018](https://arxiv.org/html/2610.12915#bib.bib22);[Miyato et al\., 2018](https://arxiv.org/html/2610.12915#bib.bib23)\)\. Nevertheless, rigorous theoretical guarantees for these methods remain elusive, particularly in the non\-convex non\-concave settings typical of neural networks in deep learning\.

While local convergence can be proven under strict assumptions—such as local convexity or linearity\([Nagarajan & Kolter, 2017](https://arxiv.org/html/2610.12915#bib.bib25);[Heusel et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib10)\), global convergence properties are often neither guaranteed nor understood\([Mescheder et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib21);[Daskalakis & Panageas, 2018](https://arxiv.org/html/2610.12915#bib.bib5);[Farnia & Ozdaglar, 2020](https://arxiv.org/html/2610.12915#bib.bib6)\)\. Furthermore, regularization\-based methods tend to introduce inductive biases that hamper model expressivity\([Mescheder et al\., 2018](https://arxiv.org/html/2610.12915#bib.bib22);[Arjovsky et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib1)\), while sophisticated dynamic adjustments frequently incur high computational overheads without ensuring stability in general cases\. Consequently, developing a solver that is both theoretically grounded and practically robust remains an open challenge\.

In this paper, we depart from the traditional view of designing better discrete optimizers and instead approach the instability problem through the lens of continuous\-time control theory\. We model the minimax training dynamics as a second\-order Ordinary Differential Equation \(ODE\), allowing us to rigorously diagnose the root cause of persistent oscillations: the specific eigenstructure of the loss function’s Hessian, which inherently precludes convergence in standard settings\. Based on this insight, we proposeControlled\-Minimax, a novel framework that treats the optimization trajectory as a controllable dynamical system\. Unlike heuristic regularizers that passively penalize model weights, our method actively injects a designed feedback controller into the dynamics\. This controller forces the system’s energy to dissipate, providing a theoretical guarantee of*local exponential stability*regardless of the underlying Hessian structure\. We validate our approach on tasks ranging from the pathological Dirac\-GAN, 2D Gaussian and Swiss Roll to high\-resolution image generation with StyleGAN2\-ada, demonstrating that Controlled\-Minimax effectively eliminates oscillatory behavior and ensures robust convergence where traditional methods fail\.

Our contribution can be summarized as:

- •We bridge the gap between discrete optimization algorithms and continuous dynamical systems by transforming the adversarial training process into a second\-order Ordinary Differential Equation \(ODE\)\. This shift in perspective allows us to apply classical tools from control theory to the typically unstable dynamics of GAN training\.
- •We identify that the instability in adversarial learning stems from specific pathological eigenstructures of the Hessian that standard gradient fields cannot correct\.
- •We introduce a novel stabilization mechanism called the Controlled\-Minimax that views optimization as a control problem\. By designing a specific feedback controller on gradient, we prove that our method achieves local exponential stability for general minimax problems, offering a rigorous guarantee that is absent in heuristic regularization methods\.
- •We demonstrate the effectiveness of our approach across a spectrum of tasks, ranging from fundamental benchmarks \(Dirac\-GAN, 2D Gaussian, Swiss Roll\) to high\-dimensional training with StyleGAN2\-ada\. Our Controlled\-Minimax consistently eliminates oscillations and accelerates convergence, validating its practical effectiveness alongside its theoretical soundness\.

### 1\.1Related Work

#### Convergence Analysis and Static Bounds

Many works on minimax optimization relies on strong assumptions regarding the curvature of the objective function\. Convergence rates have been established for settings such as convex\-concave\([Nemirovski et al\., 2009](https://arxiv.org/html/2610.12915#bib.bib26)\), strongly\-convex\-strongly\-concave\([Palaniappan & Bach, 2016](https://arxiv.org/html/2610.12915#bib.bib27)\), and specific nonconvex\-nonconcave case\([Yang et al\., 2020](https://arxiv.org/html/2610.12915#bib.bib31)\)\. Within these structured landscapes,[Daskalakis & Panageas \(2018\)](https://arxiv.org/html/2610.12915#bib.bib5)and[Gidel et al\. \(2019\)](https://arxiv.org/html/2610.12915#bib.bib7)provide rigorous analysis of Gradient Descent\-Ascent \(GDA\), though requiring specific step\-size conditions or negative momentum\. Similarly,[Mertikopoulos et al\. \(2018\)](https://arxiv.org/html/2610.12915#bib.bib20)explore extra\-gradient methods in coherent games, while[Hsieh et al\. \(2019\)](https://arxiv.org/html/2610.12915#bib.bib11)analyze convergence in non\-convex non\-concave settings\. However, these guarantees falter in deep learning because they depend on rigid curvature profiles or specific timescale separations\([Mescheder et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib21);[Heusel et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib10)\)\. Furthermore, treating optimization steps as isolated updates, ignoring the accumulated trajectory and inertia of the parameters\.

#### Heuristic Regularization and Algorithmic Variants

To address the rotational dynamics common in zero\-sum games,[Daskalakis & Panageas \(2018\)](https://arxiv.org/html/2610.12915#bib.bib5)propose the Optimistic Mirror Descent, which effectively utilizes negative momentum to cancel out interaction terms, a technique further analyzed by[Mokhtari et al\. \(2020\)](https://arxiv.org/html/2610.12915#bib.bib24)through the lens of proximal point approximations\. Regularization strategies have also been pivotal\.[Nagarajan & Kolter \(2017\)](https://arxiv.org/html/2610.12915#bib.bib25)analyze the local convergence properties of gradient penalties, introducing terms that stabilize the Jacobian of the vector field near equilibrium points\. However, a key limitation of these approaches is that their convergence guarantees often rely on local linearity or strong operator assumptions\.

#### Control Theory and Dynamical Systems

[Xu et al\. \(2020\)](https://arxiv.org/html/2610.12915#bib.bib30)and[Luo et al\. \(2023\)](https://arxiv.org/html/2610.12915#bib.bib18)have attempted to actively control training dynamics using feedback loops\.[Chavdarova et al\. \(2019\)](https://arxiv.org/html/2610.12915#bib.bib4)further explored how differential equation solvers can reduce noise in game dynamics\. Furthermore,[Li & Orabona \(2019\)](https://arxiv.org/html/2610.12915#bib.bib17)and[Su et al\. \(2016\)](https://arxiv.org/html/2610.12915#bib.bib29)have drawn connections between Nesterov acceleration and differential equations, though primarily in single\-objective settings\. While these works acknowledge the dynamic nature of learning, they often rely on complex auxiliary loss terms or heavy modifications to the standard gradient vector field\. This can make them computationally expensive or difficult to integrate with standard optimizers\.

Figure 1:Convergence analysis on Dirac\-GAN\.Top:Phase space trajectories showing the optimization path\. The baseline gets stuck in a limit cycle, while our Controlled Minimax spirals inward to the equilibrium\.Bottom:Evolution of parameters over time\. The baseline parameters oscillate indefinitely with constant amplitude, whereas our method forces the parameter values to decay to zero\.

## 2Preliminary

We collect here standard notions and results on local stability of ordinary differential equations \(ODEs\) that will be used in the analysis of our second\-order dynamics\. Throughout, we consider autonomous systems of the form

x˙=f⁡\(x\),x∈ℝd,\\dot\{x\}=f\(x\),\\qquad x\\in\\mathbb\{R\}^\{d\},\(1\)wheref:ℝd→ℝdf:\\mathbb\{R\}^\{d\}\\to\\mathbb\{R\}^\{d\}is locally Lipschitz\.

###### Definition 2\.1\(Equilibrium\)\.

A pointx⋆∈ℝdx^\{\\star\}\\in\\mathbb\{R\}^\{d\}is an*equilibrium*of system \([1](https://arxiv.org/html/2610.12915#S2.E1)\) iff⁡\(x⋆\)=0f\(x^\{\\star\}\)=0\. We recall the standard stability notions\.

###### Definition 2\.2\.

System \([1](https://arxiv.org/html/2610.12915#S2.E1)\) with equilibriumx⋆x^\{\\star\}is

1. 1\.*\(Lyapunov\) stable*if for everyε\>0\\varepsilon\>0there existsδ\>0\\delta\>0such that‖x⁡\(0\)−x⋆‖<δ\\\|x\(0\)\-x^\{\\star\}\\\|<\\deltaimplies‖x⁡\(t\)−x⋆‖<ε\\\|x\(t\)\-x^\{\\star\}\\\|<\\varepsilonfor allt≥0t\\geq 0\.
2. 2\.*asymptotically stable*if it is stable and, in addition, there existsδ′\>0\\delta^\{\\prime\}\>0such that‖x⁡\(0\)−x⋆‖<δ′\\\|x\(0\)\-x^\{\\star\}\\\|<\\delta^\{\\prime\}implieslimt→∞x⁡\(t\)=x⋆\\lim\_\{t\\to\\infty\}x\(t\)=x^\{\\star\}\.
3. 3\.*exponentially stable*if there exist constantsc\>0c\>0,α\>0\\alpha\>0, andr\>0r\>0such that‖x⁡\(0\)−x⋆‖<r\\\|x\(0\)\-x^\{\\star\}\\\|<rimplies ‖x⁡\(t\)−x⋆‖≤c​e−α​t​‖x⁡\(0\)−x⋆‖for all​t≥0\.\\\|x\(t\)\-x^\{\\star\}\\\|\\leq c\\,e^\{\-\\alpha t\}\\,\\\|x\(0\)\-x^\{\\star\}\\\|\\quad\\text\{for all \}t\\geq 0\.

We recall the classical local linearization result, which relates the stability of an equilibrium of \([1](https://arxiv.org/html/2610.12915#S2.E1)\) to the spectrum of the Jacobian offfat that equilibrium\.

###### Theorem 2\.3\(Local Linearization\)\.

Letx⋆x^\{\\star\}be an equilibrium ofx˙=f⁡\(x\),\\dot\{x\}=f\(x\),and JacobianA≔D​f​\(x⋆\)A\\coloneqq Df\(x^\{\\star\}\)\.

1. 1\.If all eigenvalues ofAAsatisfyℜ⁡\(λ\)<0\\Re\(\\lambda\)<0, thenx⋆x^\{\\star\}is*locally asymptotically stable*\.
2. 2\.If at least one eigenvalue ofAAsatisfiesℜ⁡\(λ\)\>0\\Re\(\\lambda\)\>0, thenx⋆x^\{\\star\}is*unstable*\.
3. 3\.If all eigenvalues ofAAsatisfyℜ⁡\(λ\)≤0\\Re\(\\lambda\)\\leq 0and for every eigenvalue withℜ⁡\(λ\)=0\\Re\(\\lambda\)=0the algebraic multiplicity equals the geometric multiplicity, thenx⋆x^\{\\star\}is*marginally stable*\.

## 3Minimax Dynamics as a Second Order ODE

We consider a two\-player zero\-sum minimax problem

minθ∈ℝm⁡maxϕ∈ℝn⁡L⁡\(θ,ϕ\),\\min\_\{\\theta\\in\\mathbb\{R\}^\{m\}\}\\;\\max\_\{\\phi\\in\\mathbb\{R\}^\{n\}\}L\(\\theta,\\phi\),\(2\)whereL:ℝm×ℝn→ℝL:\\mathbb\{R\}^\{m\}\\times\\mathbb\{R\}^\{n\}\\to\\mathbb\{R\}is a continuously differentiable payoff function\. The vectorθ∈ℝm\\theta\\in\\mathbb\{R\}^\{m\}denotes the strategy of the minimizing player, andϕ∈ℝn\\phi\\in\\mathbb\{R\}^\{n\}denotes the strategy of the maximizing player\. A standard continuous\-time model for gradient descent–ascent is the gradient flow

θ˙=−∇θL​\(θ,ϕ\),ϕ˙=∇ϕL​\(θ,ϕ\),\\dot\{\\theta\}=\-\\nabla\_\{\\theta\}L\(\\theta,\\phi\),\\qquad\\dot\{\\phi\}=\\nabla\_\{\\phi\}L\(\\theta,\\phi\),\(3\)where, with a slight abuse of notation, we writeθ,ϕ\\theta,\\phiinstead ofθ⁡\(t\),ϕ⁡\(t\)\\theta\(t\),\\phi\(t\)to denote the time\-dependent trajectories\.

To obtain a second\-order ODE, we differentiate \([3](https://arxiv.org/html/2610.12915#S3.E3)\) with respect to time\. Using the chain rule, we have

θ¨=−dd​t​∇θL​\(θ,ϕ\),ϕ¨=dd​t​∇ϕL​\(θ,ϕ\)\.\\ddot\{\\theta\}=\-\\frac\{d\}\{dt\}\\,\\nabla\_\{\\theta\}L\(\\theta,\\phi\),\\qquad\\ddot\{\\phi\}=\\frac\{d\}\{dt\}\\,\\nabla\_\{\\phi\}L\(\\theta,\\phi\)\.\(4\)Assuming thatLLis twice continuously differentiable, the right\-hand sides of \([4](https://arxiv.org/html/2610.12915#S3.E4)\) can be written as

\{θ¨=−∇θ​θ2​L​\(θ,ϕ\)​θ˙−∇θ​ϕ2​L​\(θ,ϕ\)​ϕ˙,ϕ¨=∇ϕ​θ2​L​\(θ,ϕ\)​θ˙\+∇ϕ​ϕ2​L​\(θ,ϕ\)​ϕ˙\.\\left\\\{\\begin\{aligned\} \\ddot\{\\theta\}&=\-\\nabla^\{2\}\_\{\\theta\\theta\}L\(\\theta,\\phi\)\\,\\dot\{\\theta\}\-\\nabla^\{2\}\_\{\\theta\\phi\}L\(\\theta,\\phi\)\\,\\dot\{\\phi\},\\\\ \\ddot\{\\phi\}&=\\nabla^\{2\}\_\{\\phi\\theta\}L\(\\theta,\\phi\)\\,\\dot\{\\theta\}\+\\nabla^\{2\}\_\{\\phi\\phi\}L\(\\theta,\\phi\)\\,\\dot\{\\phi\}\.\\end\{aligned\}\\right\.\(5\)where∇θ​θ2L,∇θ​ϕ2L,∇ϕ​θ2L,∇ϕ​ϕ2L\\nabla^\{2\}\_\{\\theta\\theta\}L,\\nabla^\{2\}\_\{\\theta\\phi\}L,\\nabla^\{2\}\_\{\\phi\\theta\}L,\\nabla^\{2\}\_\{\\phi\\phi\}Ldenote the corresponding second derivatives ofLL\. This formulation represents the uncontrolled second\-order dynamics\. In Section[3\.1](https://arxiv.org/html/2610.12915#S3.SS1), we provide a stability analysis of this original system, followed by the design of a stabilizing structure\-aware damping controller in Section[4](https://arxiv.org/html/2610.12915#S4)\.

### 3\.1Local Stability Analysis

In this section, we analyze the local stability of the dynamical system derived in Equation \([5](https://arxiv.org/html/2610.12915#S3.E5)\) and formally derive the conditions for three cases of stability in theorem[3\.2](https://arxiv.org/html/2610.12915#S3.Thmtheorem2)\.

Let the parameter spaces beθ∈ℝnθ\\theta\\in\\mathbb\{R\}^\{n\_\{\\theta\}\}andϕ∈ℝnϕ\\phi\\in\\mathbb\{R\}^\{n\_\{\\phi\}\}\. To conserve space, we define the full state vector𝐮∈ℝ2​\(nθ\+nϕ\)\\mathbf\{u\}\\in\\mathbb\{R\}^\{2\(n\_\{\\theta\}\+n\_\{\\phi\}\)\}using transpose notation𝐮=\[θ⊤,ϕ⊤,θ˙⊤,ϕ˙⊤\]⊤\.\\mathbf\{u\}=\\left\[\\theta^\{\\top\},\\phi^\{\\top\},\\dot\{\\theta\}^\{\\top\},\\dot\{\\phi\}^\{\\top\}\\right\]^\{\\top\}\.Then the dynamics can be transformed to

𝐮˙=\[θ˙ϕ˙−∇θ​θ2​L​\(θ,ϕ\)​θ˙−∇θ​ϕ2​L​\(θ,ϕ\)​ϕ˙∇ϕ​θ2​L​\(θ,ϕ\)​θ˙\+∇ϕ​ϕ2​L​\(θ,ϕ\)​ϕ˙\]\.\\dot\{\\mathbf\{u\}\}=\\begin\{bmatrix\}\\dot\{\\theta\}\\\\ \\dot\{\\phi\}\\\\ \-\\nabla^\{2\}\_\{\\theta\\theta\}L\(\\theta,\\phi\)\\dot\{\\theta\}\-\\nabla^\{2\}\_\{\\theta\\phi\}L\(\\theta,\\phi\)\\dot\{\\phi\}\\\\ \\nabla^\{2\}\_\{\\phi\\theta\}L\(\\theta,\\phi\)\\dot\{\\theta\}\+\\nabla^\{2\}\_\{\\phi\\phi\}L\(\\theta,\\phi\)\\dot\{\\phi\}\\end\{bmatrix\}\.\(6\)
To determine the long\-term convergence of the training process, we seek fixed points where the system ceases to evolve\. An equilibrium is defined by the condition𝐮˙=𝟎\\dot\{\\mathbf\{u\}\}=\\mathbf\{0\}, requiring both that the parameter updates halt and that the loss gradients vanish\. We formalize this stationary state as follows\.

###### Definition 3\.1\(Zero\-Velocity Equilibrium\)\.

An equilibrium point of the system is defined as a state𝐮∗\\mathbf\{u\}^\{\*\}where the time derivatives of the parameters vanish\. We denote this state as:

𝐮∗=\[\(θ∗\)⊤,\(ϕ∗\)⊤,𝟎⊤,𝟎⊤\]⊤,\\mathbf\{u\}^\{\*\}=\\left\[\(\\theta^\{\*\}\)^\{\\top\},\(\\phi^\{\*\}\)^\{\\top\},\\mathbf\{0\}^\{\\top\},\\mathbf\{0\}^\{\\top\}\\right\]^\{\\top\},\(7\)whereθ∗\\theta^\{\*\}andϕ∗\\phi^\{\*\}are fixed points in the parameter space, and𝟎\\mathbf\{0\}represents the zero vector of appropriate dimension\.

To determine local stability, we examine the Jacobian matrix ofℱ\\mathcal\{F\}evaluated at𝐮∗\\mathbf\{u\}^\{\*\}\. Computing the partial derivatives and noting that Hessian derivative terms vanish at zero velocity, we obtain the block\-structured matrix:

𝐉⁡\(𝐮∗\)=\[𝟎𝟎𝐈𝟎𝟎𝟎𝟎𝐈𝟎𝟎−∇θ​θ2​L−∇θ​ϕ2​L𝟎𝟎∇ϕ​θ2​L∇ϕ​ϕ2​L\],\\mathbf\{J\}\(\\mathbf\{u\}^\{\*\}\)=\\begin\{bmatrix\}\\mathbf\{0\}&\\mathbf\{0\}&\\mathbf\{I\}&\\mathbf\{0\}\\\\ \\mathbf\{0\}&\\mathbf\{0\}&\\mathbf\{0\}&\\mathbf\{I\}\\\\ \\mathbf\{0\}&\\mathbf\{0\}&\-\\nabla^\{2\}\_\{\\theta\\theta\}L&\-\\nabla^\{2\}\_\{\\theta\\phi\}L\\\\ \\mathbf\{0\}&\\mathbf\{0\}&\\nabla^\{2\}\_\{\\phi\\theta\}L&\\nabla^\{2\}\_\{\\phi\\phi\}L\\end\{bmatrix\},\(8\)whereLLis evaluated at\(θ∗,ϕ∗\)\(\\theta^\{\*\},\\phi^\{\*\}\)\. The stability is determined by the lower\-right block, defined as the interaction matrix𝐌\\mathbf\{M\}:

𝐌=\[−∇θ​θ2​L​\(θ∗,ϕ∗\)−∇θ​ϕ2​L​\(θ∗,ϕ∗\)∇ϕ​θ2​L​\(θ∗,ϕ∗\)∇ϕ​ϕ2​L​\(θ∗,ϕ∗\)\]\.\\mathbf\{M\}=\\begin\{bmatrix\}\-\\nabla^\{2\}\_\{\\theta\\theta\}L\(\\theta^\{\*\},\\phi^\{\*\}\)&\-\\nabla^\{2\}\_\{\\theta\\phi\}L\(\\theta^\{\*\},\\phi^\{\*\}\)\\\\ \\nabla^\{2\}\_\{\\phi\\theta\}L\(\\theta^\{\*\},\\phi^\{\*\}\)&\\nabla^\{2\}\_\{\\phi\\phi\}L\(\\theta^\{\*\},\\phi^\{\*\}\)\\end\{bmatrix\}\.\(9\)
###### Theorem 3\.2\(Local Stability Classification\)\.

Letσ⁡\(𝐌\)\\sigma\(\\mathbf\{M\}\)denote the spectrum of the interaction matrix𝐌\\mathbf\{M\}\. The local stability of the system \([5](https://arxiv.org/html/2610.12915#S3.E5)\) around𝐮∗\\mathbf\{u\}^\{\*\}is:

1. 1\.Asymptotically Stable if∀λ∈σ⁡\(𝐌\),Re​\(λ\)<0\\forall\\lambda\\in\\sigma\(\\mathbf\{M\}\),\\text\{Re\}\(\\lambda\)<0\. Small perturbations in velocity decay exponentially\.
2. 2\.Lyapunov Stable \(Marginally Stable\) if∀λ∈σ⁡\(𝐌\),Re​\(λ\)≤0\\forall\\lambda\\in\\sigma\(\\mathbf\{M\}\),\\text\{Re\}\(\\lambda\)\\leq 0, and for any eigenvalue withRe​\(λ\)=0\\text\{Re\}\(\\lambda\)=0the algebraic multiplicity equals the geometric multiplicity\. Velocity perturbations remain bounded\.
3. 3\.Unstable if∃λ∈σ⁡\(𝐌\)\\exists\\lambda\\in\\sigma\(\\mathbf\{M\}\)such thatRe​\(λ\)\>0\\text\{Re\}\(\\lambda\)\>0\. Arbitrarily small perturbations will grow exponentially over time\.

## 4Controlled Dynamics Formulation

In this section, we design a feedback controller for the second\-order dynamics derived in Equation \([5](https://arxiv.org/html/2610.12915#S3.E5)\)\. We provide a rigorous proof that the resulting system achieves local exponential stability, ensuring robust convergence to the equilibrium\. Building on this theoretical foundation, we propose the Controlled Minimax algorithm, which integrates our control strategy into a practical optimization framework\.

To facilitate the design of the controller, we extend the natural dynamics derived in \([5](https://arxiv.org/html/2610.12915#S3.E5)\) by introducing auxiliary control inputsuθu\_\{\\theta\}anduϕu\_\{\\phi\}\. For notational compactness, we denote the Hessian blocks ofLLasHθ​θ=∇θ​θ2LH\_\{\\theta\\theta\}=\\nabla^\{2\}\_\{\\theta\\theta\}L,Hθ​ϕ=∇θ​ϕ2LH\_\{\\theta\\phi\}=\\nabla^\{2\}\_\{\\theta\\phi\}L, and so forth\. The resulting controlled dynamical system is given by:

θ¨\\displaystyle\\ddot\{\\theta\}=−Hθ​θ​\(θ,ϕ\)​θ˙−Hθ​ϕ​\(θ,ϕ\)​ϕ˙\+uθ​\(θ,ϕ,θ˙,ϕ˙\),\\displaystyle=\-H\_\{\\theta\\theta\}\(\\theta,\\phi\)\\,\\dot\{\\theta\}\-H\_\{\\theta\\phi\}\(\\theta,\\phi\)\\,\\dot\{\\phi\}\+u\_\{\\theta\}\(\\theta,\\phi,\\dot\{\\theta\},\\dot\{\\phi\}\),\(10\)ϕ¨\\displaystyle\\ddot\{\\phi\}=Hϕ​θ​\(θ,ϕ\)​θ˙\+Hϕ​ϕ​\(θ,ϕ\)​ϕ˙\+uϕ​\(θ,ϕ,θ˙,ϕ˙\)\.\\displaystyle=\\phantom\{\-\}H\_\{\\phi\\theta\}\(\\theta,\\phi\)\\,\\dot\{\\theta\}\+H\_\{\\phi\\phi\}\(\\theta,\\phi\)\\,\\dot\{\\phi\}\+u\_\{\\phi\}\(\\theta,\\phi,\\dot\{\\theta\},\\dot\{\\phi\}\)\.
We focus on the behavior of this system near a stationary point\(θ∗,ϕ∗\)\(\\theta^\{\\ast\},\\phi^\{\\ast\}\)satisfying the first\-order optimality conditions∇θL​\(θ∗,ϕ∗\)=0\\nabla\_\{\\theta\}L\(\\theta^\{\\ast\},\\phi^\{\\ast\}\)=0and∇ϕL​\(θ∗,ϕ∗\)=0\\nabla\_\{\\phi\}L\(\\theta^\{\\ast\},\\phi^\{\\ast\}\)=0\. To ensure the problem is well\-posed for control synthesis, we introduce the following regularity assumptions\.

###### Assumption 4\.1\.

The objective functionLLis twice continuously differentiable \(L∈C2L\\in C^\{2\}\)\. Furthermore, there exist constantsr\>0r\>0andC\>0C\>0such that for all states in the neighborhood

\(θ,ϕ\)∈𝒩:=\{\(θ,ϕ\):‖\(θ,ϕ\)−\(θ∗,ϕ∗\)‖≤r\},\(\\theta,\\phi\)\\in\\mathcal\{N\}:=\\\{\(\\theta,\\phi\):\\\|\(\\theta,\\phi\)\-\(\\theta^\{\\ast\},\\phi^\{\\ast\}\)\\\|\\leq r\\\},the Hessian blocks satisfy the uniform bound:

‖Hθ​θ​\(θ,ϕ\)‖,‖Hθ​ϕ​\(θ,ϕ\)‖≤C,\\displaystyle\\\|H\_\{\\theta\\theta\}\(\\theta,\\phi\)\\\|,\\ \\\|H\_\{\\theta\\phi\}\(\\theta,\\phi\)\\\|\\leq C,‖Hϕ​θ​\(θ,ϕ\)‖,‖Hϕ​ϕ​\(θ,ϕ\)‖≤C,\\displaystyle\\\|H\_\{\\phi\\theta\}\(\\theta,\\phi\)\\\|,\\ \\\|H\_\{\\phi\\phi\}\(\\theta,\\phi\)\\\|\\leq C,where∥⋅∥\\\|\\cdot\\\|denotes the spectral norm \(or any matrix norm induced by the Euclidean vector norm\)\.

###### Assumption 4\.2\.

We assume system \([5](https://arxiv.org/html/2610.12915#S3.E5)\) is locally Lipschitz continuous in the aggregate state\(θ,ϕ,θ˙,ϕ˙\)\(\\theta,\\phi,\\dot\{\\theta\},\\dot\{\\phi\}\)on𝒩\\mathcal\{N\}\. Consequently, solutions to this system exist and are unique in a neighborhood of the equilibrium\(θ∗,ϕ∗,0,0\)\(\\theta^\{\\ast\},\\phi^\{\\ast\},0,0\)\.

### 4\.1Controller Design

We propose a linear velocity feedback controller designed to introduce damping and cross\-coupling between theθ\\thetaandϕ\\phidynamics\. The controllers are given by:

uθ​\(θ˙,ϕ˙\)\\displaystyle u\_\{\\theta\}\(\\dot\{\\theta\},\\dot\{\\phi\}\)=−λθ​θ˙\+β​Ac​ϕ˙,\\displaystyle=\-\\lambda\_\{\\theta\}\\dot\{\\theta\}\+\\beta A\_\{c\}\\dot\{\\phi\},\(11\)uϕ​\(θ˙,ϕ˙\)\\displaystyle u\_\{\\phi\}\(\\dot\{\\theta\},\\dot\{\\phi\}\)=−β​Ac⊤​θ˙−λϕ​ϕ˙,\\displaystyle=\-\\beta A\_\{c\}^\{\\top\}\\dot\{\\theta\}\-\\lambda\_\{\\phi\}\\dot\{\\phi\},whereλθ,λϕ\>0\\lambda\_\{\\theta\},\\lambda\_\{\\phi\}\>0are scalar damping gains,β∈ℝ\\beta\\in\\mathbb\{R\}is a cross\-coupling gain, andAc∈ℝdθ×dϕA\_\{c\}\\in\\mathbb\{R\}^\{d\_\{\\theta\}\\times d\_\{\\phi\}\}is a fixed coupling matrix\.

In our settings, we define the effective rankr≔min⁡\{dθ,dϕ\}r\\coloneqq\\min\\\{d\_\{\\theta\},d\_\{\\phi\}\\\}and selectAcA\_\{c\}to provide identity coupling on the leading coordinates:

\(Ac\)i​j=\{1,if​1≤i=j≤r,0,otherwise\.\(A\_\{c\}\)\_\{ij\}=\\begin\{cases\}1,&\\text\{if \}1\\leq i=j\\leq r,\\\\ 0,&\\text\{otherwise\}\.\\end\{cases\}This choice ensures direct interaction between the firstrrcomponents ofθ\\thetaandϕ\\phiwhile leaving any remaining dimensions uncoupled\.

To provide intuition for the proposed control structure, we examine the local behavior of the controlled system \([10](https://arxiv.org/html/2610.12915#S4.E10)\) near an equilibrium point\(θ∗,ϕ∗\)\(\\theta^\{\\ast\},\\phi^\{\\ast\}\)\. Letv⁡\(t\)≔\(θ˙​\(t\),ϕ˙​\(t\)\)⊤v\(t\)\\coloneqq\(\\dot\{\\theta\}\(t\),\\dot\{\\phi\}\(t\)\)^\{\\top\}denote the combined velocity vector\. The linearized velocity dynamics are governed byv˙=M​v,\\dot\{v\}=Mv,where the closed\-loop system matrixMMtakes the form

M=\(−Hθ​θ∗−λθ​I−Hθ​ϕ∗\+β​AcHϕ​θ∗−β​Ac⊤Hϕ​ϕ∗−λϕ​I\)\.M=\\begin\{pmatrix\}\-H\_\{\\theta\\theta\}^\{\\ast\}\-\\lambda\_\{\\theta\}I&\-H\_\{\\theta\\phi\}^\{\\ast\}\+\\beta A\_\{c\}\\\\\[4\.0pt\] H\_\{\\phi\\theta\}^\{\\ast\}\-\\beta A\_\{c\}^\{\\top\}&H\_\{\\phi\\phi\}^\{\\ast\}\-\\lambda\_\{\\phi\}I\\end\{pmatrix\}\.\(12\)Here,Hα​γ∗≔Hα​γ​\(θ∗,ϕ∗\)H\_\{\\alpha\\gamma\}^\{\\ast\}\\coloneqq H\_\{\\alpha\\gamma\}\(\\theta^\{\\ast\},\\phi^\{\\ast\}\)denotes the block components of the Hessian evaluated at the equilibrium\.

The structure ofMMillustrates the two distinct mechanisms introduced by the controller\. First, the terms−λθ​I\-\\lambda\_\{\\theta\}Iand−λϕ​I\-\\lambda\_\{\\phi\}Iprovide diagonal damping to dissipate energy\. Second, the termβ​Ac\\beta A\_\{c\}introduces a skew\-symmetric component that modifies the natural coupling betweenθ\\thetaandϕ\\phi\(represented by the off\-diagonal Hessian blocks\)\. This decomposition highlights how the controller augments the inherent geometry of the system\. We provide a rigorous stability proof for the full nonlinear system in the following section\.

### 4\.2Exponential Stability of the Controlled System

In this section, we establish the stability properties of the closed\-loop system \([10](https://arxiv.org/html/2610.12915#S4.E10)\) under the proposed controller \([11](https://arxiv.org/html/2610.12915#S4.E11)\)\. Our primary focus is to determine the conditions on the damping gainsλθ\\lambda\_\{\\theta\}andλϕ\\lambda\_\{\\phi\}that guarantee the convergence of the velocity dynamics to zero\. By analyzing the spectral properties of the system matrixMMdefined in \([12](https://arxiv.org/html/2610.12915#S4.E12)\), we derive a sufficient condition for exponential stability that is robust to the coupling parameterβ\\beta\. The following theorem formalizes this result, providing an explicit decay rate for the velocity vectorv⁡\(t\)v\(t\)\.

###### Theorem 4\.3\(Local Exponential stability\)\.

Suppose Assumptions[4\.1](https://arxiv.org/html/2610.12915#S4.Thmtheorem1)–[4\.2](https://arxiv.org/html/2610.12915#S4.Thmtheorem2)hold and letC\>0C\>0be as in Assumption[4\.1](https://arxiv.org/html/2610.12915#S4.Thmtheorem1)\. Consider the closed\-loop system \([10](https://arxiv.org/html/2610.12915#S4.E10)\) under the controller \([11](https://arxiv.org/html/2610.12915#S4.E11)\)\. If the damping gains are chosen such that

γ:=min⁡\{λθ,λϕ\}−2​C\>0,\\gamma:=\\min\\\{\\lambda\_\{\\theta\},\\lambda\_\{\\phi\}\\\}\-2C\>0,\(13\)then for anyβ∈ℝ\\beta\\in\\mathbb\{R\},

‖v⁡\(t\)‖≤e−γ​t​‖v⁡\(0\)‖for all​t≥0\.\\\|v\(t\)\\\|\\leq e^\{\-\\gamma t\}\\,\\\|v\(0\)\\\|\\qquad\\text\{for all \}t\\geq 0\.Consequently, the full nonlinear system is locally exponentially stable at the equilibrium\(θ∗,ϕ∗,0,0\)\(\\theta^\{\\ast\},\\phi^\{\\ast\},0,0\)\.

While Theorem[4\.3](https://arxiv.org/html/2610.12915#S4.Thmtheorem3)provides a solid theoretical guarantee for the convergence of the closed\-loop system, it relies on the assumption that the control input is available continuously and instantaneously\. In a realistic experimental setup, the control law must be discretized and computed iteratively\. Therefore, we now turn our attention to the algorithmic implementation\. Section[5](https://arxiv.org/html/2610.12915#S5)details the specific iterative scheme used to approximate the optimal control inputs and discusses its convergence properties and computational complexity\.

\(a\)Target Distribution\(b\)Baseline \(Standard GAN\)\(c\)Ours \(Controlled\-Minimax\)
Figure 2:Qualitative results on the Swiss Roll dataset\. While the standard GAN baseline struggles to learn the full manifold structure and exhibits disjointed segments, our proposed Controlled\-Minimax successfully generates a continuous and smooth spiral that closely matches the target distribution\.

## 5The Controlled\-Minimax Algoirthm

In this section, we propose theControlled\-Minimaxalgorithm to solve the minimax optimization problem while ensuring robust stability\. This method augments standard gradient descent\-ascent dynamics by integrating the cross\-coupling controller design defined in \([11](https://arxiv.org/html/2610.12915#S4.E11)\)\. By substituting this controllers into the update steps, the algorithm effectively modifies the natural gradient flow, introducing both dissipative damping and skew\-symmetric coupling to suppress oscillatory behavior and stabilize the trajectory toward the equilibrium\. The complete procedure is detailed in Algorithm[1](https://arxiv.org/html/2610.12915#alg1)\.

Algorithm 1Controlled\-Minimax Optimization0:Objective function

L⁡\(θ,ϕ\)L\(\\theta,\\phi\), learning rate

η\>0\\eta\>0
0:Damping gains

λθ,λϕ\\lambda\_\{\\theta\},\\lambda\_\{\\phi\}, coupling parameter

β\\beta
0:Initial parameters

θ0,ϕ0\\theta\_\{0\},\\phi\_\{0\}
1:

k←0k\\leftarrow 0
2:whileconvergence criteria not metdo

3:1\. Compute Gradients:

4:

gθ←∇θL​\(θk,ϕk\)g\_\{\\theta\}\\leftarrow\\nabla\_\{\\theta\}L\(\\theta\_\{k\},\\phi\_\{k\}\)
5:

gϕ←∇ϕL​\(θk,ϕk\)g\_\{\\phi\}\\leftarrow\\nabla\_\{\\phi\}L\(\\theta\_\{k\},\\phi\_\{k\}\)
6:2\. Compute Control Inputs:

7:

uθ←λθ​gθ\+β​Ac​gϕu\_\{\\theta\}\\leftarrow\\lambda\_\{\\theta\}g\_\{\\theta\}\+\\beta A\_\{c\}g\_\{\\phi\}
8:

uϕ←β​Ac⊤​gθ−λϕ​gϕu\_\{\\phi\}\\leftarrow\\beta A\_\{c\}^\{\\top\}g\_\{\\theta\}\-\\lambda\_\{\\phi\}g\_\{\\phi\}
9:3\. Update Parameters:

10:

θk\+1←θk−η​gθ\+η​uθ\\theta\_\{k\+1\}\\leftarrow\\theta\_\{k\}\-\\eta g\_\{\\theta\}\+\\eta u\_\{\\theta\}
11:

ϕk\+1←ϕk\+η​gϕ\+η​uϕ\\phi\_\{k\+1\}\\leftarrow\\phi\_\{k\}\+\\eta g\_\{\\phi\}\+\\eta u\_\{\\phi\}
12:

k←k\+1k\\leftarrow k\+1
13:endwhile

14:return

θk,ϕk\\theta\_\{k\},\\phi\_\{k\}

The proposed Controlled\-Minimax Optimization algorithm stabilizes the oscillatory dynamics inherent in standard Gradient Descent\-Ascent \(GDA\) by modeling the optimization as a controlled dynamical system\. We introduce two feedback mechanisms to counteract the rotational vector fields that typically prevent convergence to Nash equilibria:

- •Dissipative Damping: The terms parameterized byλθ\\lambda\_\{\\theta\}andλϕ\\lambda\_\{\\phi\}introduce artificial friction into the update laws\. This negative feedback dampens the kinetic energy accumulation that drives trajectories into limit cycles, effectively improving the local conditioning of the optimization landscape\.
- •Cross\-Coupled Correction: The terms involvingβ\\betaandAcA\_\{c\}inject the opponent’s gradient information into the current player’s update\. This cross\-coupling applies a corrective force orthogonal to the standard gradient flow, neutralizing the rotational component of the vector field and straightening the trajectory toward the saddle point\.

Collectively, these controls transform the update laws from simple steepest descent/ascent into a stabilized trajectory that balances local gradient exploitation with improved convergence stability\.

Figure 3:Training evolution on the 8\-Gaussian mixture dataset\. The top row displays the distribution generated by the Standard GAN baseline, while the bottom row illustrates the proposed Controlled\-Minimax\. Columns correspond to training iterationst=0,1000,3000,4999t=\{0,1000,3000,4999\}\.
## 6Experiments

In this section, we conduct a comprehensive experimental analysis ranging from low\-dimensional pathological examples to high\-dimensional image synthesis tasks to empirically validate the theoretical guarantees of the Controlled\-Minimax framework\. We begin by examining convergence behavior on synthetic benchmarks known to induce rotational dynamics\. Specifically, we visualize optimization trajectories and verify local exponential stability on the Dirac\-GAN\([Mescheder et al\., 2018](https://arxiv.org/html/2610.12915#bib.bib22)\)in Figure[1](https://arxiv.org/html/2610.12915#S1.F1), the Swiss Roll manifold in Figure[2](https://arxiv.org/html/2610.12915#S4.F2), and the 2D Gaussian mixtures\([Rasmussen, 1999](https://arxiv.org/html/2610.12915#bib.bib28)\)in Figure[3](https://arxiv.org/html/2610.12915#S5.F3)\.

Subsequently, we evaluate the scalability and robustness of our method on large\-scale adversarial image generation using the StyleGAN2\-ada\([Karras et al\., 2020](https://arxiv.org/html/2610.12915#bib.bib13)\)architecture across three standard datasets of varying complexity: CIFAR\-10\([Krizhevsky et al\., 2009](https://arxiv.org/html/2610.12915#bib.bib15)\), LSUN \(Bedroom and Cat categories\)\([Yu et al\., 2015](https://arxiv.org/html/2610.12915#bib.bib32)\), and Flickr\-Faces\-HQ \(FFHQ\)\([Karras et al\., 2019](https://arxiv.org/html/2610.12915#bib.bib12)\)\. By employing the Fréchet Inception Distance \(FID\) score\([Heusel et al\., 2017](https://arxiv.org/html/2610.12915#bib.bib10)\)to quantitatively assess sample quality and diversity, we demonstrate that our Controlled\-Minimax effectively suppresses oscillatory behavior and accelerates convergence relative to baseline methods without compromising generation fidelity in Figure[4](https://arxiv.org/html/2610.12915#S6.F4)and Table[1](https://arxiv.org/html/2610.12915#S6.T1)\.

### 6\.1Dirac\-GAN

Problem Setting\.The real data distribution is a Dirac\-delta function centered at the origin,pd​a​t​a=δ0p\_\{data\}=\\delta\_\{0\}\. The generator is parameterized by a single scalarθ\\thetaand is defined asGθ​\(z\)=θG\_\{\\theta\}\(z\)=\\theta\. The discriminator is modeled as a linear functionDψ​\(x\)=ψ⋅xD\_\{\\psi\}\(x\)=\\psi\\cdot x, parameterized by a scalarψ\\psi\. The unique Nash equilibrium for this game is located at the origin\(θ,ψ\)=\(0,0\)\(\\theta,\\psi\)=\(0,0\)\.

Result\.Figure[1](https://arxiv.org/html/2610.12915#S1.F1)presents a comparison of the optimization trajectories in parameter space\. As expected, the standard Dirac\-GAN baseline exhibits persistent oscillatory behavior, withθ\\thetaandψ\\psicycling indefinitely around the origin without converging\. In contrast, our Controlled Minimax method effectively dampens these oscillations, driving the parameters\(θ,ψ\)\(\\theta,\\psi\)rapidly to the unique Nash equilibrium at\(0,0\)\(0,0\)\.

### 6\.22D Gaussian

Dataset\. To evaluate the model’s ability to capture multimodal distributions and resist mode collapse, we utilize a synthetic 2D dataset consisting of a mixture of eight isotropic Gaussians\. The modes are arranged in a circle with a radius of 2\.0 around the origin, positioned at intervals ofπ/4\\pi/4\(specifically at coordinates\(±2,0\)\(\\pm 2,0\),\(0,±2\)\(0,\\pm 2\), and\(±2,±2\)\(\\pm\\sqrt\{2\},\\pm\\sqrt\{2\}\)\)\. Each Gaussian mode has a standard deviation ofσ=0\.02\\sigma=0\.02, creating distinct, separated clusters with minimal overlap\. During training, real data samples are drawn uniformly from these eight mixture components, while the generator attempts to map a standard normal latent distribution to this target manifold\.

Network Architecture\.Both the generator \(GG\) and the discriminator \(DD\) are parameterized as fully connected Multi\-Layer Perceptrons \(MLPs\)\. The generator maps a 2\-dimensional latent vectorz∼𝒩⁡\(0,I\)z\\sim\\mathcal\{N\}\(0,I\)to the 2\-dimensional data space using a network with two hidden layers, each containing 128 units\. Rectified Linear Unit \(ReLU\) activation functions are applied after each hidden layer, while the output layer is linear\. The discriminator mirrors this architecture, accepting 2\-dimensional data points as input and processing them through two hidden layers of 128 units with ReLU activations\. The final layer of the discriminator projects the features to a scalar scalar value, which is passed through a Sigmoid activation function to output the probability of the input being real\.

Result\.The qualitative results of the 2D Gaussian experiment are shown in Figure[3](https://arxiv.org/html/2610.12915#S5.F3)\. The standard GAN baseline exhibits severe mode collapse; despite the target distribution containing eight distinct modes, the generator fails to capture the full multimodal structure\. In contrast, our Controlled Minimax approach successfully stabilizes the training dynamics\. The generator learns to distribute its probability mass evenly across all eight Gaussian components, accurately recovering the geometry and diversity of the underlying distribution without dropping modes\.

\(a\)Cifar 10\(b\)LSUN\-Cat\(c\)FFHQ
Figure 4:Training convergence analysis across different datasets\. We plot the FID score as a function of training time \(in days\)\. The curves compare the performance of various hyperparameter configurations against the StyleGAN2\-ada baseline\.
### 6\.3Swiss Roll

Dataset\.We evaluate our method on the 2D Swiss Roll dataset, a classic benchmark for manifold learning\. The target distributionpd​a​t​a​\(x\)p\_\{data\}\(x\)is generated by samplingN=3,000N=3,000points according tox=t​cos⁡\(t\)x=t\\cos\(t\)andy=t​sin⁡\(t\)y=t\\sin\(t\), wheret∼𝒰⁡\[1\.5​π,4\.5​π\]t\\sim\\mathcal\{U\}\[1\.5\\pi,4\.5\\pi\]\. We add Gaussian noiseϵ∼𝒩⁡\(0,0\.252\)\\epsilon\\sim\\mathcal\{N\}\(0,0\.25^\{2\}\)to simulate real\-world variance\.

Network Architectures\.To ensure a fair comparison, both the baseline and our method share identical backbones\. The generator has an MLP with 3 hidden layers \(128 units, ReLU, Batch Norm\) mappingz∈ℝ2∼𝒩⁡\(0,I\)z\\in\\mathbb\{R\}^\{2\}\\sim\\mathcal\{N\}\(0,I\)to data space, while the discriminator has an MLP with 3 hidden layers \(128 units, LeakyReLUα=0\.2\\alpha=0\.2\) outputting a scalar probability\.

Result\.The generated distributions for the Swiss Roll dataset are visualized in Figure[2](https://arxiv.org/html/2610.12915#S4.F2)\. The Baseline GAN struggles to learn the continuous, spiraling manifold, and exhibits mode collapse\. Conversely, our Controlled Minimax method successfully generates samples that closely adhere to the true spiral trajectory, maintaining uniform coverage along the manifold and effectively reproducing the noise characteristics of the target distribution\.

### 6\.4StyleGAN2\-ada

Setting\.We strictly adhere to the experimental settings described in the original paper\([Karras et al\., 2020](https://arxiv.org/html/2610.12915#bib.bib13)\)to ensure a direct and fair comparison\. We evaluate performance on three standard benchmarks: CIFAR\-10 \(32×3232\\times 32\), LSUN Cat \(256×256256\\times 256\), and FFHQ \(1024×10241024\\times 1024\)\.

Result\.We compare the FID score of our Controlled\-Mimimax against the standard StyleGAN2\-ada baseline\. Figure[4](https://arxiv.org/html/2610.12915#S6.F4)illustrates the convergence speed, plotting FID scores against training time on various hyperparameter choices\. Our method demonstrates faster convergence and greater stability across all three datasets\. Table[1](https://arxiv.org/html/2610.12915#S6.T1)presents the quantitative evaluation after 7 days of training\. Our approach consistently outperforms the baseline, achieving the lowest FID scores across all tested resolutions\. Specifically, on Cifar10, our optimal configuration yields an FID of2\.912\.91, a marked improvement over the baseline’s3\.373\.37\. This performance advantage is further amplified in high\-resolution domains; on LSUN Cat and FFHQ, we reduce the FID from10\.4810\.48to8\.188\.18and from7\.817\.81to5\.465\.46, respectively\. Furthermore, the reduced oscillation ranges observed over the final 24 hours demonstrate superior stability, contrasting sharply with the high variance exhibited by the baseline near convergence\. These findings serve to validate our theoretical stability guarantees, demonstrating that the Controlled\-Minimax framework effectively improves both model performance and convergence behavior\.

Table 1:FID scores for each dataset after training for 7 days\. \(a±ba\\pm bin this table should be interpret asaais the FID score after 7 days,bbis the range of oscillation on the 24 hours span of the seventh day\)\. Throughout, we assume a unified parameterλ=λθ=λϕ\\lambda=\\lambda\_\{\\theta\}=\\lambda\_\{\\phi\}\.

## 7Conclusion and Discussion

In this work, we have addressed the fundamental instability of minimax optimization through the lens of continuous\-time dynamical systems\. We identified that the persistent limit cycles and divergence often observed in adversarial learning stem from the problematic eigenstructure of the loss function’s Hessian near equilibrium points\. To overcome these inherent structural limitations, we introduced the Controlled\-Minimax framework\. Our approach leverages control theory to derive a specific feedback controller that actively steers the optimization trajectory\.

Theoretical analysis confirms that this framework guarantees local exponential stability regardless of the underlying Hessian spectrum, effectively stabilizing problems where standard gradient descent\-ascent methods fail\. Empirically, our extensive validation ranging from pathological synthetic benchmarks like the Dirac\-GAN, 2D Gaussian and Swiss Roll to high\-dimensional image synthesis with StyleGAN2\-ada demonstrates that our Controlled\-Minimax not only suppresses oscillatory behavior but also ensures robust and rapid convergence\. Collectively, these results bridge the gap between control theory and deep learning, offering a principled solution to one of the most persistent challenges in adversarial training\.

Several promising directions remain for future exploration\. The current framework relies on fixed control parameters which guarantee local stability; however, designing an adaptive controller that adjusts to the local geometry could offer a pathway toward global stability guarantees\. Furthermore, acknowledging the inherent discrepancy between continuous ODE models and iterative algorithms, future work should rigorously address discrete\-time stability\. Establishing stability conditions directly for the discretized update rules will be essential to fully reconcile the theoretical control perspective with practical algorithmic behavior\.

## Broader Impact

This paper presents work whose goal is to advance the field of machine learning\. There are many potential societal consequences of our work, none of which we feel must be specifically highlighted here\.

## References

- Arjovsky et al\. \(2017\)Arjovsky, M\., Chintala, S\., and Bottou, L\.Wasserstein generative adversarial networks\.In*International conference on machine learning*, pp\. 214–223\. PMLR, 2017\.
- Bottou \(2010\)Bottou, L\.Large\-scale machine learning with stochastic gradient descent\.In*Proceedings of COMPSTAT’2010: 19th International Conference on Computational StatisticsParis France, August 22\-27, 2010 Keynote, Invited and Contributed Papers*, pp\. 177–186\. Springer, 2010\.
- Busoniu et al\. \(2008\)Busoniu, L\., Babuska, R\., and De Schutter, B\.A comprehensive survey of multiagent reinforcement learning\.*IEEE Transactions on Systems, Man, and Cybernetics, Part C \(Applications and Reviews\)*, 38\(2\):156–172, 2008\.
- Chavdarova et al\. \(2019\)Chavdarova, T\., Gidel, G\., Fleuret, F\., and Lacoste\-Julien, S\.Reducing noise in gan training with variance reduced extragradient\.*Advances in Neural Information Processing Systems*, 32, 2019\.
- Daskalakis & Panageas \(2018\)Daskalakis, C\. and Panageas, I\.The limit points of \(optimistic\) gradient descent in min\-max optimization\.*Advances in neural information processing systems*, 31, 2018\.
- Farnia & Ozdaglar \(2020\)Farnia, F\. and Ozdaglar, A\.Do gans always have nash equilibria?In*International Conference on Machine Learning*, pp\. 3029–3039\. PMLR, 2020\.
- Gidel et al\. \(2019\)Gidel, G\., Hemmat, R\. A\., Pezeshki, M\., Le Priol, R\., Huang, G\., Lacoste\-Julien, S\., and Mitliagkas, I\.Negative momentum for improved game dynamics\.In*The 22nd international conference on artificial intelligence and statistics*, pp\. 1802–1811\. PMLR, 2019\.
- Goodfellow et al\. \(2020\)Goodfellow, I\., Pouget\-Abadie, J\., Mirza, M\., Xu, B\., Warde\-Farley, D\., Ozair, S\., Courville, A\., and Bengio, Y\.Generative adversarial networks\.*Communications of the ACM*, 63\(11\):139–144, 2020\.
- Gulrajani et al\. \(2017\)Gulrajani, I\., Ahmed, F\., Arjovsky, M\., Dumoulin, V\., and Courville, A\. C\.Improved training of wasserstein gans\.*Advances in neural information processing systems*, 30, 2017\.
- Heusel et al\. \(2017\)Heusel, M\., Ramsauer, H\., Unterthiner, T\., Nessler, B\., and Hochreiter, S\.Gans trained by a two time\-scale update rule converge to a local nash equilibrium\.*Advances in neural information processing systems*, 30, 2017\.
- Hsieh et al\. \(2019\)Hsieh, Y\.\-P\., Liu, C\., and Cevher, V\.Finding mixed nash equilibria of generative adversarial networks\.In*International Conference on Machine Learning*, pp\. 2810–2819\. PMLR, 2019\.
- Karras et al\. \(2019\)Karras, T\., Laine, S\., and Aila, T\.A style\-based generator architecture for generative adversarial networks\.In*Proceedings of the IEEE/CVF conference on computer vision and pattern recognition*, pp\. 4401–4410, 2019\.
- Karras et al\. \(2020\)Karras, T\., Aittala, M\., Hellsten, J\., Laine, S\., Lehtinen, J\., and Aila, T\.Training generative adversarial networks with limited data\.*Advances in neural information processing systems*, 33:12104–12114, 2020\.
- Korpelevich \(1976\)Korpelevich, G\. M\.The extragradient method for finding saddle points and other problems\.*Matecon*, 12:747–756, 1976\.
- Krizhevsky et al\. \(2009\)Krizhevsky, A\., Hinton, G\., et al\.Learning multiple layers of features from tiny images\.2009\.
- Langley \(2000\)Langley, P\.Crafting papers on machine learning\.In Langley, P\. \(ed\.\),*Proceedings of the 17th International Conference on Machine Learning \(ICML 2000\)*, pp\. 1207–1216, Stanford, CA, 2000\. Morgan Kaufmann\.
- Li & Orabona \(2019\)Li, X\. and Orabona, F\.On the convergence of stochastic gradient descent with adaptive stepsizes\.In*The 22nd international conference on artificial intelligence and statistics*, pp\. 983–992\. PMLR, 2019\.
- Luo et al\. \(2023\)Luo, T\., Zhu, Z\., Chen, J\., and Zhu, J\.Stabilizing gans’ training with brownian motion controller\.In*International Conference on Machine Learning*, pp\. 23140–23156\. PMLR, 2023\.
- Madry et al\. \(2017\)Madry, A\., Makelov, A\., Schmidt, L\., Tsipras, D\., and Vladu, A\.Towards deep learning models resistant to adversarial attacks\.*arXiv preprint arXiv:1706\.06083*, 2017\.
- Mertikopoulos et al\. \(2018\)Mertikopoulos, P\., Lecouat, B\., Zenati, H\., Foo, C\.\-S\., Chandrasekhar, V\., and Piliouras, G\.Optimistic mirror descent in saddle\-point problems: Going the extra \(gradient\) mile\.*arXiv preprint arXiv:1807\.02629*, 2018\.
- Mescheder et al\. \(2017\)Mescheder, L\., Nowozin, S\., and Geiger, A\.The numerics of gans\.*Advances in neural information processing systems*, 30, 2017\.
- Mescheder et al\. \(2018\)Mescheder, L\., Geiger, A\., and Nowozin, S\.Which training methods for gans do actually converge?In*International conference on machine learning*, pp\. 3481–3490\. PMLR, 2018\.
- Miyato et al\. \(2018\)Miyato, T\., Kataoka, T\., Koyama, M\., and Yoshida, Y\.Spectral normalization for generative adversarial networks\.*arXiv preprint arXiv:1802\.05957*, 2018\.
- Mokhtari et al\. \(2020\)Mokhtari, A\., Ozdaglar, A\., and Pattathil, S\.A unified analysis of extra\-gradient and optimistic gradient methods for saddle point problems: Proximal point approach\.In*International Conference on Artificial Intelligence and Statistics*, pp\. 1497–1507\. PMLR, 2020\.
- Nagarajan & Kolter \(2017\)Nagarajan, V\. and Kolter, J\. Z\.Gradient descent gan optimization is locally stable\.*Advances in neural information processing systems*, 30, 2017\.
- Nemirovski et al\. \(2009\)Nemirovski, A\., Juditsky, A\., Lan, G\., and Shapiro, A\.Robust stochastic approximation approach to stochastic programming\.*SIAM Journal on optimization*, 19\(4\):1574–1609, 2009\.
- Palaniappan & Bach \(2016\)Palaniappan, B\. and Bach, F\.Stochastic variance reduction methods for saddle\-point problems\.*Advances in neural information processing systems*, 29, 2016\.
- Rasmussen \(1999\)Rasmussen, C\.The infinite gaussian mixture model\.*Advances in neural information processing systems*, 12, 1999\.
- Su et al\. \(2016\)Su, W\., Boyd, S\., and Candes, E\. J\.A differential equation for modeling nesterov’s accelerated gradient method: Theory and insights\.*Journal of Machine Learning Research*, 17\(153\):1–43, 2016\.
- Xu et al\. \(2020\)Xu, K\., Li, C\., Zhu, J\., and Zhang, B\.Understanding and stabilizing gans’ training dynamics using control theory\.In*International conference on machine learning*, pp\. 10566–10575\. PMLR, 2020\.
- Yang et al\. \(2020\)Yang, J\., Kiyavash, N\., and He, N\.Global convergence and variance reduction for a class of nonconvex\-nonconcave minimax problems\.*Advances in neural information processing systems*, 33:1153–1165, 2020\.
- Yu et al\. \(2015\)Yu, F\., Seff, A\., Zhang, Y\., Song, S\., Funkhouser, T\., and Xiao, J\.Lsun: Construction of a large\-scale image dataset using deep learning with humans in the loop\.*arXiv preprint arXiv:1506\.03365*, 2015\.

## Appendix AProof of Theorem[4\.3](https://arxiv.org/html/2610.12915#S4.Thmtheorem3)

###### Proof\.

Consider the quadratic Lyapunov function

V⁡\(v\)=12​\(‖θ˙‖2\+‖ϕ˙‖2\)=12​‖v‖2\.V\(v\)=\\frac\{1\}\{2\}\\big\(\\\|\\dot\{\\theta\}\\\|^\{2\}\+\\\|\\dot\{\\phi\}\\\|^\{2\}\\big\)=\\frac\{1\}\{2\}\\\|v\\\|^\{2\}\.Along trajectories of the linear system \(\) we have

V˙=v⊤​v˙=v⊤​M​v\.\\dot\{V\}=v^\{\\top\}\\dot\{v\}=v^\{\\top\}Mv\.Writingv=\(θ˙,ϕ˙\)v=\(\\dot\{\\theta\},\\dot\{\\phi\}\)in block form yields

V˙\\displaystyle\\dot\{V\}=θ˙⊤​\(−Hθ​θ∗−λθ​I\)​θ˙\+θ˙⊤​\(−Hθ​ϕ∗\+β​Ac\)​ϕ˙\\displaystyle=\\dot\{\\theta\}^\{\\top\}\(\-H\_\{\\theta\\theta\}^\{\\ast\}\-\\lambda\_\{\\theta\}I\)\\dot\{\\theta\}\+\\dot\{\\theta\}^\{\\top\}\(\-H\_\{\\theta\\phi\}^\{\\ast\}\+\\beta A\_\{c\}\)\\dot\{\\phi\}\+ϕ˙⊤​\(Hϕ​θ∗−β​Ac⊤\)​θ˙\+ϕ˙⊤​\(Hϕ​ϕ∗−λϕ​I\)​ϕ˙\.\\displaystyle\\quad\+\\dot\{\\phi\}^\{\\top\}\(H\_\{\\phi\\theta\}^\{\\ast\}\-\\beta A\_\{c\}^\{\\top\}\)\\dot\{\\theta\}\+\\dot\{\\phi\}^\{\\top\}\(H\_\{\\phi\\phi\}^\{\\ast\}\-\\lambda\_\{\\phi\}I\)\\dot\{\\phi\}\.\(14\)
*Diagonal terms\.*By Assumption[4\.1](https://arxiv.org/html/2610.12915#S4.Thmtheorem1), and using the induced matrix norm, we have

‖θ˙⊤​Hθ​θ∗​θ˙‖≤‖Hθ​θ∗‖​‖θ˙‖2≤L​‖θ˙‖2,\\\|\\dot\{\\theta\}^\{\\top\}H\_\{\\theta\\theta\}^\{\\ast\}\\dot\{\\theta\}\\\|\\leq\\\|H\_\{\\theta\\theta\}^\{\\ast\}\\\|\\,\\\|\\dot\{\\theta\}\\\|^\{2\}\\leq L\\\|\\dot\{\\theta\}\\\|^\{2\},and similarly for theϕ\\phi\-block\. Hence

θ˙⊤​\(−Hθ​θ∗−λθ​I\)​θ˙\\displaystyle\\dot\{\\theta\}^\{\\top\}\(\-H\_\{\\theta\\theta\}^\{\\ast\}\-\\lambda\_\{\\theta\}I\)\\dot\{\\theta\}≤\(L−λθ\)​‖θ˙‖2,\\displaystyle\\leq\(L\-\\lambda\_\{\\theta\}\)\\\|\\dot\{\\theta\}\\\|^\{2\},ϕ˙⊤​\(Hϕ​ϕ∗−λϕ​I\)​ϕ˙\\displaystyle\\dot\{\\phi\}^\{\\top\}\(H\_\{\\phi\\phi\}^\{\\ast\}\-\\lambda\_\{\\phi\}I\)\\dot\{\\phi\}≤\(L−λϕ\)​‖ϕ˙‖2\.\\displaystyle\\leq\(L\-\\lambda\_\{\\phi\}\)\\\|\\dot\{\\phi\}\\\|^\{2\}\.
*Cross terms\.*Define the cross contribution

Tcross:=θ˙⊤​\(−Hθ​ϕ∗\+β​Ac\)​ϕ˙\+ϕ˙⊤​\(Hϕ​θ∗−β​Ac⊤\)​θ˙\.T\_\{\\text\{cross\}\}:=\\dot\{\\theta\}^\{\\top\}\(\-H\_\{\\theta\\phi\}^\{\\ast\}\+\\beta A\_\{c\}\)\\dot\{\\phi\}\+\\dot\{\\phi\}^\{\\top\}\(H\_\{\\phi\\theta\}^\{\\ast\}\-\\beta A\_\{c\}^\{\\top\}\)\\dot\{\\theta\}\.Expanding gives

Tcross=−θ˙⊤​Hθ​ϕ∗​ϕ˙\+β​θ˙⊤​Ac​ϕ˙\+ϕ˙⊤​Hϕ​θ∗​θ˙−β​ϕ˙⊤​Ac⊤​θ˙\.T\_\{\\text\{cross\}\}=\-\\dot\{\\theta\}^\{\\top\}H\_\{\\theta\\phi\}^\{\\ast\}\\dot\{\\phi\}\+\\beta\\dot\{\\theta\}^\{\\top\}A\_\{c\}\\dot\{\\phi\}\+\\dot\{\\phi\}^\{\\top\}H\_\{\\phi\\theta\}^\{\\ast\}\\dot\{\\theta\}\-\\beta\\dot\{\\phi\}^\{\\top\}A\_\{c\}^\{\\top\}\\dot\{\\theta\}\.Sinceϕ˙⊤​Ac⊤​θ˙=θ˙⊤​Ac​ϕ˙\\dot\{\\phi\}^\{\\top\}A\_\{c\}^\{\\top\}\\dot\{\\theta\}=\\dot\{\\theta\}^\{\\top\}A\_\{c\}\\dot\{\\phi\}, the antisymmetric controller terms cancel exactly, i\.e\.,

β​θ˙⊤​Ac​ϕ˙−β​ϕ˙⊤​Ac⊤​θ˙=0,\\beta\\dot\{\\theta\}^\{\\top\}A\_\{c\}\\dot\{\\phi\}\-\\beta\\dot\{\\phi\}^\{\\top\}A\_\{c\}^\{\\top\}\\dot\{\\theta\}=0,and hence

Tcross=−θ˙⊤​Hθ​ϕ∗​ϕ˙\+ϕ˙⊤​Hϕ​θ∗​θ˙\.T\_\{\\text\{cross\}\}=\-\\dot\{\\theta\}^\{\\top\}H\_\{\\theta\\phi\}^\{\\ast\}\\dot\{\\phi\}\+\\dot\{\\phi\}^\{\\top\}H\_\{\\phi\\theta\}^\{\\ast\}\\dot\{\\theta\}\.
Again using Assumption[4\.1](https://arxiv.org/html/2610.12915#S4.Thmtheorem1)and Cauchy–Schwarz,

\|Tcross\|≤\(‖Hθ​ϕ∗‖\+‖Hϕ​θ∗‖\)​‖θ˙‖​‖ϕ˙‖≤2​L​‖θ˙‖​‖ϕ˙‖\.\|T\_\{\\text\{cross\}\}\|\\leq\\big\(\\\|H\_\{\\theta\\phi\}^\{\\ast\}\\\|\+\\\|H\_\{\\phi\\theta\}^\{\\ast\}\\\|\\big\)\\\|\\dot\{\\theta\}\\\|\\,\\\|\\dot\{\\phi\}\\\|\\leq 2L\\,\\\|\\dot\{\\theta\}\\\|\\,\\\|\\dot\{\\phi\}\\\|\.
*Combining bounds\.*Substituting these estimates into \([14](https://arxiv.org/html/2610.12915#A1.E14)\), we obtain

V˙≤\(L−λθ\)​‖θ˙‖2\+\(L−λϕ\)​‖ϕ˙‖2\+2​L​‖θ˙‖​‖ϕ˙‖\.\\dot\{V\}\\leq\(L\-\\lambda\_\{\\theta\}\)\\\|\\dot\{\\theta\}\\\|^\{2\}\+\(L\-\\lambda\_\{\\phi\}\)\\\|\\dot\{\\phi\}\\\|^\{2\}\+2L\\\|\\dot\{\\theta\}\\\|\\,\\\|\\dot\{\\phi\}\\\|\.Using the inequality2​a​b≤a2\+b22ab\\leq a^\{2\}\+b^\{2\}for alla,b≥0a,b\\geq 0, we bound

2​L​‖θ˙‖​‖ϕ˙‖≤L​‖θ˙‖2\+L​‖ϕ˙‖2,2L\\\|\\dot\{\\theta\}\\\|\\,\\\|\\dot\{\\phi\}\\\|\\leq L\\\|\\dot\{\\theta\}\\\|^\{2\}\+L\\\|\\dot\{\\phi\}\\\|^\{2\},and thus

V˙≤\(2​L−λθ\)​‖θ˙‖2\+\(2​L−λϕ\)​‖ϕ˙‖2\.\\dot\{V\}\\leq\(2L\-\\lambda\_\{\\theta\}\)\\\|\\dot\{\\theta\}\\\|^\{2\}\+\(2L\-\\lambda\_\{\\phi\}\)\\\|\\dot\{\\phi\}\\\|^\{2\}\.If \([13](https://arxiv.org/html/2610.12915#S4.E13)\) holds, then

γ:=min⁡\{λθ−2​L,λϕ−2​L\}\>0,\\gamma:=\\min\\\{\\lambda\_\{\\theta\}\-2L,\\ \\lambda\_\{\\phi\}\-2L\\\}\>0,and we have

V˙≤−γ⁡\(‖θ˙‖2\+‖ϕ˙‖2\)=−2​γ​V\.\\dot\{V\}\\leq\-\\gamma\\big\(\\\|\\dot\{\\theta\}\\\|^\{2\}\+\\\|\\dot\{\\phi\}\\\|^\{2\}\\big\)=\-2\\gamma V\.Integrating yields

V⁡\(t\)≤V⁡\(0\)​e−2​γ​t,‖v⁡\(t\)‖≤e−γ​t​‖v⁡\(0\)‖,V\(t\)\\leq V\(0\)e^\{\-2\\gamma t\},\\qquad\\\|v\(t\)\\\|\\leq e^\{\-\\gamma t\}\\,\\\|v\(0\)\\\|,which shows exponential stability ofv=0v=0for the linear system \([10](https://arxiv.org/html/2610.12915#S4.E10)\)\.

For the nonlinear system \([10](https://arxiv.org/html/2610.12915#S4.E10)\), the closed\-loop dynamics areC1C^\{1\}and locally Lipschitz by Assumption[4\.2](https://arxiv.org/html/2610.12915#S4.Thmtheorem2)and the linear controller \([11](https://arxiv.org/html/2610.12915#S4.E11)\)\. The Jacobian of the nonlinear system at\(θ∗,ϕ∗,0,0\)\(\\theta^\{\\ast\},\\phi^\{\\ast\},0,0\)coincides with the matrixMMin \([12](https://arxiv.org/html/2610.12915#S4.E12)\)\. Since the linearization is exponentially stable, standard nonlinear stability theory implies that the equilibrium\(θ∗,ϕ∗,0,0\)\(\\theta^\{\\ast\},\\phi^\{\\ast\},0,0\)of the nonlinear system is locally exponentially stable\. ∎