From Jensen–Shannon GANs to Wasserstein and Parallel Tempering
Objectives, geometry, and an eight-mode Gaussian experiment
1 Overview
The original generative adversarial network admits an exact population characterization in terms of Jensen–Shannon divergence: after optimizing the discriminator over all measurable functions, its minimax value is \(2\operatorname{JS}(P,P_\theta)-\log 4\). This identity does not extend unchanged to practical alternating optimization with a finite discriminator, and the commonly used non-saturating generator loss is not literally the original minimax/Jensen–Shannon objective.
A Wasserstein GAN changes the discrepancy. It replaces binary classification between real and generated observations with the Kantorovich–Rubinstein dual of the earth-mover distance. This yields informative gradients even when the real and generated distributions have disjoint supports, provided the critic is constrained to be 1-Lipschitz.
The parallelly tempered GAN (PTGAN) of Sohn and Song (2025) keeps the adversarial framework but changes the path of distributions learned during training. Instead of learning only the target law \(P_X\), one shared conditional generator learns the family
\[ Q^\alpha = \alpha X_1 + (1-\alpha)X_2, \qquad X_1,X_2\stackrel{\mathrm{iid}}{\sim}P_X, \qquad \alpha\in[0,1]. \]
Interior temperatures average independent target draws and place probability between separated modes. A coherency penalty then couples critic learning across temperatures. The method is called “parallel tempering,” but it is not the Markov-chain Monte Carlo algorithm: there are no chains, swaps, or Metropolis corrections. It is simultaneous conditional GAN training along a convex distribution path.
2 Common setup
Let \((\mathcal X,\|\cdot\|)\) be the sample space, \(P\) the data distribution, \(Z\sim P_Z\) a reference variable, and \(G_\theta:\mathcal Z\to\mathcal X\) a generator. The induced model distribution is the pushforward
\[ P_\theta = (G_\theta)_\#P_Z, \qquad P_\theta(A)=P_Z\{z:G_\theta(z)\in A\}. \]
Training is difficult because we ordinarily cannot evaluate a density \(p_\theta(x)\): we can sample \(G_\theta(Z)\), but the neural transformation need not be invertible. A GAN estimates a discrepancy between \(P\) and \(P_\theta\) using a second network and differentiates that estimated discrepancy through \(G_\theta\).
3 Traditional GAN: classification and Jensen–Shannon divergence
3.1 The minimax game
The original GAN of Goodfellow et al. (2014) uses a discriminator \(D_\phi:\mathcal X\to(0,1)\) and solves
\[ \min_\theta\max_\phi V(\phi,\theta), \qquad V(\phi,\theta) =\mathbb E_{X\sim P}\log D_\phi(X) +\mathbb E_{Z\sim P_Z}\log\{1-D_\phi(G_\theta(Z))\}. \]
For equal prior probabilities of “real” and “fake,” \(D_\phi(x)\) is a probabilistic classifier for the label real. Suppose \(P\) and \(P_\theta\) have densities \(p\) and \(q\) with respect to a common dominating measure \(\mu\). For a fixed generator, the discriminator objective separates pointwise:
\[ V(D,q)=\int \left[p(x)\log D(x)+q(x)\log\{1-D(x)\}\right]d\mu(x). \]
At a point where \(p(x)+q(x)>0\), differentiate the integrand with respect to \(d=D(x)\):
\[ \frac{\partial}{\partial d} \{p\log d+q\log(1-d)\} =\frac{p}{d}-\frac{q}{1-d}. \]
The first-order condition is \(p(1-d)=qd\), so
\[ D^*(x)=\frac{p(x)}{p(x)+q(x)}. \]
The second derivative, \(-p/d^2-q/(1-d)^2\), is nonpositive, hence this is the pointwise maximum. Define the mixture \(M=(P+P_\theta)/2\), with density \(m=(p+q)/2\). Substitution gives
\[ \begin{aligned} V(D^*,q) &=\int p\log\frac{p}{p+q}\,d\mu +\int q\log\frac{q}{p+q}\,d\mu\\ &=\int p\log\frac{p}{2m}\,d\mu +\int q\log\frac{q}{2m}\,d\mu\\ &=\operatorname{KL}(P\|M)+\operatorname{KL}(P_\theta\|M)-2\log 2\\ &=2\operatorname{JS}(P\|P_\theta)-\log 4, \end{aligned} \]
where
\[ \operatorname{JS}(P\|Q) =\frac12\operatorname{KL}\!\left(P\middle\|\frac{P+Q}{2}\right) +\frac12\operatorname{KL}\!\left(Q\middle\|\frac{P+Q}{2}\right). \]
Thus an unrestricted, fully optimized discriminator converts the generator’s minimax problem into minimizing Jensen–Shannon divergence. The global optimum is \(P_\theta=P\), at which \(D^*(x)=1/2\) and \(V=-\log4\).
3.2 Why the qualification matters
The equality above is a population identity for \(D^*\). Alternating stochastic gradient updates with a finite \(D_\phi\) do not optimize \(D\) over all measurable functions. Moreover, practitioners usually replace the minimax generator loss
\[ \mathcal L_G^{\mathrm{minimax}}(\theta) =\mathbb E_Z\log\{1-D_\phi(G_\theta(Z))\} \]
with the non-saturating loss
\[ \mathcal L_G^{\mathrm{NS}}(\theta) =-\mathbb E_Z\log D_\phi(G_\theta(Z)). \]
They have the same desired equilibrium but different gradients. If \(D=\sigma(a)\) and a generated point is confidently rejected, then \(a\ll0\). The logit derivative of the minimax loss is
\[ \frac{\partial}{\partial a}\log(1-\sigma(a))=-\sigma(a)\approx0, \]
whereas the non-saturating derivative is
\[ \frac{\partial}{\partial a}\{-\log\sigma(a)\} =-(1-\sigma(a))\approx-1. \]
The second loss therefore sends a usable signal to a poor generator, but its finite-critic training dynamics should not be described as direct numerical minimization of Jensen–Shannon divergence.
3.3 The support-mismatch pathology
Let \(P=\delta_0\) and \(P_\theta=\delta_\theta\). For every \(\theta\ne0\), the two supports are disjoint and
\[ \operatorname{JS}(\delta_0\|\delta_\theta)=\log2. \]
It jumps to zero only at \(\theta=0\). The perfect discriminator assigns 1 to the real point and 0 to the generated point; the divergence is locally flat in \(\theta\) almost everywhere. This idealized example captures a high-dimensional problem: data and generator distributions can lie on thin, initially disjoint manifolds, allowing classification to become perfect before the generator has learned how to move toward the data.
4 Wasserstein GAN: transport rather than classification
4.1 Primal and dual distances
For probability measures with finite first moments, the Wasserstein-1 distance is
\[ W_1(P,Q) =\inf_{\gamma\in\Pi(P,Q)} \mathbb E_{(X,Y)\sim\gamma}\|X-Y\|, \]
where \(\Pi(P,Q)\) is the set of all couplings with marginals \(P\) and \(Q\). The primal asks for the cheapest joint transportation plan. The Kantorovich–Rubinstein dual is
\[ W_1(P,Q) =\sup_{\|f\|_{\mathrm{Lip}}\le1} \left\{\mathbb E_P f(X)-\mathbb E_Q f(Y)\right\}, \]
where
\[ \|f\|_{\mathrm{Lip}} =\sup_{x\ne y}\frac{|f(x)-f(y)|}{\|x-y\|}. \]
The WGAN critic \(f_\phi:\mathcal X\to\mathbb R\) produces an unrestricted real score rather than a class probability. Its population game is
\[ \min_\theta\max_{\phi:\|f_\phi\|_{\mathrm{Lip}}\le1} \left[\mathbb E_P f_\phi(X) -\mathbb E_Z f_\phi(G_\theta(Z))\right]. \]
The Dirac example now has
\[ W_1(\delta_0,\delta_\theta)=|\theta|, \]
which changes continuously with the generator location and has derivative \(\operatorname{sign}(\theta)\) away from zero. That is the central geometric advantage over the saturated Jensen–Shannon value.
4.2 Enforcing the Lipschitz constraint
The original WGAN clipped network weights (Arjovsky, Chintala, and Bottou 2017). WGAN-GP instead penalizes critic gradients at interpolated real–fake points (Gulrajani et al. 2017):
\[ \widehat H_{\mathrm{GP}}(\phi) =\frac1b\sum_{i=1}^b \left(\|\nabla_x f_\phi(\widehat x_i)\|_2-1\right)^2, \]
\[ \widehat x_i=\epsilon_i x_i+(1-\epsilon_i)\widetilde x_i, \qquad \epsilon_i\sim\operatorname{Unif}(0,1), \qquad \widetilde x_i=G_\theta(z_i). \]
Trex’s benchmark uses the one-sided version
\[ \widehat H_{\mathrm{GP}}^+ =\frac1b\sum_i \left[\max\{\|\nabla_xf_\phi(\widehat x_i)\|_2-1,0\}\right]^2, \]
so gradients smaller than one are not penalized. In minimization convention, the Trex critic and generator losses are
\[ \widehat{\mathcal L}_D =\frac1b\sum_i f_\phi(\widetilde x_i) -\frac1b\sum_i f_\phi(x_i) +\lambda_{\mathrm{GP}}\widehat H_{\mathrm{GP}}^+, \]
\[ \widehat{\mathcal L}_G =-\frac1b\sum_i f_\phi(G_\theta(z_i)). \]
Wasserstein geometry fixes the particular support-mismatch problem, but it does not turn the adversarial game into convex optimization. A finite critic, stochastic gradients, uneven progress across modes, and generator capacity can still produce oscillation or mode collapse. The gradient penalty controls local slopes on real–fake segments; it does not explicitly coordinate the critic’s learning across different levels of multimodality.
5 PTGAN: learn a convex path of target laws
5.1 The tempered distribution
PTGAN begins with two independent target draws and defines
\[ Q^\alpha=\alpha X_1+(1-\alpha)X_2. \]
For \(0<\alpha<1\), if \(P\) has a density \(p\) on \(\mathbb R^d\), the density of \(Q^\alpha\) is the scaled convolution
\[ p_\alpha(q) =\int_{\mathbb R^d} p(x)\,(1-\alpha)^{-d} p\!\left(\frac{q-\alpha x}{1-\alpha}\right)dx. \]
Equivalently, its characteristic function is
\[ \varphi_{Q^\alpha}(t) =\varphi_X(\alpha t)\varphi_X((1-\alpha)t). \]
If \(\mathbb E X=\mu\) and \(\operatorname{Var}(X)=\Sigma\), independence implies
\[ \mathbb E Q^\alpha=\mu, \qquad \operatorname{Var}(Q^\alpha) =\{\alpha^2+(1-\alpha)^2\}\Sigma. \]
The variance multiplier is minimized at \(\alpha=1/2\), where it equals \(1/2\). Also,
\[ Q^\alpha\overset d=Q^{1-\alpha}, \]
because swapping \(X_1\) and \(X_2\) leaves their joint law unchanged.
For a \(K\)-component Gaussian mixture
\[ P_X=\sum_{k=1}^K\pi_k\mathcal N(\mu_k,\Sigma_k), \]
the tempered law is available exactly:
\[ P_{Q^\alpha} =\sum_{k=1}^K\sum_{\ell=1}^K \pi_k\pi_\ell\, \mathcal N\!\left( \alpha\mu_k+(1-\alpha)\mu_\ell, \alpha^2\Sigma_k+(1-\alpha)^2\Sigma_\ell \right). \]
The \(K^2\) component means include points on segments between the original centers. Thus convex tempering does more than add independent noise: it creates bridges between modes while averaging within-component noise.
5.2 Joint objective and temperature sampling
One shared generator and critic learn the joint law of \((Q^\alpha,\alpha)\). The paper samples
\[ \alpha\sim p_\alpha =r\delta_1+(1-r)\operatorname{Unif}(0,1). \]
The point mass \(r\) ensures frequent direct training on the target slice \(\alpha=1\); the uniform part supplies smoother bridge distributions. The symmetry \(Q^\alpha\overset d=Q^{1-\alpha}\) is imposed through
\[ t(\alpha)=1-2|\alpha-1/2|\in[0,1], \qquad t(\alpha)=t(1-\alpha). \]
Writing \(G_\theta(z,t)\) and \(D_\phi(q,t)\), the Wasserstein-style joint objective is
\[ \min_\theta\max_\phi \mathcal L(\phi,\theta) =\mathbb E_{\alpha,X_1,X_2} D_\phi(Q^\alpha,t(\alpha)) -\mathbb E_{\alpha,Z} D_\phi(G_\theta(Z, t(\alpha)),t(\alpha)). \]
In the implementation, real and fake terms in a minibatch share the same \(\alpha\). This common-random-number coupling leaves each expectation unbiased and reduces irrelevant Monte Carlo differences between temperature mixtures.
The paper also recommends reference interpolation
\[ Z^\alpha=\alpha Z_1+(1-\alpha)Z_2, \qquad Z_1,Z_2\stackrel{\mathrm{iid}}{\sim}P_Z, \]
so the reference path has the same algebraic structure as the target path. Trex uses \(G_\theta(Z^\alpha,t(\alpha))\). At inference, setting \(\alpha=1\) gives \(Z^1=Z_1\) and requests the original target distribution.
5.3 Why parallel training alone is not enough
Sharing parameters across temperatures does not force progress at one temperature to agree with progress at another. Let \(\alpha_1\sim p_\alpha\), \(\alpha_2\sim\operatorname{Unif}(0,1)\), and use the same pair \((X_1,X_2)\) to construct
\[ q_1=\alpha_1X_1+(1-\alpha_1)X_2, \qquad q_2=\alpha_2X_1+(1-\alpha_2)X_2. \]
For \(\nu\sim\operatorname{Unif}(0,1)\), define
\[ \widetilde q=\nu q_1+(1-\nu)q_2, \qquad \widetilde\alpha=\nu\alpha_1+(1-\nu)\alpha_2. \]
The desired heuristic is \(D(q_1,t(\alpha_1))\approx D(q_2,t(\alpha_2))\). Set \(q(s)=sq_1+(1-s)q_2\) and \(\alpha(s)=s\alpha_1+(1-s)\alpha_2\). Away from the cusp of \(t\) at \(1/2\), the complete path derivative includes both a spatial term and a temperature term:
\[ \frac{dD}{ds} =\nabla_qD^\top(q_1-q_2) +\frac{\partial D}{\partial t}\, t'(\alpha(s))(\alpha_1-\alpha_2). \]
If one instead connects the transformed endpoints linearly, the second factor is \(t(\alpha_1)-t(\alpha_2)\). PTGAN deliberately regularizes only the spatial term. Its coherency penalty is
\[ H(\phi) =\lambda\, \mathbb E\left[ \left\{ \nabla_qD_\phi(\widetilde q,t(\widetilde\alpha))^\top (q_1-q_2) \right\}^2 \right]. \]
This is a heuristic first-order synchronization penalty, not an exact equality constraint. Its weighting has a useful identity:
\[ q_1-q_2=(\alpha_1-\alpha_2)(X_1-X_2). \]
Pairs from distant modes therefore receive a larger squared directional penalty, as do temperatures farther apart. In the minimization convention used by Trex, the PTGAN updates are
\[ \widehat{\mathcal L}_D =\frac1b\sum_iD_\phi(g_i,t_i) -\frac1b\sum_iD_\phi(q_{1i},t_i) +\lambda\widehat H, \]
\[ \widehat{\mathcal L}_G =-\frac1b\sum_iD_\phi(G_\theta(z_i^{\alpha_i},t_i),t_i), \]
where \(g_i=G_\theta(z_i^{\alpha_i},t_i)\). The paper default is \(\lambda=100\) with one critic update per generator update. The full PTGAN benchmark uses the coherency penalty and no WGAN gradient penalty, so its effect is not confounded with GP. Consequently, this finite-network critic uses the Wasserstein sign convention but is not guaranteed to be 1-Lipschitz; its critic value should not be read as a certified numerical value of \(W_1\).
5.4 What the convergence argument does and does not say
The paper’s mechanism is a bias–variance trade-off. Interior temperatures do not have the same critic gradient expectation as the target slice, so \(r<1\) introduces bias for the immediate \(\alpha=1\) update. But smoother, better connected intermediate distributions can reduce the stochastic critic-gradient variance. Under the two-sided Polyak–Łojasiewicz conditions studied by Yang, Kiyavash, and He (2020), a class of nonconvex–nonconcave minimax algorithms has an optimization error of order
\[ O\!\left(\frac{\sigma_{\mathrm{grad}}^2}{T}\right), \]
so a smaller variance constant can mean faster local optimization. This is not a proof that arbitrary neural GAN training satisfies the PL assumptions. Nor does PTGAN improve the global statistical order: the paper retains an \(n^{-1/2}\)-type minimax estimation rate, with somewhat different constants and approximation terms. The proposed gain is primarily stabilization against bad local dynamics and mode collapse.
6 Numerical experiment: eight Gaussian modes
6.1 Data-generating process
The target follows the paper’s toy geometry:
\[ X\sim\frac18\sum_{k=0}^7 \mathcal N(\mu_k,0.01I_2), \qquad \mu_k=1.5 \begin{pmatrix} \cos(2\pi k/8)\\ \sin(2\pi k/8) \end{pmatrix}. \]
Each of 10 paired replications uses 4,000 training observations and a separate 4,000-observation evaluation sample. All methods receive the same target data, seed, batch size 100, 1,500 critic/generator updates, two hidden layers of 64 ReLU units, latent dimension 2, Adam learning rate \(10^{-4}\), and Adam coefficients \((\beta_1,\beta_2)=(0,0.9)\). There is one critic update per generator update and no dropout.
The compared methods are:
- WGAN-GP: the ordinary target only, with one-sided GP weight 10.
- PTGAN-GP: convex tempering with \(r=0.9\), but no coherency penalty; ordinary GP weight 10 is retained. This isolates whether interpolation alone suffices.
- PTGAN-CP: the full proposal with \(r=0.9\), coherency weight 100, and no ordinary GP.
6.2 Evaluation metrics
Every checkpoint evaluates the generated distribution at \(\alpha=1\).
For \(L=256\) random unit directions \(u_\ell\), sliced Wasserstein distance is
\[ \operatorname{SW}_{256}(P,Q) =\frac1{256}\sum_{\ell=1}^{256} W_1(u_\ell^\top X,u_\ell^\top Y). \]
With equal empirical sample sizes \(m=4{,}000\), each one-dimensional term is computed exactly from order statistics:
\[ \widehat W_1 =\frac1m\sum_{j=1}^m \left|x_{(j)}-y_{(j)}\right|. \]
A generated point is assigned to its nearest true center. A mode counts as captured if at least \(\max\{5,0.005m\}=20\) generated points lie within \(3\sigma=0.3\) of that center. If \(\widehat\pi_k\) is the nearest-center share, mode-mass total variation is
\[ \operatorname{TV}_{\mathrm{mode}} =\frac12\sum_{k=1}^8 \left|\widehat\pi_k-\frac18\right|. \]
Lower sliced Wasserstein and total variation are better; higher mode coverage is better.
| Sliced W1 | Modes / 8 | Mode TV | Full coverage | Train seconds | |
|---|---|---|---|---|---|
| Method | |||||
| PTGAN-CP | 0.219 (0.034) | 8.00 (0.00) | 0.068 (0.017) | 10/10 | 1.092 (0.004) |
| PTGAN-GP | 0.728 (0.244) | 2.40 (1.17) | 0.587 (0.194) | 0/10 | 1.318 (0.005) |
| WGAN-GP | 0.808 (0.315) | 2.50 (2.22) | 0.553 (0.223) | 1/10 | 1.260 (0.006) |
The divergence appears after roughly 400 updates. By step 1,200, full PTGAN has mean sliced Wasserstein 0.265 and covers 7.7 modes; WGAN-GP is at 0.739 and 2.4 modes. At step 1,500, full PTGAN recovers all eight modes in every seed. WGAN-GP does so in one seed, while tempered training with ordinary GP does so in none. Relative to WGAN-GP, the full method reduces final sliced Wasserstein by about 73% and mode-mass total variation by about 88%.
The paired PTGAN-minus-WGAN sliced-Wasserstein difference is -0.588, with a 95% t interval [-0.814, -0.362] and paired \(p=0.00023\). This is a small simulation comparison, not a general performance guarantee.
6.3 What a representative run looks like
The PTGAN-GP ablation is decisive for interpretation. It has the same convex temperature path and interpolated reference noise as full PTGAN but lacks the coherency penalty; it fails similarly to WGAN-GP. The result therefore does not support the claim that data interpolation by itself solves mode collapse. For this architecture and target, cross-temperature critic synchronization is the operative ingredient.
6.4 Wall-clock convergence
The methods receive equal numbers of critic and generator updates, but the regularizers have different costs. At 1,500 updates, mean training-only CPU times are 1.092 seconds for PTGAN-CP, 1.260 for WGAN-GP, and 1.318 for PTGAN-GP on this machine. Evaluations are excluded from those timings. The absolute values are not portable; the useful observation is that the directional coherency penalty is not more expensive than the gradient-norm penalty in this small network.
7 Edge case: one Gaussian
The paper warns that ordinary training may be more efficient when the target is already simple. We repeat the design for \(X\sim\mathcal N(0,0.01I_2)\), using 10 seeds and 1,000 updates.
| Sliced W1 | Nearest-center distance | Train seconds | |
|---|---|---|---|
| Method | |||
| PTGAN-CP | 0.0147 (0.0030) | 0.1244 (0.0070) | 0.736 (0.002) |
| WGAN-GP | 0.0194 (0.0059) | 0.1304 (0.0112) | 0.847 (0.005) |
Both methods converge. PTGAN’s final sliced distance is 0.0147 versus 0.0194 for WGAN-GP, an absolute difference of only 0.0047. Mode coverage is uninformative here because there is only one mode and both generators place enough mass near it even at initialization. This edge case does not reproduce a vanilla-WGAN speed advantage, but it does show that PTGAN’s dramatic multimodal benefit mostly disappears when there is no mode-collapse problem.
8 Implementation in Trex
The public estimator is trex.TabularPTGAN. Its paper-specific defaults are one critic step, Adam betas \((0,0.9)\), coherency weight 100, symmetric temperature input, and interpolated reference noise. The default temperature_ratio=0.5 follows the paper’s tabular experiments; the harder Gaussian-ring benchmark above uses \(r=0.9\).
from trex import TabularPTGAN
model = TabularPTGAN(
hidden_dims=(128, 128, 128),
max_steps=2_000,
temperature_ratio=0.5,
coherency_weight=100.0,
interpolate_noise=True,
seed=0,
)
model.fit(x_train)
x_synthetic = model.sample(5_000, alpha=1.0)The exact benchmark is reproducible with:
python benchmarks/ptgan_convergence.py \
--steps 1500 \
--checkpoints 0,50,100,200,400,800,1200,1500 \
--seeds 10 \
--n-train 4000 \
--n-eval 4000 \
--hidden-dims 64 64 \
--batch-size 100 \
--temperature-ratio 0.9 \
--coherency-weight 100 \
--include-ablation \
--output-dir tmp/ptgan-convergence-main9 Comparison and implications
The three methods change different mathematical objects:
- Traditional GAN: a real-versus-fake classification game; under an ideal discriminator, the original minimax value is \(2\operatorname{JS}-\log4\).
- WGAN: an integral probability metric over 1-Lipschitz critics, equal to Wasserstein-1 distance at the population optimum.
- PTGAN: a joint conditional GAN over a convex family of target laws, plus a directional coherency penalty that transfers progress across temperatures.
On the eight-mode toy problem, WGAN geometry alone is not enough for this small network and training budget. Full PTGAN reaches all modes in all ten runs; both the WGAN baseline and tempering-without-coherency ablation usually collapse. That is strong evidence for the local optimization mechanism in the setting it was designed for. It is not yet evidence that PTGAN dominates on mixed tabular data or on downstream econometric estimands. External validity should be assessed on paired tabular benchmarks, such as the Lalonde/CPS simulator, while tracking both distributional metrics and bias in target causal estimands.
10 References
Arjovsky, Martín, Soumith Chintala, and Léon Bottou. 2017. “Wasserstein Generative Adversarial Networks.” In International Conference on Machine Learning.
Goodfellow, Ian J., Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. 2014. “Generative Adversarial Nets.” In Advances in Neural Information Processing Systems.
Gulrajani, Ishaan, Faruk Ahmed, Martín Arjovsky, Vincent Dumoulin, and Aaron Courville. 2017. “Improved Training of Wasserstein GANs.” In Advances in Neural Information Processing Systems.
Sohn, Jinwon, and Qifan Song. 2025. “Parallelly Tempered Generative Adversarial Nets: Toward Stabilized Gradients.” arXiv Preprint arXiv:2411.11786. https://arxiv.org/abs/2411.11786.
Yang, Junchi, Negar Kiyavash, and Niao He. 2020. “Global Convergence and Variance Reduction for a Class of Nonconvex-Nonconcave Minimax Problems.” In Advances in Neural Information Processing Systems.