PEP Lyapunov Stability Controller

(1) Integration point: add the monitor immediately after each generator and discriminator optimizer update. Keep the last N+1 projected parameter vectors and gradients. To avoid storing full-dimensional matrices, draw one fixed Gaussian sketch R in R^{m by d}, with m=128 or 256, and use projected quantities R theta and R gradient. (2) Pseudocode: solve the PEP offline on a quadratic saddle family and save P_x, P_y, and rho; at training step k, append projected parameters and gradients to the history; construct z_x,k=[x_k,...,x_{k-N},g_k^x,...,g_{k-N}^x] and z_y,k analogously; compute V_k=z_x,k^T P_x z_x,k+z_y,k^T P_y z_y,k; after the update compute V_{k+1}, r_k=(V_{k+1}-rho^2 V_k)/(V_k+epsilon), and eta <- eta exp(-gamma max(0,r_k)); clip eta to a preset minimum and maximum. If r_k is positive for q consecutive steps, clear momentum and history. (3) The PEP supplies P_x, P_y, and rho. The runtime computes only matrix-vector products and a scalar ratio. Cross-gradient vectors h can be approximated with one Jacobian-vector product probe per player, or omitted by synthesizing a reduced certificate using only iterates and gradients. Calibrate the sketch scale on a held-out batch. (4) First experiment: train an MLP GAN on eight-Gaussians and a small DCGAN on MNIST. Compare fixed Adam, OGDA, and extragradient with the Lyapunov controller at equal gradient budgets. Track positive-decrease violations, gradient spikes, mode coverage, FID, and collapse frequency. Success is fewer unstable runs and better final distribution quality without materially increasing wall-clock cost; falsification is frequent unnecessary learning-rate suppression or no reduction in oscillation.