Midpoint Ergodic Readout

1. Integration point: maintain a shadow copy of adversarial parameters during training and use the averaged copy for validation, checkpointing, and final inference. Apply this to GAN generators, discriminators, actor policies, critics, or any two-player module with visible cycling. Keep the optimizer unchanged initially so the experiment isolates the readout effect. 2. Pseudocode: initialize s = 0 and n = 0. After each synchronized update, increment n and set s = (1 - 1/n) s + x_n / n. For midpoint averaging, after obtaining x_t and x_{t+1}, form m_t = (x_t + x_{t+1}) / 2 and update the shadow model with the running mean of m_t. At evaluation, copy the shadow parameters into the model, compute metrics, and restore the live parameters. 3. The paper gives the exact modal attenuation 1 / sqrt(1 + eta^2 omega^2 / 4). Estimate dominant omega by power iteration on an empirical skew interaction matrix or by fitting the angle between successive parameter differences. Treat the bound as a local diagnostic for nonlinear networks, not as a global theorem. 4. First experiment: train a small CIFAR-10 GAN and compare final-iterate, exponential moving average, uniform average, and midpoint average with identical update sequences. Measure FID, Inception score, discriminator-loss variance, gradient cosine oscillation, and distance to an empirical stationary point. Success means lower evaluation variance and improved FID at equal training steps, with the largest gain when the estimated eta times omega is large.