Discounted Saddle-Gap Controller

(1) Exact integration point: apply this to a GAN, adversarial representation learner, or policy-versus-discriminator game. Let x denote generator or policy parameters and y denote discriminator or adversary parameters. Every M ordinary training minibatches, freeze the current network parameters and run K cheap best-response probe steps to estimate both extrema in the saddle gap. For bounded toy parameters use projection onto the feasible box; for ordinary neural parameters use gradient clipping or a trust-region norm cap. Do not backpropagate through the probe steps.

(2) Pseudocode:

initialize x, y, R = 0, previous_R = None
for block t = 1, 2, ...:
    for m = 1, ..., M:
        x <- x - eta_x * optimizer_x(grad_x f_t(x,y))
        y <- y + eta_y * optimizer_y(grad_y f_t(x,y))
    x_probe <- stop_gradient(x)
    y_probe <- stop_gradient(y)
    repeat K times:
        y_probe <- project_Y(y_probe + eta_probe * grad_y f_t(x, y_probe))
    repeat K times:
        x_probe <- project_X(x_probe - eta_probe * grad_x f_t(x_probe, y))
    gap <- f_t(x, y_probe) - f_t(x_probe, y)
    R <- rho * R + (1-rho) * gap
    if previous_R is not None and R > previous_R * (1+tau):
        eta_x <- beta_down * eta_x
        eta_y <- beta_down * eta_y
        clear optimizer momentum/state for x and y
    else if R < previous_R * (1-tau) for P consecutive blocks:
        eta_x <- min(eta_x_max, beta_up * eta_x)
        eta_y <- min(eta_y_max, beta_up * eta_y)
    previous_R <- R

(3) Computed directly from the paper's mathematics: the saddle-gap definition, the discounted recursion, and the horizon 1/(1-rho). Estimated empirically: the max and min using K probe steps, payoff values on minibatches, and the trend thresholds. Normalize the gap by abs(f_t(x,y)) plus epsilon or by a running payoff scale. Use hysteresis tau so minibatch noise does not cause rapid control switching. Sweep rho in {0.9, 0.99, 0.999}, K in {1, 3, 5}, beta_down in {0.5, 0.8}, and beta_up in {1.05, 1.2}. Compare fixed-step Adam, ordinary loss-EMA control, and extragradient with and without the controller.

(4) First cheap experiment: train a DCGAN or small MLP GAN on MNIST with a deliberately nonstationary stream, such as alternating two-class subsets every 500 minibatches or slowly rotating the augmentation distribution. Measure held-out FID or classifier accuracy, generator diversity, collapse frequency, loss oscillation, and discounted saddle gap at equal FLOPs. The expected signal is a lower recent saddle gap, faster recovery after distribution changes, and fewer collapse episodes. On stationary MNIST, verify that the controller does not substantially reduce final quality; increasing rho should recover behavior closer to a long-memory metric.