Polynomial-depth derivative regularization

(1) Integration point: add the displayed derivative loss to the training objective of a wide scalar-output tanh MLP, evaluated with respect to the input tensor x. Use Xavier Gaussian initialization for every affine layer. For vector outputs, either sum over output coordinates or apply the penalty to randomly selected coordinates. Keep the task loss unchanged and use the derivative penalty only during training.

(2) Pseudocode:

for x, y in minibatch:
    pred = f_theta(x)
    loss = task_loss(pred, y)
    g = grad_x(f_theta(x), x)                 # first order
    reg = mean(norm(g, dim=-1)**2 / tau1**2)
    for k in K:                               # e.g. K={2}
        for u in random_coordinate_subsets(k):
            d_u = mixed_derivative(f_theta(x), x, u)
            target = tau_k * factorial(k) * (L + eps)**(k-1)
            reg += mean(d_u**2 / target**2) / number_of_subsets
    (loss + lambda * reg).backward()
    optimizer.step()

Use forward-mode JVPs for sampled mixed derivatives when input dimension is moderate; use nested reverse-mode autodiff otherwise. The formula's factor k!(L+eps)^(k-1) is the adapted version of the theorem's C_0|u|!(C_1L)^(|u|-1) bound.

(3) Computed from the mathematics: the width scaling L^3 n_0^2 and the factorial/polynomial depth dependence. Estimated empirically: tau_1 and tau_k, because the extracted theorem does not provide numerical C_0, C_1, or beta_j. Initialize tau values from the 90th percentile of derivative magnitudes on a few untrained minibatches, then optionally update them with an exponential moving average. Estimate the Lipschitz proxy by the maximum observed ||grad_x f|| on validation samples.

(4) First experiment: use a depth-8, width-512 tanh MLP on a 20-dimensional smooth synthetic regression problem with a known target. Compare Xavier initialization with no penalty against the same network with k=1 and k=2 penalties. Repeat for depths 4, 8, and 16. Measure validation MSE, maximum sampled Jacobian norm, sampled mixed-derivative magnitudes, robustness to projected-gradient input perturbations, and wall-clock training cost. Success is derivative growth that remains approximately depth-independent for k=1 and polynomial rather than exponential for k=2, with improved perturbation robustness and less than 20% training overhead.