Dissipative drift parameterization

(1) Integration point: modify only the drift prediction in a conditional diffusion policy. Let the current diffusion action state be A_t. Add the exact analytic term -mu*A_t to the network output and let an MLP f_theta predict the residual. Apply spectral normalization to layers influencing the action input, or use a separate residual branch with target action-Lipschitz budget L_f<mu. Observation x and time t remain inputs to the residual network.

(2) Pseudocode:

choose mu > 0 and residual budget Lf < mu
spectrally normalize f_theta so its action-Lipschitz upper bound <= Lf
for each training batch:
    sample t and construct noised action state A_t
    drift = -mu*A_t + f_theta(t, x, A_t)
    loss = diffusion_training_loss(drift, target_drift_or_noise)
    update theta using the loss
for inference step:
    drift = -mu*A + f_theta(t, x, A)
    A = A + drift*dt + sigma*sqrt(dt)*randn()

The paper's mathematical ingredient is the one-sided dissipativity inequality and its sharper sample-size regime; the contraction margin is mu-L_f. Estimate the residual action-Lipschitz bound from layer spectral norms and activation constants, then verify it empirically with finite differences on normalized random inputs. Log ||A_t||, drift norms, pairwise contraction ratios, and sensitivity to the Euler step size. First experiment: compare a standard diffusion policy, a globally spectrally normalized policy, and this dissipative parameterization on a two-dimensional or six-dimensional multimodal behavior-cloning benchmark using 1k, 10k, and 100k demonstrations. Evaluate held-out negative log-likelihood, action-mode coverage, rollout return, Euler-step sensitivity, and performance versus diffusion-step count. Success is lower variance across seeds and stable sampling with fewer Euler steps while matching or exceeding multimodal validation likelihood; the predicted statistical advantage should become clearer as dataset size increases.