Directional Almost-Dominance OT Loss

(1) Integration point: use this as an auxiliary loss on a minibatch of model outputs whose distribution should not exceed a reference distribution. Examples are predicted risk distributions versus a safety target, distributional-RL returns versus a baseline, or generated samples that should be no worse than another generator. Let x_1,...,x_n be model samples and y_1,...,y_m reference samples. For multidimensional outputs choose w_j from domain importance or set w_j=1. (2) Pseudocode: compute pairwise costs C_xy[i,k]=Σ_j w_j·softplus((x_i[j]-y_k[j])/τ)·τ and C_yx[k,i]=Σ_j w_j·softplus((y_k[j]-x_i[j])/τ)·τ; run log-domain Sinkhorn with entropic coefficient η to obtain plans Π_xy and Π_yx; set ot_xy=Σ_i,k Π_xy[i,k]C_xy[i,k], ot_yx=Σ_k,i Π_yx[k,i]C_yx[k,i]; set gamma_hat=ot_xy/(ot_xy+ot_yx+ε); set loss=relu(gamma_hat-γ0)^2; add λ·loss to the task objective and backpropagate through the Sinkhorn iterations. (3) The mathematical quantities computed directly are the directional cost, coupling optimization, and γ ratio; τ, η, ε, λ, and γ0 are engineering hyperparameters. Estimate w_j from validation sensitivity if desired, but do not adapt them from the same minibatch. For one-dimensional outputs, sort samples and replace Sinkhorn by the exact empirical quantile formula, which is O(n log n). (4) First experiment: train a small heteroscedastic MLP on synthetic regression where the target conditional distribution is deliberately shifted upward in 20% of regions. Compare ordinary NLL, symmetric Wasserstein-1 regularization, and NLL plus this loss at equal batch size and wall-clock budget. Measure violation probability P[X>Y], estimated gamma_hat, calibration error, and test NLL. A useful success signal is substantially lower directional violation and calibration error at matched NLL, without destabilizing training; in distributional RL, additionally test whether return distributions improve under stochastic dominance rather than merely having lower mean.