Constant-sum ordinal preference loss

Integrate this method at the output loss for a model that predicts a scalar utility, such as a text preference model, recommender embedding model, graph-node scorer, or learned ranking function. Given two examples x_i and x_j, compute z_i=f_theta(x_i) and z_j=f_theta(x_j), then form d=z_i-z_j+b_ij. Choose L ordinal outcome levels and initialize a point vector p satisfying p_y+p_{L-1-y}=C. Enforce the constraint by parameterization: for each complementary pair, set p_y=C/2+v_y and p_{L-1-y}=C/2-v_y; for the middle category when L is odd, set p_{(L-1)/2}=C/2. For each adjacent pair, compute inc[y]=a[y]+p[y]*d. Set log_r[0]=0 and recursively compute log_r[y]=log_r[y-1]+inc[y]. Apply log_softmax to log_r and minimize the negative log probability of the observed ordinal label. Pseudocode: z_i=f(x_i); z_j=f(x_j); d=z_i-z_j+b; log_r=[0]; for y in 1..L-1: log_r.append(log_r[-1]+a[y]+p[y]*d); loss=-log_softmax(log_r)[label]. Learn theta, a, v, and context parameters jointly. The mathematical quantities are the constant-sum constraint and adjacent log-odds recursion; estimate all neural scores and intercepts empirically by gradient descent. For a first cheap experiment, train a small MLP on MovieLens pairwise preferences converted into three ordered labels, comparing this loss against one-hot cross-entropy, standard cumulative-link ordinal regression, and Bradley-Terry loss. Also create repeated comparisons with balanced and skewed pair frequencies. Aggregate each item's observations with S_i=sum_m p_{y_m}, then measure held-out ordinal log-likelihood, calibration, Kendall tau of rankings, and performance versus the number of labels per pair. Success means improved ranking and calibration at equal training examples, especially for repeated or imbalanced comparisons, while balanced schedules should reduce the gap to ordinary baselines.