Bidirectional Conditional Cycle Loss

Integrate the method at the loss level in a model exposing both conditional directions, such as a discrete latent-variable model, a bidirectional imputer, or two transformer heads sharing an encoder. For each minibatch, compute logits and log-softmax outputs for q_theta(x|y) and r_phi(y|x). Sample two distinct X values and two distinct Y values per training example, either from observed values, the model's marginal samples, or uniformly from the categorical support. Gather the eight log-probability entries and compute Delta exactly from the displayed equation. Add lambda times the mean squared or Huber Delta to the ordinary task loss. Pseudocode: log_q = model_q(y); log_r = model_r(x); sample x1,x2,y1,y2; Delta = log_q[x1,y1] + log_r[y1,x2] + log_q[x2,y2] + log_r[y2,x1] - log_r[y1,x1] - log_q[x1,y2] - log_r[y2,x2] - log_q[x2,y1]; loss = task_loss + lambda * huber(Delta); loss.backward(). Use log-softmax directly, and restrict the first experiment to full-support categorical outputs. The network estimates probabilities; no mathematical quantities need to be estimated externally. Test first on synthetic data sampled from a known joint over X and Y, comparing independently trained conditionals against the same architecture with the cycle loss at equal parameters, steps, and FLOPs. Then test discrete latent codes on MNIST. Pre-register that the held-out 95th-percentile absolute cycle residual must decrease by at least 10x, while conditional calibration or recovered-joint error improves at fixed task loss. Measure residuals on 10,000 unseen quadruples. Ablate only lambda by setting lambda to zero, plus a shuffled-quadruple control. The transfer is falsified if training residuals decrease but held-out residuals do not, or if compatibility improves without any improvement in joint recovery, calibration, or sampling consistency.