Farthest-cell balance regularizer

Integrate this at the embedding output of a classifier, metric-learning encoder, graph neural network, or retrieval tower, before the task head. In every training step, sample T triples of prototype vectors from the current minibatch or from a maintained queue. For each triple and each embedding z_b, compute the three distances and the soft-farthest weights w_bj above. Average weights over the query minibatch to obtain q_hat_j, compute entropy, and subtract alpha H_FC from the ordinary loss. Use Euclidean distance initially; for normalized embeddings, squared Euclidean distance or cosine-derived distance is sufficient because the hard statistic depends only on ordering. Start with beta equal to 5 divided by the running median pairwise distance, ramp alpha from zero over the first 10% of training, and clip the regularizer so it cannot exceed 10% of the task loss. Keep a detached hard-label metric using j=argmax_j d(z_b,p_j) to verify that the soft surrogate reflects actual farthest-cell balance. The paper supplies the entropy construction and boundedness; minibatch sampling, softening, and weighting are engineering adaptations. First test on CIFAR-10 with a small ResNet-18 and supervised contrastive loss, comparing no regularizer, ordinary nearest-cell entropy, and this farthest-cell entropy at equal compute. Track hard H_FC/log 3, class accuracy, embedding collapse indicators, and robustness under monotone distance rescaling. Success means higher hard farthest entropy without accuracy loss, fewer dominant prototypes, and improved retrieval or corruption performance.