Service-Cost-Independent Admission

Attach this regularizer to the admission probability or router probability, not to the final classifier loss. For each request i, record its type features Y_i, predicted or measured service requirement S_i, current workload W_i, and gate probability p_i in [0,1]. Partition the minibatch into K workload bins B_b using fixed quantiles or fixed ranges; within each bin compute the centered covariance estimate Cov_hat_b(p,S) shown above. Add lambda_ind times the sum of squared covariances to the normal loss. If service cost is unknown before execution, use a lightweight cost predictor S_hat_i trained from actual latency or FLOP measurements, and stop its gradient when computing the independence penalty. A stronger version can use random permutations of S within each workload bin: compare the observed gate-cost product mean to the permuted product mean and penalize their difference. Do not apply the penalty globally without workload conditioning, because a sensible gate should legitimately admit more work when the queue is empty. Begin with a 4-expert Transformer or dynamic batching simulator, where request types are sequence lengths and S_i is measured expert compute time; compare standard load-balancing loss, this covariance penalty, and both together. Evaluate mean and p99 latency, overflow or dropped-token rate, expert utilization variance, validation perplexity or accuracy, and the empirical inequality E[W_t^2] versus E[W_hat_t^2]. The expected signal is lower tail congestion and fewer delayed expensive requests at matched average admission rate, with no material task-loss increase.