Three-Class Fuzzy Multi-Loss Scalarizer

Integrate this at the loss-aggregation point in a multitask, constrained, or preference-trained neural model. Choose two to four component losses \(L_i(\theta)\), such as task cross-entropy, calibration error, latency proxy, fairness violation, or reconstruction error. Maintain detached exponential-moving estimates of \(\ell_i\) and \(u_i\), preferably approximating the 5th and 95th percentiles of recent minibatch losses; compute normalized \(c_i\) with stop-gradient through these bounds. Select \(\tau_i\) as the desired operating point, \(\delta_i<\tau_i<\upsilon_i\) as desirable and unacceptable thresholds, and use \(\epsilon=10^{-6}\), \(\beta\in[5,20]\). Each step performs: (1) compute all \(L_i\); (2) normalize to \(c_i\); (3) compute \(m_{d,i},m_{t,i},m_{u,i}\); (4) accumulate \(R_k=\sum_i m_{k,i}\); (5) calculate \(p_k\) and \(F\); and (6) backpropagate \(F\) with AdamW. Warm up for 1,000 steps using an ordinary weighted sum, then linearly increase the fuzzy-loss weight. The mathematical quantities are the triangular memberships, class accumulations, consequents, and ambiguity term; percentile bounds, thresholds, \(\lambda\), and \(\beta\) are empirical hyperparameters. Log class probabilities, gradient norms, and the fraction of batches with \(\max_k p_k<0.55\), which diagnoses plateau-like ambiguity. First test a small shared-bottom two-task MLP on a synthetic conflicting-regression dataset or CIFAR-10 with accuracy plus calibration loss. Compare against fixed weighted sums and uncertainty weighting at equal steps and FLOPs. Success is equal primary-task quality with lower secondary loss or constraint violation, reduced ambiguity frequency, and fewer gradient conflicts.