A Gradient-Level Diagnosis of Extreme Class Imbalance in Multiple Instance Learning via q-Calculus

Training under extreme class imbalance (>1:100) remains an open problem in weakly supervised learning. The standard remedy—loss-level reweighting (focal loss, asymmetric loss, class-balanced loss)—is widely adopted, yet its behavior at extreme ratios in Multiple Instance Learning (MIL) is poorly understood. On digital breast tomosynthesis (attention-based pooling over frozen EfficientNet-B3 features), we study the optimization bounds under extreme bag-level MIL imbalance (1:251), intervening at two levels: the loss surface (via reweighting) and the gradient dynamics (via a novel q-calculus gradient modification using the Jackson q-derivative). All three reweighting strategies degrade classification relative to unweighted binary cross-entropy (BCE), monotonically, eliminating the loss surface as the bottleneck. Extended evaluation (n=20 seeds) shows q-calculus gradient smoothing matches vanilla BCE (p=0.632, Cohen’s d=0.003) despite provably reducing gradient variance, establishing an empirical ceiling on the optimization-achievable area under the precision-recall curve (AUPRC) of 0.0912; loss reweighting defines the floor at 0.055. Focal loss is additionally catastrophically miscalibrated (ECE > 0.44 vs. 0.036 for vanilla BCE), a collapse that persists under adaptive binning. In this regime, exceeding the ceiling points to the data-representation level, not the optimizer. We further identify ratio-invariant safety—non-degradation at any imbalance ratio, satisfied by vanilla BCE and q-calculus but violated by all reweighting methods—and give recommendations spanning moderate to extreme imbalance.