Logistic-Gated Operators Enable Auditable Unit-Aware Thresholds in Symbolic Regression

TL;DR

The paper tackles the gap between symbolic regression and the need for auditable, unit-aware decision boundaries in health contexts. It proposes logistic-gated operators (LGO) as differentiable gates with learnable location $b$ and steepness $a$, embedded as typed primitives and inverted back to physical units for audit. Across ICU and NHANES data, hard gates yield clinically plausible, sparse thresholds (e.g., lactate, MAP, SBP, HDL, waist) while maintaining competitive predictive accuracy, and gates are pruned on predominantly smooth tasks, enabling parsimonious, governance-ready models. Thresholds are explicitly mapped to natural units, enabling direct audit against guidelines and facilitating deployment as standalone or safety-overlay components for clinical decision support and beyond. The approach closes the interpretability gap by turning thresholds into first-class, auditable model levers, providing a practical calculus for regime switching, auditability, and governance in real-world health AI systems, with extensions to other scientific domains. $LGO_{soft}(x;a,b)=x\sigma(a(x-b))$ and $LGO_{hard}(x;a,b)=\sigma(a(x-b))$ illustrate the gate forms, while $\hat{b}_{raw}=\mu_x+\sigma_x\hat{b}$ enables unit-aware threshold interpretation.$

Abstract

Symbolic regression promises readable equations but struggles to encode unit-aware thresholds and conditional logic. We propose logistic-gated operators (LGO) -- differentiable gates with learnable location and steepness -- embedded as typed primitives and mapped back to physical units for audit. Across two primary health datasets (ICU, NHANES), the hard-gate variant recovers clinically plausible cut-points: 71% (5/7) of assessed thresholds fall within 10% of guideline anchors and 100% within 20%, while using far fewer gates than the soft variant (ICU median 4.0 vs 10.0; NHANES 5.0 vs 12.5), and remaining within the competitive accuracy envelope of strong SR baselines. On predominantly smooth tasks, gates are pruned, preserving parsimony. The result is compact symbolic equations with explicit, unit-aware thresholds that can be audited against clinical anchors -- turning interpretability from a post-hoc explanation into a modeling constraint and equipping symbolic regression with a practical calculus for regime switching and governance-ready deployment.

Figures (4)

• Figure 1: Performance comparison across methods and datasets. Violin plots show the distribution over up to ten seeds per method (Hydraulic excludes 2 anomalous seeds; see SI Table S10). The black dots are individual seeds; violin width reflects probability density. The baseline metric is R$^2$ for regression datasets and AUROC for CTG; the mean and standard deviation for each violin are printed underneath (mean on the top line, $\pm$ std on the bottom line). The figure provides a high-level accuracy overview before parsimony and interpretability analyses.
• Figure 2: LGO threshold alignment with clinical guidelines (ICU and NHANES). Panels A and C: agreement heatmaps (green $\leq 10\%$, yellow $\leq 20\%$, red $>20\%$). Only anchored features with valid median thresholds are shown, so grey (N/A) cells are suppressed. Each cell reports the median threshold (natural units) and the relative deviation $\Delta$. Panels B and D: distribution of the same thresholds -- horizontal bars show the IQR, black dots mark medians, and red vertical lines mark anchors; the $x$-axis is in natural units. The figure conveys agreement (A,C) and stability (B,D) without duplicating the numerical details tabulated in Table \ref{['tab:thres-detail']}.
• Figure 3: Gating mechanism usage across datasets. Top panels (one subplot per dataset): for LGOsoft and LGOhard, the blue bars show gate usage % (fraction of the top-$100$ models that include at least one LGO gate), and the orange bars show the median number of gates per model (zeros included). Bottom panel: gate usage % aggregated side-by-side across datasets, contrasting LGOsoft (light) vs. LGOhard (dark). Hard gates keep usage high where thresholding is needed (ICU, NHANES, CTG) while pruning superfluous gates (e.g., on Cleveland), and they realize fewer gates per model than soft gates, supporting parsimony and auditability.
• Figure S1: Extended Pareto fronts (CV loss vs. symbolic complexity). One panel per dataset. Points are non‑dominated candidates aggregated across seeds. ICU / NHANES: LGOhard yields competitive low‑loss candidates at moderate complexities; LGOsoft is broader and typically less favorable. CTG: near‑separable, fronts are saturated. Cleveland: LGOhard reaches the lowest loss at higher complexity, Operon trades slightly higher loss for lower complexity. Hydraulic: smooth relations favor RILS-ROLS/Operon. Complexity is computed with a unified node count; CV‑loss uses the same proxy across methods (details in Table \ref{['tab:hyp']}).