E-optimal-ranking (EOR): Design & Fairness

• E-optimal-ranking (EOR) is defined in two contexts: robust simulation-based sampling for nonlinear ODE models and a fairness criterion for group-wise equitable ranking.
• In experiment design, it uses Monte Carlo sampling, sensitivity propagation, and SDP-based selection to improve parameter estimation with measurable error reduction.
• For fair ranking, it employs a group-wise merge algorithm to balance relevance mass across groups, achieving computational efficiency with theoretical fairness guarantees.

E-optimal-ranking (EOR) refers to two distinct formal methodologies in the academic literature: one in simulation-based optimal experiment design for dynamical systems (Ha et al., 10 Nov 2025), and one as a fairness criterion for group-wise equitable ranking under relevance uncertainty (Rastogi et al., 2023). Both approaches, despite sharing a naming convention, address fundamentally different problems—sample point selection for optimal parameter estimation versus unfair burden mitigation in ranked selection processes. Each leverages a ranking or optimization mechanism to realize its underlying criterion, and both introduce practical algorithms with quantified theoretical guarantees and empirical validation.

1. E-optimal-ranking in Simulation-based Optimal Sampling Design

The E-optimal-ranking (EOR) method in systems biology targets robust optimal sampling design for parameter estimation in nonlinear ODE models. Given a dynamical system

$\dot X(t) = f(X(t), \theta), \quad X(t_0) = X_0,$

with observations

$$y(t_k) = g(X(t_k)) + w_k, \quad w_k \sim \mathcal{N}(0, \Sigma),$$

the traditional design objective is to maximize the smallest eigenvalue $\lambda_{\min}$ of the Fisher information matrix (FIM)

$I(\theta) = \sum_{k=1}^N S_k^\top S_k,$

where $$S_k = \frac{\partial X(t_k; \theta)}{\partial \theta}$$ denotes the sensitivity at time $t_k$.

Classical E-optimal design requires a plug-in parameter $\theta$, rendering it sensitive to prior misspecification. The EOR approach circumvents this by integrating over a parameter prior, yielding a ranking-based consensus robust to uncertainty. It proceeds as follows:

• Monte Carlo Sampling: For $j = 1, \ldots, K$, sample $\theta^{(j)}$ uniformly from a parameter box $\Theta$.
• Sensitivity Propagation: For each draw, solve $X$ and sensitivity ODEs to obtain $S_k^{(j)}$ for all candidate times $t_k$.
• SDP-based Selection: For each $\theta^{(j)}$, solve the convex semi-definite program

$$\min_{\lambda, t} -t \quad \textrm{s.t.} \quad \sum_{k=1}^N \lambda_k S_k^{(j)\top} S_k^{(j)} \succeq t I_m, \quad \lambda_k \ge 0, \quad \sum_k \lambda_k = 1,$$

ranking times in descending order by $\lambda_k^{(j)}$.

• Consensus Aggregation: Compute $\bar r(k)$, the average rank of each time $k$ across all draws.
• Design Extraction: Select the $n$ lowest average rank times as the final sample schedule.

A typical implementation uses $K \approx 1\,000$ Monte Carlo draws and candidate grids $N \approx 50\text{–}200$. SDP solvers like CVXPY+MOSEK or Gurobi, warm-starting, and high-order ODE integrators are standard.

2. Statistical Properties and Numerical Performance in Systems Biology

EOR's statistical robustness emerges from its use of the empirical prior over $\Theta$, converting parameter uncertainty into sampling design consensus and obviating the need for post-selection bootstrapping or plug-in estimates. The only approximation comes from finite Monte Carlo sampling; stabilization occurs for $K \gtrsim 500$ in observed practice.

Empirical studies using Lotka-Volterra and three-compartment pharmacokinetic models, with $n=5$ out of $N=101$ times selected, show that EOR achieves a mean squared parameter error reduction of approximately 30% compared to both random and plug-in E-optimal selection in the LV model, and matches the best classical E-optimal in the PK model for $1\,000$ simulated datasets. Tukey’s HSD tests at FWER=0.05 confirm statistically significant improvements on LV and parity with the classical method on PK.

Method	LV model mean (std)	3-comp. mean (std)
Random	1.63 (0.61)	1.08 (0.61)
E-optimal	1.76 (0.61)	0.55 (0.27)
EOR	1.22 (0.44)	0.55 (0.26)
At-LSTM	1.27 (0.39)	0.77 (0.50)

Performance gains indicate that EOR can yield a single design robust to parameter realization and at least as efficient as classical approaches.

3. Complexity, Implementation Guidelines, and Limitations

The main computational cost of EOR arises from $K$ folds of sensitivity ODE solving and solution of $K$ SDPs. Each SDP, with naive routines, scales as $O(N^3)$ per solve, which can become significant if $N$ or parameter dimension $m$ is large—though sparsity-aware solvers offer practical mitigations.

For grid size $N\approx50$–$200$ and $K \approx 1\,000$, standard computational resources are sufficient. Best practices include:

• Monitoring average rank convergence as a stopping criterion,
• Warm-starting SDPs,
• Using adaptive ODE integrators,
• Selecting sample size $n$ based on experimental constraints.

A plausible implication is that EOR is tractable and effective for moderate-scale experimental regimes but may be computationally demanding for very large grids or high-dimensional parameter spaces.

4. E-optimal-ranking as a Fairness Criterion in Group-wise Ranking under Uncertainty

In ranking under uncertainty, particularly with relevance-score disparity between groups, EOR is defined as a fairness criterion ensuring that each group’s relevant mass appears at similar rates throughout all ranking prefixes. Let $n$ candidates split into protected groups $A,B$, each with model-based expected relevance $p_i = P(r_i=1 | D)$.

The EOR criterion seeks a deterministic ranking $\sigma$ such that, for all $k$,

$$\delta(\sigma_k) = \left| \frac{nRel(A|\sigma_k)}{nRel(A)} - \frac{nRel(B|\sigma_k)}{nRel(B)} \right| \le \delta,$$

where $nRel(g|\sigma_k)$ is the cumulative relevance mass from group $g$ in the first $k$ slots, and $\delta=0$ achieves perfect group fairness matching a fair lottery.

The corresponding integer program minimizes overall missed-relevance cost while enforcing the EOR fairness constraint for any prefix size $k$.

5. Algorithmic Realization and Approximation Guarantees in Ranking

Efficient computation of EOR rankings is enabled via a group-wise merge algorithm:

• Independently sort each group’s candidates by $p_i$ (local PRP).
• Greedily select the group whose next candidate yields the smallest $\delta$ increment when appended to the mixed ranking.
• Continue until $k$ slots are filled, repairing as needed if a group is exhausted.

This yields $O(n\log n)$ runtime for two groups, and $O(n\log n + Gn)$ for $G$ groups. The method admits an explicit additive approximation bound: for every prefix $k$, principal cost is within $\phi \cdot \Delta_k$ of the ILP optimum, with $\Delta_k$ the maximal achieved imbalance and $\phi$ a function of last-selected $p_i$'s and groupwise normalized scores.

For $G>2$ groups, the merge generalizes by always picking the group whose addition minimizes the maximal-minimal groupwise coverage gap.

6. Comparative Evaluation in Ranking and Empirical Outcomes

EOR’s fairness-by-mass property stands in contrast to other ranking fairness approaches:

• Probability Ranking Principle (PRP): maximizes expected relevance but ignores groupwise equity, potentially yielding high burden on minority groups.
• Demographic Parity (DP): enforces group count parity in top-$k$ but not coverage of relevant mass, often failing with disparate uncertainty.
• Proportional Rooney Rule (FA*IR): prioritizes headcount constraints for a pre-designated group, without balancing relevance mass.
• Exposure-based Fairness: averages representation across entire ranking, which may be insufficient for finite prefix or practical review scenarios.

Empirical studies on synthetic data, US Census predictions (e.g., Black/White in Alabama, multi-racial in NY), and Amazon product search logs demonstrate that EOR achieves near-zero maximal groupwise burden difference ($\Delta(\sigma_k) \approx 0$) and equalizes group outranking costs. Principal performance metrics (recall@k, nDCG) remain competitive with PRP. EOR also proves valuable in audit settings, highlighting disparities in existing deployed rankings where access to ground-truth or calibrated models is available.

7. Synthesis: Distinct Contexts for E-optimal-ranking

The E-optimal-ranking (EOR) nomenclature encapsulates two mathematically rigorous approaches advancing the state-of-the-art in their respective domains:

• In systems biology, EOR transforms plug-in FIM-based optimal experiment design into a robust, simulation-based consensus ranking for sampling, eliminating critical dependence on prior parameter estimates and demonstrating superior empirical performance with quantifiable efficiency/cost trade-offs.
• In machine learning fairness, EOR provides a principled ranking mechanism that equalizes fairness costs across protected groups under disparate uncertainty, is computationally efficient, and offers theoretical guarantees bounding the additional total cost relative to classical, group-agnostic ranking.

The shared foundation is the conversion of an optimality or fairness criterion into a practical, tractable ranking algorithm—either over time points for ODE sampling or candidate orderings for sensitive human/machine decision tasks. The terminology EOR thus serves both as a precise descriptor and a unifying concept for robust and equitable selection under uncertainty.