Nonlinear Subjective Weighting Model

• Non-linear Subjective Weighting Model is a framework where weights are derived via non-linear functions applied to subjective inputs such as expert opinions, gradients, or rankings.
• It leverages mappings like Gompertz functions, softmax, and OWA operators to transform input signals into adaptive and normalized weights for robust aggregation in various applications.
• Empirical results show improvements in efficiency and accuracy, reducing communication rounds in federated learning and enhancing Pareto-optimal solutions in reinforcement learning and finance.

A non-linear subjective weighting model is a framework in which the weights assigned to criteria, contributions, or alternatives are determined by a non-linear transformation of subjective or system-derived factors, rather than by static linear assignment. In these models, the final (normalized or aggregated) weight for each element is a non-linear function of inputs such as rankings, gradients, expert opinions, or contextual factors. This non-linearity enables richer modeling expressivity, augmenting classical weighting schemes in federated learning, multi-criteria decision analysis, reinforcement learning, and market equilibrium. These models increase robustness to inconsistencies, enable dynamic adaptation, and can represent complex interactions or preferences beyond simple linear aggregation.

1. Key Principles and Definitions

Non-linear subjective weighting models are characterized by three core properties:

• Non-linearity: The weighting function—mapping features, gradients, or subjective judgments to weights—is non-linear, and thus sensitive to the pattern and interaction of its inputs.
• Subjectivity: Inputs to the weighting process incorporate subjective assessments, such as decision maker preferences, node-specific contribution signals, or investor beliefs.
• Normalization/Aggregation: Weights are typically controlled to sum to one or to operate within a simplex, ensuring comparability and interpretability in aggregation.

Central primitives include:

• A non-linear mapping φ (e.g., Gompertz, softmax of angles or ranks, hypervolume transforms).
• A context- or sample-dependent extraction of subjective signals (such as contribution alignment, pairwise comparison scores, or behavioral/market adjustments).

2. Non-Linear Subjective Weighting in Federated Learning

Federated Adaptive Weighting (FedAdp) exemplifies non-linear subjective weighting in distributed optimization (Wu et al., 2020). At each round, the server computes the alignment of each client's local gradient $g_i$ with the aggregated global gradient $g$, measured by the angle

$$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$

This metric quantifies the alignment (contribution) of individual client updates. The model then applies a diminishing Gompertz-like non-linear mapping:

$$\varphi(\theta) = \alpha \cdot (1 - \exp\{-\exp[-\alpha(\theta-1)]\})$$

with domain $\theta\in[0, \pi/2]$ and $\alpha>0$. The resulting scores are normalized using a (possibly data-size-weighted) softmax:

$$w_i = \frac{D_i \exp[\varphi(\theta_i)]}{\sum_{j\in S_t} D_j \exp[\varphi(\theta_j)]}.$$

This mechanism sharply up-weights clients whose updates are well aligned with global decrease direction and down-weights those that are not, dynamically adapting to non-IID data and provably tightening the per-round improvement bound compared to uniform averaging. Empirically, FedAdp reduces communication rounds by up to 54.1% (MNIST) and 45.4% (FashionMNIST) over FedAvg (Wu et al., 2020).

3. Non-Linear Subjective Weighting in Multi-Criteria Decision Analysis

In the field of multi-criteria decision making (MCDM), the non-linear Best-Worst Method (BWM) and the Ordered Weighted Average (OWA) operator are canonical non-linear subjective weighting models (Ratandhara et al., 8 Aug 2025, Renaud et al., 2024).

Nonlinear Best-Worst Method (BWM)

BWM starts with pairwise subjective preference scores: a best-to-others vector $A_b$ and an others-to-worst vector $A_w$. The problem is formulated as a constrained non-linear program seeking weights $\{w_i\}$ that minimize the maximum deviation from these preference ratios. The solution is a closed-form interval for each $g$0 determined by:

• Single- and pairwise deviation computations,
• Closed-form formulas for lower and upper bounds of optimal weights. The Consistency Index (CI) and Consistency Ratio (CR) further provide analytic indicators of the input’s internal coherence, characterizing the reliability of the elicited subjective preferences (Ratandhara et al., 8 Aug 2025).

OWA and Choquet Integral

The OWA operator allocates weights $g$1 not to criteria, but to their performance ranks:

$g$2

where $g$3 is the $g$4th largest utility value. This architecture is inherently non-linear due to the sorting step and thus encodes subjectivity at the rank rather than criterion level. Extensions include hybrid models (OWA + MAUT) and the Choquet integral, which further captures interactions between criteria through a capacity function $g$5 on all subsets (Renaud et al., 2024). D-optimal experimental design is recommended for parameter identification to minimize estimation error in these non-linear weight models.

4. Dynamic Non-Linear Weighting in Reinforcement Learning

In multi-objective reinforcement learning, dynamic non-linear subjective weighting circumvents the limitations of fixed linear scalarization, enabling the discovery of Pareto-optimal solutions even in non-convex reward landscapes (Lu et al., 14 Sep 2025). The general scalarized reward takes the form

$g$6

where $g$7 is the K-objective return vector. Two schemes are prominent:

• Hypervolume-guided weighting: A meta-weight is computed as a non-linear function of the gain in Pareto-front hypervolume,

$g$8

scaling the subjective weight vector $g$9 according to recent improvement.

• Gradient-based adaptive weighting: The weight vector $$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$0 is updated via a mirror-descent fixed-point with exponential non-linearity driven by per-objective policy-gradient influence signals,

$$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$1

Empirical evaluations confirm that these adaptive, non-linear models yield consistently Pareto-dominant solutions and accelerate front convergence relative to fixed-weight baselines (Lu et al., 14 Sep 2025).

5. Subjective Weighting in Asset Pricing Under Imperfect Information

Asset pricing under imperfect information and subjective views features a non-linear system for equilibrium weights, integrating market structure, shadow-costs (informational imperfections), and investors’ subjective views (Lalioui et al., 21 Jan 2025). The expected excess return vector is given by:

$$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$2

with $$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$3. Solving for the equilibrium market portfolio $$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$4 requires resolving the implicit non-linear dependence of $$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$5 on $$\theta_i(t) = \arccos \left( \frac{\langle g, g_i \rangle}{\|g\|\,\|g_i\|} \right).$$6. Posterior beliefs are integrated through Bayesian updating, resulting in a multivariate normal for the excess returns, weighted by the precision of subjective investor views and market information. Explicit sensitivity formulas detail how responses change with respect to inputs, and numerical examples illustrate substantial portfolio tilting under changed subjective views (Lalioui et al., 21 Jan 2025).

6. Practical Implications and Model Selection

Across domains, the advantages of non-linear subjective weighting models include:

• Robustness to input inconsistency (e.g., analytic CI/CR in BWM (Ratandhara et al., 8 Aug 2025))
• Finer control of allocation to contributors or criteria as a function of alignment or preference coherence (FedAdp (Wu et al., 2020), RL dynamic weighting (Lu et al., 14 Sep 2025))
• Capacity to express non-additive, interactive, or order-dependent aggregation (OWA, Choquet (Renaud et al., 2024))
• Ability to reconcile conflicting signals, integrate subjective and informational factors (market equilibrium (Lalioui et al., 21 Jan 2025))

Model selection should weigh expressivity against identifiability; practical recommendations include:

• D-optimal sample selection in estimation (Renaud et al., 2024)
• Ensuring regularization or constraints in parameter identification for high-parameter models (Choquet, market equilibrium).
• Empirical validation of predicted aggregate scores or allocations against observed subject-matter judgments, with careful quantification of error metrics and sensitivity.

7. Representative Examples and Empirical Results

Domain	Model/Operator	Key Empirical Finding
Federated Learning	FedAdp	30–54% reduction in comms rounds over FedAvg (Wu et al., 2020)
MCDM	OWA + D-optimality	2× reduction in RMSE vs. random sampling (Renaud et al., 2024)
Reinforcement Learning	Grad.-based weighting	Cuts convergence steps by ~10% vs. fixed-weight (Lu et al., 14 Sep 2025)
Finance	Market equilibrium	Portfolio tilting by integrating subjective views (Lalioui et al., 21 Jan 2025)

These models enable state-of-the-art performance in federated optimization, multi-objective learning, structured decision analysis, and asset allocation in settings where linear weights are insufficient to encode the inherent complexity, adaptivity, or perceived value of system components or alternatives.