Factor Attribution Paradigm: A Structured Overview

• Factor Attribution Paradigm is a framework that decomposes outputs into interpretable, granular factors by assigning quantitative contribution scores to different system components.
• It employs formal axiomatic methods and diverse families of techniques—such as gradient-based, perturbation, and surrogate methods—to ensure robust and faithful attribution.
• The paradigm underpins applications in machine learning, finance, and cognitive science, enhancing model explainability and guiding credit assignment in complex systems.

The Factor Attribution Paradigm formalizes the problem of assigning responsibility, credit, or quantitative contribution scores to components—whether features in machine learning, actions in agents, demonstrations in in-context learning, creative blocks in generative systems, or causal factors in explanation tasks. It reframes attribution as a structured decomposition of system output, performance, or explanation into interpretable, granular “factors,” and provides methodologies for quantifying, validating, and leveraging attribution scores. The paradigm pervades modern explainable AI, finance, agent design, generative media, and cognitive science, uniting them under rigorous theoretical and algorithmic frameworks.

1. Formal Foundations and Motivating Contexts

The Factor Attribution Paradigm originates from diverse subfields that share an underlying requirement: determining the extent to which specific factors (features, actions, blocks, beliefs) drive or explain observed behavior or outcomes. In machine learning, factors are typically input features; in systematic investing, they are risk premia or trading actions; in generative AI, reusable content blocks; in causal inference and agent cognition, interventions or beliefs.

Canonical definitions rely on one of several formal devices:

• For ML, an attribution method is a mapping assigning to each input factor a real-valued score reflecting influence on a model’s prediction (Deng et al., 11 Aug 2025).
• In financial attribution, relative returns are decomposed into passive factor bets, rebalancing/trading profits, and leakage/reconstitution drag, each operationalizing distinct economic "factors" (Papathanakos, 2016).
• In XAI and formal verification, attributions aggregate over all minimal sufficient explanations (AXp’s), ensuring logical rigor and completeness (Yu et al., 2023).
• In cognitive science, explanations are scored by precision, informativeness, and causal relevance (2505.19376).
• In RL agents and LLM-driven systems, attribution layers structure agent reasoning over multi-dimensional factor taxonomies (Yu et al., 8 Jan 2026).

These formalizations provide a mathematically-grounded basis for all subsequent attributions, enabling both algorithmic computation and theoretical analysis.

2. Families of Attribution Methods and Decomposition Principles

Attribution methods can be unified under formulation-driven and reformulation-based frameworks:

• Formulation-driven families:
  • Gradient-based: Attributions via (possibly modified) backpropagation; includes gradient saliency, Guided Backprop, SmoothGrad (Deng et al., 11 Aug 2025).
  • Gradient × Input / Path methods: Integrated Gradients (IG), Expected Gradients (EG), Grad×Input, DeepLIFT, which compute scores along paths from a baseline (Deng et al., 11 Aug 2025, Erion et al., 2019).
  • Perturbation/feature-removal: Quantify the effect of masking or permuting features (e.g., Shapley, Banzhaf, occlusion) (Deng et al., 11 Aug 2025).
  • Local surrogate: LIME and variants fit an interpretable model in a local neighborhood (Deng et al., 11 Aug 2025).
  • Subsets and interaction models: Taylor-expansion and meaningful/influential-perturbation frameworks attempt to recover minimal explanatory subsets (Deng et al., 11 Aug 2025, Afchar et al., 2021, Yu et al., 2023).
  • Attention-based: Uses self-attention weights as surrogates for feature importance (Deng et al., 11 Aug 2025).
• Reformulation-based families:
  • Feature-additive linear models: Any method that reduces to linear weights over an additive basis (Shapley, DeepLIFT-Res, LIME) (Deng et al., 11 Aug 2025).
  • Game-theoretic weighting: Weighted sums over all feature subsets, ensuring axiomatic uniqueness (Shapley, Aumann–Shapley) (Deng et al., 11 Aug 2025).
  • Path-integral approaches: Line integrals in feature space that generalize IG and similar methods (Deng et al., 11 Aug 2025).

Axiomatic frameworks define the properties attribution methods should satisfy, such as local accuracy (additivity), missingness (nullity), consistency (monotonicity), implementation invariance, and sensitivity (Deng et al., 11 Aug 2025, Taimeskhanov et al., 30 May 2025, Erion et al., 2019). The first-principles framework casts all attributions as Lebesgue–Stieltjes integrals over atomic attributions, showing that classical methods are special cases parameterized by integration measures (Taimeskhanov et al., 30 May 2025).

3. Faithfulness, Evaluation, and Theoretical Guarantees

Evaluating attribution constructions remains a central concern. Recent work introduced dual metrics:

• Soundness: Fraction of total attribution mass assigned to truly predictive features.
• Completeness: Fraction of predictive information captured by the set of attributed features (Li et al., 2023).

These metrics operate under the faithfulness paradigm, which probes the impact of masking/removing attributed factors on model output or accuracy, but address previous ranking-only limitations by leveraging attribution magnitudes (Li et al., 2023).

Theoretical studies show that most gradient-based and backpropagation methods do not, in general, satisfy all desired axioms. For example, LRP and Deconv may be insensitive to predicted class, while only IG, Expected Gradients, DeepLIFT-Res, Shapley, DeepSHAP fully satisfy allocation completeness and effect allocation correctness (Deng et al., 11 Aug 2025). Empirical and synthetic evaluations underline the divergence between formal, axiomatically sound methods and popular heuristics (e.g., LIME, SHAP), with formal enumeration approaches (e.g., FFA) repeatedly outperforming on both error and ranking-based metrics (Yu et al., 2023).

Relaxed functional dependence frameworks provide a more general view, requiring that valid attributions satisfy set-theoretic (complementary-dependence) and inclusion (hierarchy) properties; only a handful of methods meet these on synthetic tasks with known ground-truth attributions (Afchar et al., 2021).

4. Extensions: Structured, Contextual, and High-Level Attribution

The paradigm extends beyond standard feature attribution:

• Controllable-factor attribution explicitly partitions features into controllable and uncontrollable, conditioning attributions on the controllable subspace while holding uncontrollable features fixed—allowing actionable, policy-relevant explanations (CAFA) (Kovvuri et al., 2022).
• Attribution in in-context learning (e.g., DETAIL) treats demonstrations as factors, computing influence function-based scores for reordering and curation to optimize prediction accuracy and stability (Zhou et al., 2024).
• High-level attribution priors directly regularize network explanations during training, shaping properties such as smoothness, sparsity, or graph-conformality by penalizing global functionals of attribution maps (e.g., via Expected Gradients), enhancing robustness and interpretability (Erion et al., 2019).
• Block-level attribution in generative AI underlies provenance and economic credit allocation for creative “blocks” in collaborative music and content platforms; attribution events are logged at retrieval/composition time and linked to settlement/payments (Kim et al., 23 Oct 2025).
• Causal and agentic attribution as in 4D-ARE, which decomposes agent reasoning across four dimensions (Results, Process, Support, Long-term) and operationalizes attribution-completeness in LLM agents and decision-support systems (Yu et al., 8 Jan 2026).

The paradigm further links to cognitive and social explanation, where attributions are scored by accuracy, informativeness, and causal relevance—with empirical results showing that causal relevance best predicts human explanation choice in theory-of-mind experiments (2505.19376).

5. Practical Considerations, Limitations, and Algorithmic Implications

Algorithmic instantiations of the Factor Attribution Paradigm face hardness results (e.g., Σ₂^P-completeness for exact formal feature attribution (Yu et al., 2023)) but benefit from anytime enumeration and scalable sampling-based approximations. High-fidelity approximations are feasible for models with compact logical representations (tree ensembles, logistic models).

Domain-specific adaptations are needed for settings where controllable/uncontrollable variable partitions or structured representations (block, graph, hierarchical factors) are central. Scalability, faithfulness under realistic data distributions, handling of feature dependence, and OOD-robustness remain open challenges (Kovvuri et al., 2022, Deng et al., 11 Aug 2025).

Contextual validity of attributions requires scrutinizing the assumptions underlying each method (e.g., conditioning in CAFA, or theoretical baselines in IG/EG). In particular, selection of the measure μ_{j,x} in first-principles frameworks (Taimeskhanov et al., 30 May 2025) or calibration of leakage in trading attribution (Papathanakos, 2016) may introduce modeling risk if not carefully chosen or calibrated.

6. Synthesis, Broader Impacts, and Open Problems

The Factor Attribution Paradigm serves as a cross-cutting explanatory architecture for complex systems, uniting local feature/saliency attributions, formal subset-based abduction, structured (block/control) credit assignment, causal reasoning, and high-level function-regularization. It underpins the interpretability, auditability, and accountability of modern ML and agentic systems.

Open theoretical issues include formalizing sufficient (not just necessary) conditions for faithfulness, expanding evaluation frameworks to attention- and surrogate-based explanations, developing causally-grounded attributions beyond Shapley variants, and integrating distributional or real-world context into atomic attribution construction (Deng et al., 11 Aug 2025, Taimeskhanov et al., 30 May 2025). Practically, the paradigm motivates the design of modular, compositional systems—whether for AI-driven media, policy support, or collaborative agent platforms—where credit, responsibility, and explanation must be distributed and logged at atomic granularity (Kim et al., 23 Oct 2025, Yu et al., 8 Jan 2026).

Through systematic factor attribution, researchers achieve interpretability, robustness, and actionable insight across the statistical, economic, cognitive, and agentic dimensions of modern intelligent systems.