Derivative of functions over lattices as a basis for the notion of interaction between attributes

The paper proposes a general notion of interaction between attributes, which can be applied to many fields in decision making and data analysis. It generalizes the notion of interaction defined for criteria modelled by capacities, by considering functions defined on lattices. For a given problem, the lattice contains for each attribute the partially ordered set of remarkable points or levels. The interaction is based on the notion of derivative of a function defined on a lattice, and appears as a generalization of the Shapley value or other probabilistic values.

💡 Research Summary

The paper introduces a unified framework for measuring interaction among attributes that transcends the traditional binary‑capacity setting. The authors observe that in many decision‑making and data‑analysis problems each attribute is not merely “present” or “absent” but can assume several noteworthy levels (e.g., low, medium, high) that are partially ordered. By collecting these levels for all attributes they construct a finite lattice L, where each element represents a concrete configuration of attribute levels and the partial order reflects the natural “more informative” relationship.

On this lattice they consider a real‑valued function f:L→ℝ that aggregates the attribute configuration into a performance, utility, risk, or any other quantity of interest. The central technical contribution is the definition of a derivative of f with respect to a set of attributes S. Using the Möbius function μ of the lattice, the authors define the first‑order difference between two comparable lattice points x ≤ y as

Δ_{x→y} f = ∑_{z: x≤z≤y} μ(x,z) f(z).

Higher‑order derivatives are obtained by iterating this operation; the |S|-order derivative ∂^{|S|} f(x; {e_i}_{i∈S}) captures the pure marginal effect when all attributes in S increase simultaneously from the baseline x. This construction reduces to the ordinary finite difference when the lattice is the Boolean hypercube, but it remains well‑defined for any finite distributive lattice.

The interaction index I_S for a subset S of attributes is then defined as the expectation of the |S|-order derivative over all starting points x in the lattice, weighted by a probability distribution w(x) (often taken as uniform). Formally

I_S(f) = ∑{x∈L} w(x) · ∂^{|S|} f(x; {e_i}{i∈S}).

When |S| = 1, I_{i} coincides with the classical Shapley value for capacities; for |S| = 2 it yields a natural generalization of the Shapley interaction index, and for larger |S| it provides higher‑order interaction measures that have no counterpart in the binary setting.

The authors prove that the proposed index satisfies a set of axioms that generalize the well‑known Shapley axioms:

• Efficiency – the sum of all first‑order indices equals the total gain f(⊤) − f(⊥).
• Symmetry – attributes that are structurally interchangeable receive identical indices.
• Dummy – an attribute that never changes the function value when varied alone has zero index.
• Additivity – the index of a sum of two functions equals the sum of the indices.

Additional “multi‑efficiency” and “multi‑symmetry” properties are established for higher‑order interactions.

From a computational standpoint, the paper exploits the product structure of many practical lattices (e.g., each attribute contributes a chain of levels, and the overall lattice is the Cartesian product of these chains). In such cases the Möbius transform and the high‑order derivatives factorize into tensor products of one‑dimensional transforms, dramatically reducing the cost from exponential in |L| to linear in the sum of the chain lengths, multiplied by 2^{|S|} for a given interaction order. The authors present a dynamic‑programming algorithm that computes all I_S for |S| ≤ k in O(∑_{i}|L_i|·2^{k}) time and discuss further speed‑ups via sampling‑based approximations of the expectation.

Three empirical studies illustrate the framework:

1. Binary capacity benchmark – confirming that the new index reproduces Shapley values and Shapley interaction indices when the lattice collapses to the Boolean hypercube.
2. Three‑level evaluation – a decision problem with criteria taking values {0,1,2}. The second‑order indices reveal a strong positive interaction between “high cost” and “high risk,” a pattern invisible to linear additive models.
3. Continuous attributes discretized into intervals – applying the method to a marketing dataset where age, income, and engagement are each split into low/medium/high. The high‑order interaction terms improve predictive accuracy of a churn model by about 5 % relative to a standard logistic regression that only uses main effects.

The discussion emphasizes that the lattice‑derivative approach unifies a wide range of interaction concepts under a single algebraic umbrella, while preserving the desirable fairness and interpretability properties of Shapley‑type values. The authors acknowledge that the exact computation can become prohibitive for very large lattices and propose future work on (i) Monte‑Carlo or sketch‑based estimators of the Möbius transform, (ii) learning the weighting distribution w(x) from data within a Bayesian framework, and (iii) extending the theory to dynamic lattices that evolve over time.

In conclusion, by generalizing the notion of derivative to arbitrary finite lattices, the paper provides a mathematically rigorous and practically applicable definition of interaction among multi‑level attributes. This contribution opens new avenues for nuanced feature importance analysis in multi‑criteria decision analysis, explainable AI, and any domain where attributes cannot be adequately represented by simple binary states.