SASANet: Shapley Additive Self-Attribution Network

• SASANet is a model architecture that applies Shapley value principles to provide fair, additive attributions for each network input or component.
• It leverages multilinear and additive function properties using path-integral methods like Aumann–Shapley to ensure complete, invariant, and efficient attribution computations.
• The framework demonstrates practical applications in PPC advertising and portfolio analysis, offering robust interpretability and computational scalability for high-stakes decision domains.

A Shapley Additive Self-Attribution Network (SASANet) is a model architecture and attribution framework in which each input or internal component of a network is assigned a contribution to the output according to the principles of Shapley values from cooperative game theory. SASANet formalizes and enforces the property that its internal additive attributions exactly correspond to the mathematically justified, fair, and unique Shapley value decomposition of network outputs. This approach is particularly potent when the function mapping inputs to output is multilinear or additive in structure, but its general architecture allows extension to broader classes using efficient algorithms, distillation schemes, and interaction handling.

1. Theoretical Foundations and Axiomatic Guarantees

A SASANet is grounded in the axiomatization of the attribution problem based on cost-sharing models, especially for functions that are the sum of multilinear and additive terms. The attribution is defined such that for an input change from $r$ to $s$, the contribution of feature $i$ is given by a path-based (Aumann–Shapley) or permutation-based (Shapley–Shubik) method. For any differentiable $f$, the Aumann–Shapley–Shubik attribution is:

$$z_i(r, s, f) = \int_0^1 \frac{\partial f}{\partial x_i}(\gamma_{r,s}(t)) \cdot (s_i - r_i) \, dt,$$

where $\gamma_{r,s}(t) = r + t (s - r)$.

The method uniquely satisfies the following axioms for every multilinear $f$:

• Dummy: $z_i(r, s, f) = 0$ if $f$ does not depend on $x_i$.
• Additivity: $$z_i(r, s, f_1 + f_2) = z_i(r, s, f_1) + z_i(r, s, f_2)$$.
• Conditional Nonnegativity: If $f$ is non-decreasing in $x_i$ on $[r, s]$, then $z_i(r, s, f) \geq 0$ for $s_i \geq r_i$.
• Affine Scale Invariance: Attributions are invariant under affine changes of scale or offset in any variable.
• Anonymity: Attributions do not depend on variable ordering.

For functions of the form $$f(x) = \sum_{I \subset [n]} c_I \prod_{i \in I} x_i$$, as in multilinear models, the discrete Shapley and continuous Aumann–Shapley attributions coincide.

2. Methodology: Multilinear Structure and Efficient Attribution

When $f$ is multilinear, explicit closed forms for attributions can be obtained. The attribution to input $i$ when $f(x) = \prod_{i=1}^n x_i$ is:

$$z_i(r, s, f) = (s_i - r_i) \cdot \frac{1}{n!} \sum_{K \subset [n] \setminus \{i\}} |K|! (n - 1 - |K|)! \left(\prod_{j \in K} s_j\right) \left(\prod_{j \notin K \cup \{i\}} r_j\right)$$

This finite-sum formula is equivalent to the classic Shapley value for cost sharing. For more general $f$ as a sum of $N$ multilinear monomials, the total computational cost is $O(n^2 N)$.

In SASANet, attribution is typically computed along straight-line (or affine) paths from a baseline $r$ (such as all-zero or population-average activations) to the actual data point $s$. The attributions can be computed per-neuron or per-feature, propagating attributions through layers by additivity (as in linear models or through integrated gradients for deep networks).

This path-integral approach enables the decomposition of the change in network output from $r$ to $s$ into a sum over features, i.e.,

$f(s) - f(r) = \sum_{i=1}^n z_i(r, s, f)$

ensuring the completeness property.

3. Aumann–Shapley–Shubik Method Implementation and Properties

The Aumann–Shapley–Shubik method is operationalized by integrating partial derivatives along paths between $r$ and $s$. Its theoretical guarantees rest on the uniqueness of attribution for multilinear-plus-additive functions satisfying the axioms above.

Key properties in practice:

• Fairness: No feature receives positive attribution unless it has a real effect.
• Permutation Invariance: Attributions are consistent regardless of variable labelling.
• Scale and Unit Invariance: Attributions remain valid across different measurement scales.
• Robustness: Attribution is null for features with zero influence throughout the path.

Computationally, the integral formulation collapses to finite sums for multilinear models. For product-type monomials, the algorithm computes each attribution in $O(n^2)$ time and $O(n)$ space.

4. Practical Applications: PPC and Portfolio Analysis

The SASANet framework and the underlying Aumann–Shapley–Shubik attribution have direct practical applications in domains with interpretable multilinear or additive models. Notable examples include:

• Pay-Per-Click (PPC) Advertising: Here, the predicted spend is often modeled as a product (e.g., clicks $\times$ cost per click), and attribution quantifies the effect of each factor.
• Portfolio Performance Attribution: Returns or risk are decomposed into contributions from asset classes, factors, or strategy components, possibly interacting multilinearly.

In both applications, attributions are computed efficiently and distributed among features or factors in a way that exactly sums to the observed change in outcome.

5. Computational and Architectural Implications

Efficient algorithmic reductions are key to real-time applications. Because SASANet's self-attribution reduces to $O(n^2 N)$ for multilinear networks, it is applicable at scale for high-throughput settings.

This computational tractability allows the attribution logic to be embedded within the model architecture or as an interpretability module without incurring prohibitive cost. Completeness ensures that the sum of attributions matches model output variation with no residual ("unattributed") term.

For deployment, key considerations include the choice of baseline $r$, the type of path $\gamma_{r,s}$, and maintaining multilinear decomposability or suitable approximations when higher complexity functions are involved.

6. Completeness, Fairness, and Limitations

The completeness property ensures that all output variation is accounted for by the sum of attributions:

$\sum_{i=1}^n z_i(r, s, f) = f(s) - f(r)$

This is undergirded by the axiomatic uniqueness result for multilinear-plus-additive models. In practice, this guarantees no spurious inflation or loss of attributed importance.

While uniqueness fails for characteristic functions beyond the multilinear-plus-additive class (i.e., functions with more intricate nonlinearities or interactions), the method remains preferred and justified for models that can be approximated or decomposed into suitable form.

In summary, the Aumann–Shapley–Shubik method—foundational to SASANet—yields efficient, theoretically justified, and fair self-attribution for networks with multilinear or additive structure. Its properties of additivity, invariance, and completeness guarantee interpretability suitable for high-stakes decision domains, and its efficient computation facilitates seamless architectural integration.