Gradient-Based Relation Attribution Explanations

• The paper introduces G-RAEs, a method that quantifies the directional influence of individual edge weights on topic strength in Edge‐Weighted Quantitative Bipolar Argumentation Frameworks.
• G-RAEs leverage differentiability and backpropagation-style computations to efficiently compute influence scores and guide iterative weight updates toward a desired argument strength.
• They provide clear qualitative insights through rules like sign correctness, parity, and counterfactuality, enhancing the contestability and interpretability of AI-driven outcomes.

Searching arXiv for the cited papers to ground the article in current sources. Gradient-Based Relation Attribution Explanations (G-RAEs) are relation-level sensitivity explanations for Edge-Weighted Quantitative Bipolar Argumentation Frameworks (EW-QBAFs). They quantify how the strength of a designated topic argument changes under infinitesimal perturbations of individual edge weights, and were introduced to support contestability: the problem of modifying preferences or other relation weights so that an AI-driven outcome attains a desired argument strength in a way that remains interpretable to humans (Yin et al., 15 Jul 2025). In this setting, G-RAEs are both explanatory and operational: they attribute influence to attacks and supports, and they guide iterative weight updates toward a target strength.

1. Formal setting and contestability problem

An EW-QBAF is a quintuple

$$\mathcal{Q}=\left\langle\mathcal{A}, \mathcal{R}^{-}, \mathcal{R}^{+}, \tau, w\right\rangle,$$

where $\mathcal{A}$ is a finite set of arguments, $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$ is an attack relation, $$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$ is a support relation, $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$, $\tau: \mathcal{A} \rightarrow [0,1]$ is a base score function, and $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$ is an edge weight function. The union $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$ denotes the set of all relations.

For an argument $\alpha$, the incoming attacks, supports, and total incoming relations are

$$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$

Paths from $\mathcal{A}$0 to $\mathcal{A}$1 are denoted by $\mathcal{A}$2, and $\mathcal{A}$3 is the set of all such paths. Relative to a topic argument $\mathcal{A}$4, relations are classified as direct, indirect, multifold, or independent. A direct edge is of the form $\mathcal{A}$5. An indirect edge $\mathcal{A}$6 satisfies $\mathcal{A}$7. A multifold edge satisfies $\mathcal{A}$8. An independent edge satisfies $\mathcal{A}$9.

A quantitative gradual semantics assigns a strength

$$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$0

where $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$1 may be undefined in some cyclic cases. The main theory of G-RAEs focuses on acyclic EW-QBAFs, where strengths are well-defined and efficiently computable.

The contestability problem is formulated as follows. Given a topic argument $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$2 and a desired strength $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$3 with $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$4, the task is to find a modified weight function $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$5 such that

$$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$6

Weights are always constrained to $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$7 and are projected back into this interval after each update. The attack and support relations remain disjoint, so relation polarity is determined by membership in $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$8 or $$\mathcal{R}^{-} \subseteq \mathcal{A} \times \mathcal{A}$$9.

2. Strength semantics and differentiability

Strength computation in the relevant semantics is modular. An aggregation function combines weighted attacker and supporter contributions, and an influence function modifies the base score using the aggregate. Initialization is

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$0

For a temporary strength assignment $$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$1, the incoming weighted multisets at $$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$2 are

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$3

Two aggregation schemes are highlighted. Sum aggregation is

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$4

Product aggregation, used by DF-QuAD, is

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$5

For QE and DF-QuAD, the influence function has the form

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$6

where, for DF-QuAD,

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$7

and, for QE,

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$8

The semantics are iterated until convergence. In acyclic EW-QBAFs, this reduces to a single forward pass in a topological ordering and is computable in linear time.

Differentiability is central to G-RAEs. For acyclic EW-QBAFs under QE, REB, DF-QuAD, and MLP-based semantics, the strength function is differentiable with respect to edge weights:

$$\mathcal{R}^{+} \subseteq \mathcal{A} \times \mathcal{A}$$9

This differentiability supports directional-derivative explanations and backpropagation-style computation.

3. Definition of G-RAEs

For a topic argument $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$0 and an edge $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$1, perturb only the weight of $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$2 by $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$3, defining $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$4 and $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$5 for all $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$6. The G-RAE score is the directional derivative

$\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$7

In acyclic EW-QBAFs, this quantity is well-defined by differentiability.

The sign of $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$8 provides the basic interpretation. A positive score indicates positive influence on the topic strength, a negative score indicates negative influence, and a zero score indicates neutrality. The explanatory role of G-RAEs is therefore local and signed: they state how increasing a particular edge weight changes the topic’s current strength.

For direct relations, the polarity is aligned with edge type under edge-monotonicity: support edges have non-negative attributions and attack edges have non-positive attributions. For indirect relations, polarity depends on the parity of attacks along the path from the edge’s child to the topic. Thus a support can acquire a negative attribution, or an attack a positive attribution, when the unique downstream path contains an odd number of attacks.

The paper distinguishes analytical and perturbation-based computation. In acyclic graphs with differentiable aggregation and influence functions, one applies the chain rule along the computation graph in reverse topological order, in a backpropagation-style procedure that yields all $\mathcal{R}^{-} \cap \mathcal{R}^{+} = \emptyset$9 in time $\tau: \mathcal{A} \rightarrow [0,1]$0. When analytical computation is inconvenient, especially in cyclic settings, the score can be approximated by

$\tau: \mathcal{A} \rightarrow [0,1]$1

with a small $\tau: \mathcal{A} \rightarrow [0,1]$2, for example $\tau: \mathcal{A} \rightarrow [0,1]$3.

4. Qualitative properties

Under standard axioms such as edge-directionality, monotonicity, edge-monotonicity, neutrality, and balance, satisfied by QE, REB, DF-QuAD, and MLP-based semantics, G-RAEs satisfy several qualitative properties.

Direct influence yields sign correctness. If $\tau: \mathcal{A} \rightarrow [0,1]$4 is direct with respect to $\tau: \mathcal{A} \rightarrow [0,1]$5, then

$\tau: \mathcal{A} \rightarrow [0,1]$6

If $\tau: \mathcal{A} \rightarrow [0,1]$7 is direct, then

$\tau: \mathcal{A} \rightarrow [0,1]$8

Indirect influence obeys a parity rule. Let $\tau: \mathcal{A} \rightarrow [0,1]$9 be the number of attacks on the unique path $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$0 from the edge’s child to $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$1. Then the sign of $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$2 depends jointly on whether $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$3 is an attack or support and on whether $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$4 is even or odd. A support preserves positive sign when $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$5 is even and flips to non-positive when $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$6 is odd; an attack preserves non-positive sign when $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$7 is even and flips to non-negative when $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$8 is odd.

Irrelevance states that if an edge is independent of $$w: \mathcal{R}^{-}\cup\mathcal{R}^{+} \rightarrow [0,1]$$9, meaning that there is no path from the edge’s child to the topic, then

$\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$0

This excludes spurious attributions to relations outside the topic’s ancestor structure.

Counterfactuality gives a removal interpretation. If $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$1, then setting $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$2 cannot decrease $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$3. If $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$4, then setting $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$5 cannot increase $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$6. The claim is qualitative rather than metric: it links local derivative sign to one-step counterfactual edge removal.

Qualitative invariability states that for direct and indirect edges, the sign of $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$7 is invariant across all feasible values of $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$8, under monotonicity and edge-monotonicity. This matters for contestability because it supports stable explanatory guidance even as the optimization procedure changes weights.

5. Iterative weight adjustment and attainability

G-RAEs are used to build an iterative algorithm for contestability. Let $\mathcal{R}=\mathcal{R}^{-}\cup\mathcal{R}^{+}$9 be the desired strength of topic $\alpha$0. At each iteration, one computes the current strength $\alpha$1, estimates G-RAEs $\alpha$2 for all $\alpha$3, and updates each weight by the projected gradient step

$\alpha$4

where $\alpha$5 is a step size and

$\alpha$6

The stopping criterion is

$\alpha$7

for a small tolerance $\alpha$8, for example $\alpha$9, and the procedure is capped at $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$0 iterations to avoid runtime blow-ups or stalling. A practical schedule uses a larger $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$1 when the current strength is far from the target and reduces $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$2 near convergence.

Attainability is characterized through extremal assignments. A max case $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$3 sets all support edges to $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$4 and all attack edges to $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$5 recursively, yielding the maximal attainable strength. A min case $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$6 sets direct attacks of $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$7 to $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$8, direct supports to $$\mathcal{R}^{-}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{-}\}, \qquad \mathcal{R}^{+}(\alpha)=\{(\beta,\alpha)\in\mathcal{R}^{+}\}, \qquad \mathcal{R}(\alpha)=\mathcal{R}^{-}(\alpha)\cup\mathcal{R}^{+}(\alpha).$$9, and recursively maximizes attackers’ strengths while neutralizing their direct attacks to $\mathcal{A}$00, yielding the minimal attainable strength. With continuity, the attainable set is an interval

$\mathcal{A}$01

Accordingly, any target $\mathcal{A}$02 can be reached in principle in acyclic settings.

The update procedure is heuristic. It is described as gradient-descent-like, and may require multiple attempts with different initializations to avoid local traps. For acyclic EW-QBAFs and perturbation-based estimation, the time per iteration is

$\mathcal{A}$03

and over $\mathcal{A}$04 iterations the total cost is

$\mathcal{A}$05

Its interpretability derives from edge-local scores that indicate which relations to adjust and in which direction, such as increasing weights on relations with positive support to the topic or decreasing weights on negatively contributing attacks.

6. Worked example and empirical evaluation

A worked QE example makes the local mechanics explicit (Yin et al., 15 Jul 2025). Consider a small acyclic EW-QBAF with topic $\mathcal{A}$06, one supporter $\mathcal{A}$07, and one attacker $\mathcal{A}$08, with

$\mathcal{A}$09

and edge weights

$\mathcal{A}$10

Since there are no other edges, $\mathcal{A}$11 and $\mathcal{A}$12. Under sum aggregation,

$\mathcal{A}$13

For QE,

$\mathcal{A}$14

so the topic strength is

$\mathcal{A}$15

The analytical G-RAEs are

$\mathcal{A}$16

The direct support therefore has positive attribution and the direct attack negative attribution, exactly as the qualitative theory predicts. If the target is $\mathcal{A}$17 and the step size is $\mathcal{A}$18, then one update gives

$\mathcal{A}$19

which increases the topic strength to

$\mathcal{A}$20

A second iteration yields

$\mathcal{A}$21

and

$\mathcal{A}$22

The example is intended to show gradual movement toward the target under projected gradient steps.

The empirical study uses synthetic EW-QBAFs with two structural regimes. In PRS-like acyclic EW-QBAFs, graphs contain $\mathcal{A}$23 arguments, base scores are i.i.d. uniform in $\mathcal{A}$24, each pair $\mathcal{A}$25 with $\mathcal{A}$26 receives an edge with probability $\mathcal{A}$27, each edge is support or attack with equal probability, and each weight is uniform in $\mathcal{A}$28. The topic is $\mathcal{A}$29, and the target is the midpoint of attainable bounds. Under edge-weighted QE, REB, and DF-QuAD, validity is reported as $\mathcal{A}$30 success across all sizes and semantics. REB always succeeded in the first attempt; QE had average attempts approximately $\mathcal{A}$31 with maximum $\mathcal{A}$32; DF-QuAD had average attempts approximately $\mathcal{A}$33 with maximum $\mathcal{A}$34. Average runtimes remained under $\mathcal{A}$35 seconds across sizes and semantics, median runtime was below $\mathcal{A}$36 seconds in all cases, and REB was fastest on average.

In MLP-like EW-QBAFs, structures are $\mathcal{A}$37, $\mathcal{A}$38, and $\mathcal{A}$39, with connection probability $\mathcal{A}$40 between adjacent layers. Base scores and edge weights or polarities are sampled as in the PRS-like setting, the topic is the output-layer node, and the semantics are MLP-based with sum aggregation. Validity is again reported as $\mathcal{A}$41 success for all structures and densities. Attempts were almost always $\mathcal{A}$42, with a single case for $\mathcal{A}$43 requiring $\mathcal{A}$44 attempts. Average runtime increased with density, with a maximum average of approximately $\mathcal{A}$45 seconds, and median runtime strictly increased with density, with a maximum median of approximately $\mathcal{A}$46 seconds. The reported interpretation is polynomial-time behavior and practical runtimes even on dense, layered graphs.

The paper also provides illustrative G-RAE scores on a movie-recommendation EW-QBAF under MLP-based semantics, with topic argument Movie. The direct support $\mathcal{A}$47 receives $\mathcal{A}$48, $\mathcal{A}$49 receives $\mathcal{A}$50, more distant supports such as $\mathcal{A}$51 and $\mathcal{A}$52 receive $\mathcal{A}$53 and $\mathcal{A}$54, $\mathcal{A}$55 receives $\mathcal{A}$56, while $\mathcal{A}$57 receives $\mathcal{A}$58 and the direct edge $\mathcal{A}$59 receives $\mathcal{A}$60. These values exemplify the use of signed edge-local scores as a ranked explanation interface.

7. Relation to Shapley-based RAEs, limitations, and extensions

G-RAEs belong to a broader literature on relation-level explanation in quantitative bipolar argumentation. A 2024 technical report introduced Relation Attribution Explanations (RAEs) by adapting Shapley values to relations in QBAFs, with the value to be explained defined as the topic strength under a restricted set of relations, and established properties such as efficiency, dummy, symmetry, and dominance (Yin et al., 2024). That report also presents a principled formulation of gradient-based relation attribution within the same framework, while explicitly stating that the paper does not define or analyze gradient-based relation attributions as part of its main contribution.

The contrast between the two families is structural. Shapley RAEs are global, coalition-averaged attributions over relation subsets; G-RAEs are local sensitivities with respect to continuous edge weights. Shapley RAEs capture interaction effects by averaging marginal contributions across contexts. G-RAEs provide instantaneous directional information at the current point in weight space. This distinction explains why G-RAEs are especially useful for optimization and search over weights, whereas Shapley RAEs are especially useful when fair allocation of influence across interacting relations is the primary objective. A common misconception is to treat the two methods as interchangeable; the published comparison indicates that they answer different explanatory questions.

Several limitations are identified. Gradients are local and can be small near plateaus, including cases where the aggregate is non-positive for QE and $\mathcal{A}$61, which can impede progress. Non-convexities and discontinuities, including DF-QuAD’s ReLU-like $\mathcal{A}$62, may produce local traps, so multiple restarts or schedule tuning may be needed. Projection to $\mathcal{A}$63 can saturate weights at the bounds and limit feasible adjustments. Cycles can break convergence for some semantics; perturbation-based estimation remains applicable, but strengths may be undefined for some cyclic graphs unless continuization techniques are used.

Proposed extensions include integrated gradients or path-based attributions to ameliorate saturation or nonlinearity issues, second-order methods using approximate curvature for faster convergence, constrained optimization formulations incorporating sparsity or minimal-change objectives, and multi-contestability, where multiple targets are jointly achieved on the same topic across different base score functions. A plausible implication is that these directions aim to preserve the explanatory advantages of G-RAEs while reducing the gap between local sensitivity analysis and broader counterfactual or interaction-aware notions of attribution.