Candidate Retention for Abductive Learning

Hao-Yuan He; Yu Liu; Ming Li

cs.LGSep 30, 2026

Candidate Retention for Abductive Learning

Authors: Hao-Yuan He, Yu Liu, Ming Li

Organizations: School of Artificial Intelligence, Nanjing University · National Key Laboratory for Novel Software Technology, Nanjing University

Abstract

Abductive learning combines neural perception with symbolic reasoning, using explanations generated by abduction to supervise the perception model. Multiple valid explanations of the same symbolic target can assign conflicting labels to the same inputs. Common policies select a single candidate as a pseudo-label, which may reinforce mistaken assignments, or weight all candidates, which may spread supervision across competing labels. These risks motivate selecting a retained subset to balance supervision sharpness and model-mass coverage. To guide this choice, we bound the coordinate-level supervision error using retained uncertainty, discarded model mass, and model mismatch. For a fixed model and training pair, only the first two terms depend on the retained set. We propose Abductive Candidate Retention (ACR), which uses these terms to guide greedy additions, accepting a candidate when its recovered mass exceeds the increase in retained uncertainty. Experiments show that ACR improves concept accuracy over single-candidate baselines and A3BL in most evaluated aggregated mod-addition settings. Objective ablations support the joint use of uncertainty and posterior mass.

Figures & tables

Figure 1: Candidate retention in the ABL loop. Upper: abduction returns

{\bm{s}}({\bm{x}},y)

; supervision construction marginalizes the retained candidates’ weighted labels to train the perception model. Lower: three retention policies with schematic candidate weights

\tilde{p}

. Shading indicates which candidates are retained.

{\bm{c}}^{\dagger}\in\arg\min_{{\bm{c}}\in C_{K}\setminus R}\Delta_{{\bm{c}}}J_{\theta}(R).

Algorithm 1 Greedy Candidate Retention

	MNIST				KMNIST
Method	$n=1$	$n=2$	$n=3$	$n=4$	$n=1$	$n=2$	$n=3$	$n=4$
DeepStochLog	$99.00_{\pm\mathmakebox[l]{0.13}}$	$98.88_{\pm\mathmakebox[l]{0.02}}$	$98.94_{\pm\mathmakebox[l]{0.15}}$	$98.98_{\pm\mathmakebox[l]{0.13}}$	$94.00_{\pm\mathmakebox[l]{0.73}}$	$94.06_{\pm\mathmakebox[l]{0.30}}$	$94.00_{\pm\mathmakebox[l]{0.29}}$	$93.93_{\pm\mathmakebox[l]{0.49}}$
NeurASP	$50.01_{\pm\mathmakebox[l]{0.44}}$	$10.21_{\pm\mathmakebox[l]{0.39}}$	$10.21_{\pm\mathmakebox[l]{0.39}}$	N/A	$10.01_{\pm\mathmakebox[l]{0.58}}$	$10.05_{\pm\mathmakebox[l]{0.51}}$	N/A	N/A
DeepProbLog	$97.42_{\pm\mathmakebox[l]{0.29}}$	N/A	N/A	N/A	$80.54_{\pm\mathmakebox[l]{0.21}}$	N/A	N/A	N/A
ABL-hamming	$98.49_{\pm\mathmakebox[l]{0.22}}$	$98.65_{\pm\mathmakebox[l]{0.08}}$	$98.63_{\pm\mathmakebox[l]{0.12}}$	$60.41_{\pm\mathmakebox[l]{40.27}}$	$91.48_{\pm\mathmakebox[l]{0.54}}$	$92.43_{\pm\mathmakebox[l]{0.65}}$	$63.51_{\pm\mathmakebox[l]{36.91}}$	$47.74_{\pm\mathmakebox[l]{37.13}}$
ABL-confidence	$79.31_{\pm\mathmakebox[l]{41.35}}$	$79.32_{\pm\mathmakebox[l]{41.61}}$	$77.10_{\pm\mathmakebox[l]{43.53}}$	$79.22_{\pm\mathmakebox[l]{41.65}}$	$86.32_{\pm\mathmakebox[l]{25.40}}$	$79.80_{\pm\mathmakebox[l]{34.91}}$	$72.76_{\pm\mathmakebox[l]{38.75}}$	$54.84_{\pm\mathmakebox[l]{42.97}}$

Table 1: Character accuracy on digit addition (%; mean

\pm

std). N/A: did not finish within 24 hours.

Figure 2: Training curves for four representative modulus pairs per dataset. Annotations show ACR’s accuracy gain in percentage points over the strongest baseline. Full results in Appendix C .

Figure 3: Task-wise ablation distributions across 19 modulus pairs per dataset. Columns: character accuracy, TLM, and retained set size. ACR w/o mass and ACR w/o uncertainty (w/o M and w/o U in the panels) remove the mass and uncertainty terms.

Method	Acc. (%)	TLM (%)	Size
ACR w/o mass	20.01	7.88	1.00
ACR w/o uncertainty	65.22	40.15	18.76
ACR	94.98	94.25	16.68

Table 2: Objective ablation averaged over all aggregated mod-addition settings (19 modulus pairs

\times

2 datasets). ACR achieves the highest accuracy and TLM; removing either term degrades both.

Figure 4: Training dynamics on MNIST mod addition

(3,4)

. Top: retained uncertainty

\mathrm{U}_{\theta}

and discarded mass

1-M_{\theta}

; bottom: TLM. Lines: medians; bands: interquartile ranges.

	Chess (%)		Judicial
Method	Attack	Piece	Macro-F1	Micro-F1
ABL-hamming	69.01	17.95	0.32470	0.70256
ABL-confidence	69.01	17.95	0.32470	0.70256
$\text{A}^{3}\text{BL}$	97.53	87.19	0.83224	0.89574
ACR (ours)	99.31	98.29	0.83309	0.89870

Table 3: Results beyond arithmetic. Best in bold.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Character-accuracy trajectories for all 19 MNIST modulus pairs. The bottom-right panel summarizes ACR’s test gain over the strongest baseline.

Figure 6: Character-accuracy trajectories for all 19 KMNIST modulus pairs. The bottom-right panel summarizes ACR’s test gain over the strongest baseline.

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Assessing Reliability of Symbol Detection in Concept Bottleneck Models

Javier Fumanal-Idocin, Javier Andreu-Perez

University of Essex, Wivenhoe Park, Essex, United Kingdom

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Concept Bottleneck Models (CBMs) are a relevant tool for explainable Artificial Intelligence because they make their predictions through human-interpretable symbols. However, high task accuracy does not guarantee that these symbols are detected faithfully: jointly trained CBMs may encode task-specific shortcuts in the bottleneck, making their explanations unreliable. In this paper, we study concept-detection reliability by swapping independently trained concept detectors and classification heads that share the same symbolic vocabulary. We use the resulting performance degradation, concept-level metrics, and symbol-wise uncertainty estimates to identify concepts that are especially prone to spurious firing. Finally, we propose a reliability-aware training strategy in which a shared concept detector is optimized with multiple classification heads and penalized for relying on globally or instance-wise unreliable symbols. On CUB-200-2011 with full concept supervision, detectors and heads are almost freely interchangeable (swap drop below one accuracy point, relative retention above

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Abstract

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