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Deciphering Raw Data in Neuro-Symbolic Learning with Provable Guarantees
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Provides a theoretical analysis of abductive learning (neuro-symbolic) showing that minimal-inconsistency objectives upper-bound a location-based risk, and derives a full-rank probability-matrix criterion to diagnose whether a knowledge base will enable successful learning.
Problem
Neuro-symbolic hybrid systems work empirically, but there is no theory explaining when a symbolic knowledge base provides enough supervision to learn accurate perception models, so failures cannot be anticipated before training.
Approach
Abduction-based minimal-inconsistency objectives are shown to upper-bound a location-based risk (L-Risk/TL-Risk) that relates classifier outputs to instance positions and target concepts via a probability matrix Q; if Q has full row rank, minimising this risk recovers the true labels, giving a pre-training rank criterion for knowledge bases.
Key results
- On HED, the binary additive knowledge base gives a rank-4 probability matrix (full rank) and high accuracy, while the decimal knowledge base gives rank 7 vs 12 symbols and ABL-TL falls below 50% test accuracy.
- On ConjEq and Conjunction all four abduction strategies exceed 98% accuracy on MNIST/EMNIST/USPS/Kuzushiji/Fashion with MLP.
- On Addition with MLP, ABL-TL reaches 98.00% on MNIST and 98.41% on EMNIST, outperforming ABL-RAND and ABL-AVG.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use TL-Risk minimisation instead of random abduced-label selection | ABL on Addition · Addition (MNIST, EMNIST, USPS, Kuzushiji, Fashion) | test accuracy: 92.01 -> 98.00; 92.94 -> 98.41; 90.96 -> 94.68; 73.18 -> 90.04; 79.08 -> 88.38 | Helped |
| Use TL-Risk minimisation instead of average candidate-label loss | ABL on Addition · Addition (MNIST, EMNIST, USPS, Kuzushiji, Fashion) | test accuracy: 94.90 -> 98.00; 95.71 -> 98.41; 93.22 -> 94.68; 80.94 -> 90.04; 84.43 -> 88.38 | Helped |
| Use MaxP or MinD rather than random label selection | ABL on Addition · Addition | test accuracy | Helped |
| Use TL-Risk minimisation rather than MaxP or MinD label selection | ABL on Addition · Addition | test accuracy | Mixed |
| Use TL-Risk minimisation rather than the tested abduction strategies | ABL on ConjEq and Conjunction · ConjEq and Conjunction | test accuracy | No clear effect |
| Use the decimal-addition knowledge base, whose matrix rank is 7 rather than 12 | ABLTL on HED · HED with decimal additive equations | test accuracy: below 50% | Hurt |
| Use number bases larger than 5 | ABL on HED · HED across number systems | test accuracy | Hurt |
| Use a rank-deficient rather than full-row-rank probability matrix | ABL on random knowledge bases · Random DNF/CNF knowledge bases | test accuracy | Hurt |
Surprising findings
- On the Addition task, MaxP and MinD outperform Random abduction in later training, despite behaving like random guessing early on.
- For the single-concept conj0 case the probability matrix has rank one, so learning may fail even though the knowledge base is nontrivial.
Limitations
- Theory assumes abduced labels are chosen randomly from the candidate set, which only approximates heuristic abduction strategies.
- The rank criterion is negative for decimal addition, where the authors note learning can still succeed under extra similarity assumptions not covered by the analysis.
- Experiments use simple benchmark tasks with small candidate sets, limiting generality.
Experimental setup
Datasets: MNIST EMNIST USPS Kuzushiji Fashion HASYv2
Benchmarks: ConjEq Conjunction Addition HED random DNF/CNF knowledge bases
Metrics: test accuracy
Baselines: MaxP abduction MinD abduction Random abduction Average abduction
Builds on: abductive learning partial-label learning reasoning shortcuts
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