Stability Analysis of Various Symbolic Rule Extraction Methods from Recurrent Neural Network

This paper analyzes two competing rule extraction methodologies: quantization and equivalence query. Previous research has shown that equivalence query $L^*$ is robust DFA extraction method and LSTMs are capable of learning counter languages. However, we empirically show significant instabilities in both cases. We trained $3600$ RNN models, extracting $18000$ DFAs with a quantization approach (k-means and SOM) and $3600$ DFAs by equivalence query($L^*$) methods across $10$ initialization seeds. We sampled the datasets from $7$ Tomita and $4$ Dyck grammars and trained them on $4$ RNN cells: LSTM, GRU, O2RNN, and MIRNN. The observations from our experiments establish the superior performance of O2RNN and quantization-based rule extraction over others. $L^*$, primarily proposed for regular grammars, performs similarly to quantization methods for Tomita languages when neural networks are perfectly trained. However, for partially trained RNNs, $L^*$ shows instability in the number of states in DFA, e.g., for Tomita 5 and Tomita 6 languages, $L^*$ produced more than $100$ states. In contrast, quantization methods result in rules with a number of states very close to ground truth DFA. Among RNN cells, O2RNN produces stable DFA consistently compared to other cells. For Dyck Languages, we observe that although GRU outperforms other RNNs in network performance, the DFA extracted by O2RNN has higher performance and better stability. The stability is computed as the standard deviation of accuracy on test sets on networks trained across $10$ seeds. On Dyck Languages, quantization methods outperformed $L^*$ with better stability in accuracy and the number of states. $L^*$ often showed instability in accuracy in the order of $16\\% - 22\\%$ for GRU and MIRNN while deviation for quantization methods varied in $5\\% - 15\\%$. In many instances with LSTM and GRU, DFA's extracted by $L^*$ even failed to beat chance accuracy ($50\\%$), while those extracted by quantization method had standard deviation in the $7\\%-17\\%$ range. For O2RNN, both rule extraction methods had a deviation in the $0.5\\% - 3\\%$ range.