Stability Analysis of Various Symbolic Rule Extraction Methods from
  Recurrent Neural Network
Neisarg Dave Daniel Kifer C. Lee Giles Ankur Mali
This paper analyzes two competing rule extraction methodologies: quantization
and equivalence query. We trained $3600$ RNN models, extracting $18000$ DFA
with a quantization approach (k-means and SOM) and $3600$ DFA 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 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 deviation in the $0.5\% - 3\%$ range.