Learning a Formula of Interpretability to Learn Interpretable Formulas

Abstract

Many risk-sensitive applications require Machine Learning (ML) models to be interpretable. Attempts to obtain interpretable models typically rely on tuning, by trial-and-error, hyper-parameters of model complexity that are only loosely related to interpretability. We show that it is instead possible to take a meta-learning approach: an ML model of non-trivial Proxies of Human Interpretability (PHIs) can be learned from human feedback, then this model can be incorporated within an ML training process to directly optimize for interpretability. We show this for evolutionary symbolic regression. We first design and distribute a survey finalized at finding a link between features of mathematical formulas and two established PHIs, simulatability and decomposability. Next, we use the resulting dataset to learn an ML model of interpretability. Lastly, we query this model to estimate the interpretability of evolving solutions within bi-objective genetic programming. We perform experiments on five synthetic and eight real-world symbolic regression problems, comparing to the traditional use of solution size minimization. The results show that the use of our model leads to formulas that are, for a same level of accuracy-interpretability trade-off, either significantly more or equally accurate. Moreover, the formulas are also arguably more interpretable. Given the very positive results, we believe that our approach represents an important stepping stone for the design of next-generation interpretable (evolutionary) ML algorithms.

Mining Road Traffic Rules with Signal Temporal Logic and Grammar-Based Genetic Programming

“…Moreover, evolution prefers the ∧ and ¬ operators over the temporal operators (of which there is approximately one per formula). This finding is in line with our expectations since temporal operators are likely to be the least interpretable for a human (as confirmed for mathematical expressions in [23]). In the following, we transcribe some instances of best individuals.…”

“…However, we believe that posing a further limit on the composition of the temporal operators may make the STL formulas more readable, and not only just smaller. Our belief is corroborated by the findings of [23] for mathematical expressions: some operators, such as log and sin, make the expressions less interpretable than others, e.g., + and ÷.…”

“…In this work, we circumvent the problem of defining interpretability with a practical approach: we force our method to search in a subset of Φ containing structurally simple formulas, where both the overall size of the formula and the maximum degree of nesting of temporal operators are limited. The rationale of the latter constraint is in the fact that the interpretability of symbolic models is affected differently by different kinds of components [23].…”

Model Learning with Personalized Interpretability Estimation (ML-PIE)

“…Recently, an estimator of human-interpretability for symbolic models expressed as formulae was machine-learned from human feedback [58]. A survey was used to make users simulate the calculations of (random) formulae (an implementation of the XAI concept of simulatability [34]), and to identify the behavior of the formula when part of it would vary in some interval (an implementation of the XAI concept of decomposability [34]).…”

“…In this paper we build upon [58] and extend it in three ways: (1) we concurrently learn a model of intepretability that is specific to the user; (2) we use a more complex, non-linear estimator instead of a linear one; and (3) we exploit uncertainty estimation to require a small amount of feedback to train the estimator.…”