  It is almost always easier to find an accurate-but-complex model than an
accurate-yet-simple model. Finding optimal, sparse, accurate models of various
forms (linear models with integer coefficients, decision sets, rule lists,
decision trees) is generally NP-hard. We often do not know whether the search
for a simpler model will be worthwhile, and thus we do not go to the trouble of
searching for one. In this work, we ask an important practical question: can
accurate-yet-simple models be proven to exist, or shown likely to exist, before
explicitly searching for them? We hypothesize that there is an important reason
that simple-yet-accurate models often do exist. This hypothesis is that the
size of the Rashomon set is often large, where the Rashomon set is the set of
almost-equally-accurate models from a function class. If the Rashomon set is
large, it contains numerous accurate models, and perhaps at least one of them
is the simple model we desire. In this work, we formally present the Rashomon
ratio as a new gauge of simplicity for a learning problem, depending on a
function class and a data set. The Rashomon ratio is the ratio of the volume of
the set of accurate models to the volume of the hypothesis space, and it is
different from standard complexity measures from statistical learning theory.
Insight from studying the Rashomon ratio provides an easy way to check whether
a simpler model might exist for a problem before finding it, namely whether
several different machine learning methods achieve similar performance on the
data. In that sense, the Rashomon ratio is a powerful tool for understanding
why and when an accurate-yet-simple model might exist. If, as we hypothesize in
this work, many real-world data sets admit large Rashomon sets, the
implications are vast: it means that simple or interpretable models may often
be used for high-stakes decisions without losing accuracy.
