  We present a principled technique for reducing the lattice and matrix size in
some applications of Coppersmith's lattice method for finding roots of modular
polynomial equations. Motivated by ideas from machine learning, it relies on
extrapolating patterns from the actual behavior of Coppersmith's attack for
smaller parameter sizes, which can be thought of as "focus group" testing. When
applied to the small-exponent RSA problem, our technique reduces lattice
dimensions and consequently running times, and hence can be applied to a wider
range of exponents. Moreover, in many difficult examples our attack is not only
faster but also more successful in recovering the RSA secret key. We include a
discussion of subtleties concerning whether or not existing metrics (such as
enabling condition bounds) are decisive in predicting the true efficacy of
attacks based on Coppersmith's method. Finally, indications are given which
suggest certain lattice basis reduction algorithms (such as Nguyen-Stehl\'e's
L2) may be particularly well-suited for Coppersmith's method.
