  For many machine learning applications in science, data acquisition, not
training, is the bottleneck even when avoiding experiments and relying on
computation and simulation. Correspondingly, and in order to reduce cost and
carbon footprint, training data efficiency is key. We introduce minimal
multilevel machine learning (M3L) which optimizes training data set sizes using
a loss function at multiple levels of reference data in order to minimize a
combination of prediction error with overall training data acquisition costs
(as measured by computational wall-times). Numerical evidence has been obtained
for calculated atomization energies and electron affinities of thousands of
organic molecules at various levels of theory including HF, MP2, DLPNO-CCSD(T),
DFHFCABS, PNOMP2F12, and PNOCCSD(T)F12, and treating tens with basis sets TZ,
cc-pVTZ, and AVTZ-F12. Our M3L benchmarks for reaching chemical accuracy in
distinct chemical compound sub-spaces indicate substantial computational cost
reductions by factors of $\sim$ 1.01, 1.1, 3.8, 13.8 and 25.8 when compared to
heuristic sub-optimal multilevel machine learning (M2L) for the data sets QM7b,
QM9$^\mathrm{LCCSD(T)}$, EGP, QM9$^\mathrm{CCSD(T)}_\mathrm{AE}$, and
QM9$^\mathrm{CCSD(T)}_\mathrm{EA}$, respectively. Furthermore, we use M2L to
investigate the performance for 76 density functionals when used within
multilevel learning and building on the following levels drawn from the
hierarchy of Jacobs Ladder:~LDA, GGA, mGGA, and hybrid functionals. Within M2L
and the molecules considered, mGGAs do not provide any noticeable advantage
over GGAs. Among the functionals considered and in combination with LDA, the
three on average top performing GGA and Hybrid levels for atomization energies
on QM9 using M3L correspond respectively to PW91, KT2, B97D, and $\tau$-HCTH,
B3LYP$\ast$(VWN5), TPSSH.
