  The free energy is a key quantity of interest in Ising models, but
unfortunately, computing it in general is computationally intractable. Two
popular (variational) approximation schemes for estimating the free energy of
general Ising models (in particular, even in regimes where correlation decay
does not hold) are: (i) the mean-field approximation with roots in statistical
physics, which estimates the free energy from below, and (ii) hierarchies of
convex relaxations with roots in theoretical computer science, which estimate
the free energy from above. We show, surprisingly, that the tight regime for
both methods to compute the free energy to leading order is identical.
  More precisely, we show that the mean-field approximation is within
$O((n\|J\|_{F})^{2/3})$ of the free energy, where $\|J\|_F$ denotes the
Frobenius norm of the interaction matrix of the Ising model. This
simultaneously subsumes both the breakthrough work of Basak and Mukherjee, who
showed the tight result that the mean-field approximation is within $o(n)$
whenever $\|J\|_{F} = o(\sqrt{n})$, as well as the work of Jain, Koehler, and
Mossel, who gave the previously best known non-asymptotic bound of
$O((n\|J\|_{F})^{2/3}\log^{1/3}(n\|J\|_{F}))$. We give a simple, algorithmic
proof of this result using a convex relaxation proposed by Risteski based on
the Sherali-Adams hierarchy, automatically giving sub-exponential time
approximation schemes for the free energy in this entire regime. Our
algorithmic result is tight under Gap-ETH.
  We furthermore combine our techniques with spin glass theory to prove (in a
strong sense) the optimality of correlation rounding, refuting a recent
conjecture of Allen, O'Donnell, and Zhou. Finally, we give the tight
generalization of all of these results to $k$-MRFs, capturing as a special case
previous work on approximating MAX-$k$-CSP.
