  We develop a variant of the Monteiro-Svaiter (MS) acceleration framework that
removes the need to solve an expensive implicit equation at every iteration.
Consequently, for any $p\ge 2$ we improve the complexity of convex optimization
with Lipschitz $p$th derivative by a logarithmic factor, matching a lower
bound. We also introduce an MS subproblem solver that requires no knowledge of
problem parameters, and implement it as either a second- or first-order method
by solving linear systems or applying MinRes, respectively. On logistic
regression our method outperforms previous second-order momentum methods, but
under-performs Newton's method; simply iterating our first-order adaptive
subproblem solver performs comparably to L-BFGS.
