We propose faster methods for unconstrained optimization of emph{structured convex quartics}, which are convex functions of the form begin{equation*} f(x) = c^top x + x^top mathbf{G} x + mathbf{T}[x,x,x] + frac{1}{24} mathopen| mathbf{A} x mathclose|_4^4 end{equation*} for $c in mathbb{R}^d$, $mathbf{G} in mathbb{R}^{d times d}$, $mathbf{T} in mathbb{R}^{d times d times d}$, and $mathbf{A} in mathbb{R}^{n times d}$ such that $mathbf{A}^top mathbf{A} succ 0$. In particular, we show how to achieve an $epsilon$-optimal minimizer for such functions with only $O(n^{1/5}log^{O(1)}(mathcal{Z}/epsilon))$ calls to a gradient oracle and linear system solver, where $mathcal{Z}$ is a problem-dependent parameter. Our work extends recent ideas on efficient tensor methods and higher-order acceleration techniques to develop a descent method for optimizing the relevant quartic functions. As a natural consequence of our method, we achieve an overall cost of $O(n^{1/5}log^{O(1)}(mathcal{Z} / epsilon))$ calls to a gradient oracle and (sparse) linear system solver for the problem of $ell_4$-regression when $mathbf{A}^top mathbf{A} succ 0$, providing additional insight into what may be achieved for general $ell_p$-regression. Our results show the benefit of combining efficient higher-order methods with recent acceleration techniques for improving convergence rates in fundamental convex optimization problems.
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