Understanding the connection between higher-order Newton methods and the convex-concave procedure

Question

I've been studying a paper Ahmadi et al.$^\color{magenta}{\dagger}$ that introduces higher-degree Newton methods with improved oracle complexity compared to the classical approach. This paper seems to be very interesting and has recently appeared on Quanta Magazine$^\color{blue}{\ddagger}$.

On page $24$ of $28$, the authors make an interesting claim that I'm struggling to verify. They state:

Each iteration of the algorithms that we have presented in this paper can be interpreted as running just one iteration of the so-called "convex-concave procedure" (see, e.g., [34]) to a particular difference of convex decomposition of the Taylor expansion of $f$. Are there benefits of working with alternative difference of convex decompositions (see, e.g., [3]) of the Taylor expansion, or running more iterations of the convex-concave procedure before the Taylor polynomial is updated ?

For context, they present the following iteration for a degree 3-Newton method:

$$x_{k+1} = x_k - 2 \frac{f''(x_k)}{f^{(3)}(x_k)} - \sqrt[3]{\frac{f'(x_k) - \frac{2}{3} \frac{(f''(x_k))^2}{f^{(3)}(x_k)}}{\frac{(f^{(3)}(x_k))^2}{12 f''(x_k)}}}.$$

After examining this formula and reviewing reference [34] on the convex-concave procedure, I'm unable to see how each iteration corresponds to a single iteration of the convex-concave procedure applied to a particular difference of convex decomposition of the Taylor expansion.

Could someone please help explain the following?

1. How this higher-order Newton iteration can be interpreted in terms of the convex-concave procedure?

2. What would be the specific convex-concave decomposition of the Taylor expansion in this case?

3. How does this interpretation relate to the authors' follow-up question about "alternative difference of convex decompositions"?

I would appreciate explanations that avoid highly abstract mathematical concepts where possible, as I'm approaching this from an applied engineering perspective rather than a pure mathematics background.

Any insights or references would be greatly appreciated.

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_{$\color{magenta}{\dagger}$ Amir Ali Ahmadi, Abraar Chaudhry, Jeffrey Zhang, Higher-order Newton methods with polynomial work per iteration, 2023.}

_{$\color{blue}{\ddagger}$ Kevin Hartnett, "Three hundred years later, a tool from Isaac Newton gets an update", Quanta Magazine, March 24, 2025.}