Thank you for following up and pointing out that work by Feldman. To answer the comment, we note the following:
1. For DR-submodular setting, we begin by noting that Algorithm 1 in the previous paper you mentioned [Fazel et al.], may be considered an adaptation of Algorithm 1 of the Feldman paper (which considers set submodular functions) to the monotone DR-submodular setting where the constraint set contains the origin $0 \in \mathcal{K}$.
As we mentioned in our previous reply, any extension of the Fazel paper (with curvature and strong DR-submodularity) to settings with non-monotone functions or when the constraint set does not contains the origin, would require a non-trivial change in the update rule. However, we believe such extension to be possible.
A natural starting point for monotone DR-submodular functions with (just) bounded curvature would be to use Fazel et al. paper by "removing" the strong DR-submodularity. However, doing so is not trivial (if even possible).
Setting $\mu \gets 0$ simplifies the update formula $v_k$ in SDRFW algorithm, however, their main result is only applicable when $\mu > 0$. More precisely, we note that the approximation bounds depend on a number of iterations $K \propto 1/\mu$, which is meaningless for $\mu=0$, so we would need to modify the proof, and possibly the update rule, to obtain a result using this approach.
Another significant issue is with the algorithm design, as Fazel et al. algorithm requires as input a particular linear function $\ell_i = \min_x \nabla_i f(x)$ which in general may be challenging to compute. The function essentially finds, for each coordinate, worst case marginal gains achieved in the feasible region. It is not clear to us how we could compute that efficiently (even with exact gradient oracles) to begin with in addition to the FW steps that incorporate that. Further, with stochastic gradient estimates (via a stochastic gradient oracle or using samples from a value oracle), it is not immediately clear how robust the algorithm would be to an inexact $\ell_i$.
We remark that Fazel et al.'s gradient ascent algorithm (Algorithm 2) interestingly does not require knowledge of $\mu$ or require as input the same linear function (constructed from $F$) as their FW type algorithm SDRFW did. That result looks more promising to generalize to different feasible regions and objective oracle types. The approximation coefficient obtained $1/(1+c_f)$. Note that this result is for monotone DR-submodular functions over general convex sets and therefore their approximation coefficient is $1/2$ rather than $1-1/e$ when $c = 1$.
2. The mentioned paper (Feldman, "Guess free$\dots$") does not discuss maximization of submodular functions with bounded curvature. It considers the closely related problem of optimizing the sum of a submodular function and a linear function, which is used as a subroutine for bounded curvature (DR-)submodular maximization as in [Sviridenko et al., Mathematics of Operations Research 42.4 (2017): 1197-1218. https://pubsonline.informs.org/doi/abs/10.1287/moor.2016.0842 ] or [Fazel et al.], and will be discussed in (3) below. For a given known linear function (i.e., separate value/gradient oracles for the DR-submodular and the linear functions), we think it is plausible can likely extend our paper to the sum of a DR-submodular function and a linear function. However, a detailed investigation is needed, since the approximation ratios need to be seen if they still hold (where approximation ratio is in the front of only DR-submodular function). Further, the update rules would need modifications. We believe that a careful analysis might work out, and is a good topic for the future work.
3. As mentioned above, optimizing the sum of a submodular function and a linear function is used as a subroutine for bounded curvature (DR-)submodular maximization.
In other words, the first step for maximizing a bounded curvature (DR-)submodular function is to decompose it as a sum of a linear and a (DR-)submodular function.
[Sviridenko et al., 2017] constructs a linear function for the discrete case, which can be evaluated using multiple oracle calls. However, such an approach is shown only for monotone (set) functions. Further, in order to obtain this linear function, we may need to query outside the feasible set. For the DR-submodular case, a linear function construction is given in Fazel et al., as mentioned in point (1) above, whose computation can be challenging and it is not evident how to compute it efficiently with only oracle calls.
In summary, we believe that the problem of extending our work on the direction of bounded curvature is an interesting problem. However, we do not believe this to be an easy direction. Nevertheless, we will point this possible extension in the future works section in the final version.