I have a (black-box) function $f(\mathbf{x};\mathbf{\theta})$ which depends on $\mathbf{x} = (x_1,...,x_N)$ a set of RGB images and a set of real and discrete parameters $\mathbf{\theta}$.
I cannot evaluate the derivative directly, but I can evaluate in 30 minutes $f$ at a point by fixing $\mathbf{\theta}$, thus, evaluating a lot of derivatives by finite difference is prohibitively expensive.
$f$ is an entire algorithm with stochastic methods inside.
Q:
- I would like to know if $f$ is smooth in its parameters $\mathbf{\theta}$ for any $\mathbf{x}$.
- If it is not smooth, I would like to know if it is smooth in some parameters $\theta_i$ ?
- Are there local propreties I can exploit to know if it is smooth everywhere?
- Do you have some nice references on this problem so I can dig deeper?
- Ideally, I would like to know if it is smooth for any parameters $\mathbf{\theta}$ and any input $\mathbf{x}$ in the future by exploiting the fact that $\mathbf{x}$ is a set of RGB images.
I'm particularly interested in the analysis of the local properties of the function or in algorithmic approaches to prove almost certainly that the function is smooth.
Thanks :)