Given a set of data points, how to use gradient descent to find the minimum in the function that passes from those data points?

Question

I have a function with n parameters. I don't know the formula of the function but I can generate as many data points as I want using the function that I have. My question is, how can I find the set of parameters that generate (approximately) minimum value using that function.

In other words,...

1. I have a function F(param1, ..., paramN)

2. Given a set of arguments {arg1, ..., argN}, for which the function yields a result.

3. How can I find the set of arguments that yield a value close to minimum (a local minimum).

My idea was

1. generate some points using different arguments with F(param1, ..., paramN)

2. Apply a curve fitting method on these data points to find a function that passes from these points.

3. Use the generated function formula to apply Gradient Descent and calculate local minimum.

Thanks!

Answer 1

What's commonly used in practice is to use finite-differences methods to approximate the gradient. When that's done, you can simply use gradient descent algorithms as normal.