You could try stochastic optimization methods. The rough idea is to leverage noise to efficiently explore the landscape of $f(t)$. To quote from the linked article, the injected randomness may enable the method to escape a local optimum and eventually to approach a global optimum.
To illustrate this point, suppose you knew that the only point where $f'(t)=0$ is at the minimizer, then you could use a numerical solution to the SDE: $$ d Y = - f'(Y) dt + f'(Y) dW $$ to find the minimizer. Here $W$ is a standard Brownian motion. Note that this SDE has a fixed point at the minimizer, and away from the minimizer it efficiently explores $f(t)$.
As a concrete test, consider $f(x) = 1/2 (x-1/2)^2 + \epsilon \cos(10 x \pi)$, which has a global minimum at $x=1/2$ as shown in the figure below with $\epsilon=0.01$. Starting from the initial condition $0.92$ the method described above converges like a charm to the minimum at $0.5$ in just $32$ steps despite the fact that $f(t)$ is a bit bumpy.
The dots in this figure represent the points along a numerical solution of the SDE by the simple Euler-Maruyama scheme.