To implement adaptive step sizes during sampling, you can utilize methods that adjust the step size based on the behavior of the function being sampled. The idea is to change the size of the steps taken during the sampling process in response to the local characteristics of the function. This is often seen in techniques like Adaptive Metropolis or Hamiltonian Monte Carlo, where the objective is to achieve more efficient sampling with fewer iterations.
One common approach is to monitor the acceptance rate of proposed samples. For instance, if you're using a Markov Chain Monte Carlo (MCMC) method, you can start with an initial step size and observe how many proposed samples are accepted versus rejected. If the acceptance rate is too high (usually above 70%), it suggests the step size might be too small, and you can increase it to explore the space more effectively. Conversely, if the acceptance rate is low (below 30%), it indicates the step size may be too large, resulting in poor exploration. Adjusting these values dynamically can lead to a more balanced exploration between acceptance and rejection.
Another method involves calculating the gradients of the function as you sample. By measuring how quickly the function value changes, you can adaptively adjust the step size during the sampling process. For instance, if the function changes rapidly, a smaller step size is warranted to avoid overshooting, while in flatter regions, a larger step size could be beneficial. The combination of acceptance rates and gradient information allows for a more nuanced approach to sampling, helping improve convergence times and the quality of the samples collected.