How do you implement adaptive step sizes during sampling?

Implementing adaptive step sizes during sampling involves dynamically adjusting the step size based on real-time feedback to balance computational efficiency and accuracy. The core idea is to monitor a metric—like local error estimates or acceptance rates—and then increase or decrease the step size to maintain desired performance. For example, in numerical integration or solving differential equations, a larger step might be used in smooth regions but reduced when the system changes rapidly. Similarly, in Markov Chain Monte Carlo (MCMC) sampling, the step size (e.g., proposal distribution width) is tuned to maintain an optimal acceptance rate, ensuring the sampler explores the parameter space effectively.

A common method for adaptive step sizing in numerical methods is the embedded Runge-Kutta approach (e.g., RK45). Here, two approximations of different orders (e.g., 4th and 5th) are computed at each step. The difference between them provides an error estimate. If the error exceeds a tolerance threshold, the step is rejected, and the step size is reduced (e.g., by halving). If the error is below the threshold, the step is accepted, and the step size may increase (e.g., doubled). In MCMC, adaptation often targets an acceptance rate between 20-40%. For instance, if the acceptance rate over a window of steps is too low (e.g., 10%), the proposal step size is reduced (e.g., multiplied by 0.9) to avoid overly aggressive moves; if it’s too high (e.g., 60%), the step size increases (e.g., multiplied by 1.1) to encourage exploration.

When implementing adaptive steps, practical considerations are key. First, set bounds on minimum and maximum step sizes to prevent numerical instability or excessive computation. Second, use smoothing techniques—like exponential moving averages for acceptance rates—to avoid overreacting to short-term fluctuations. Third, balance computational overhead: frequent step size adjustments improve accuracy but add cost. For example, in MCMC, tuning might occur every 100 steps to reduce noise. Libraries like SciPy’s ODE solvers or PyMC3’s MCMC samplers automate these steps, but custom implementations require careful testing to ensure convergence and stability across diverse scenarios.