The Root Finding Problem Revisited: Beyond the Robbins-Monro procedure

Abstract

We introduce Sequential Probability Ratio Bisection (SPRB), a novel stochastic approximation algorithm that adapts to the local behavior of the (regression) function of interest around its root. We establish theoretical guarantees for SPRB's asymptotic performance, showing that it achieves the optimal convergence rate and minimal asymptotic variance even when the target function's derivative at the root is small (at most half the step size), a regime where the classical Robbins-Monro procedure typically suffers reduced convergence rates. Further, we show that if the regression function is discontinuous at the root, Robbins-Monro converges at a rate of $1/n$ whilst SPRB attains exponential convergence. If the regression function has vanishing first-order derivative, SPRB attains a faster rate of convergence compared to stochastic approximation. As part of our analysis, we derive a nonasymptotic bound on the expected sample size and establish a generalized Central Limit Theorem under random stopping times. Remarkably, SPRB automatically provides nonasymptotic time-uniform confidence sequences that do not explicitly require knowledge of the convergence rate. We demonstrate the practical effectiveness of SPRB through simulation results.