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

TL;DR

This paper introduces SPRB, a new stochastic approximation method that adapts to local function behavior, achieving optimal convergence and confidence sequences even in challenging regimes where classical methods struggle.

Contribution

The paper presents SPRB, a novel adaptive stochastic approximation algorithm with theoretical guarantees and practical advantages over Robbins-Monro, especially near flat or discontinuous roots.

Findings

SPRB achieves optimal convergence rate and minimal asymptotic variance.

SPRB attains exponential convergence for discontinuous functions.

SPRB provides nonasymptotic confidence sequences without prior knowledge of convergence rate.

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.