Learning the Optimal Stopping for Early Classification within Finite Horizons via Sequential Probability Ratio Test

TL;DR

This paper introduces FIRMBOUND, an efficient SPRT-based framework for optimal early classification in finite horizon time series, balancing speed and accuracy with theoretical guarantees and practical speedups.

Contribution

FIRMBOUND bridges optimal stopping theory and real-world deployment by efficiently estimating backward induction, employing density ratio estimation and convex functions for consistent statistical estimation.

Findings

Achieves optimal Bayes risk in early classification tasks.

Reduces decision-time variance for reliable decisions.

Offers a faster Gaussian process regression alternative with minor statistical tradeoffs.

Abstract

Time-sensitive machine learning benefits from Sequential Probability Ratio Test (SPRT), which provides an optimal stopping time for early classification of time series. However, in finite horizon scenarios, where input lengths are finite, determining the optimal stopping rule becomes computationally intensive due to the need for backward induction, limiting practical applicability. We thus introduce FIRMBOUND, an SPRT-based framework that efficiently estimates the solution to backward induction from training data, bridging the gap between optimal stopping theory and real-world deployment. It employs density ratio estimation and convex function learning to provide statistically consistent estimators for sufficient statistic and conditional expectation, both essential for solving backward induction; consequently, FIRMBOUND minimizes Bayes risk to reach optimality. Additionally, we present a faster alternative using Gaussian process regression, which significantly reduces training time while retaining low deployment overhead, albeit with potential compromise in statistical consistency. Experiments across independent and identically distributed (i.i.d.), non-i.i.d., binary, multiclass, synthetic, and real-world datasets show that FIRMBOUND achieves optimalities in the sense of Bayes risk and speed-accuracy tradeoff. Furthermore, it advances the tradeoff boundary toward optimality when possible and reduces decision-time variance, ensuring reliable decision-making. Code is publicly available at https://github.com/Akinori-F-Ebihara/FIRMBOUND