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

arXiv: https://arxiv.org/abs/2501.18059 · HTML: https://mlsanity.com/paper/2501.18059/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jan 2025.

## Built on by (most cited first, 3 of 3)

- [When Evidence is Sparse: Weakly Supervised Early Failure Alerting in Dialogs and LLM-Agent Trajectories](https://mlsanity.com/paper/2606.05414/) (2026)
- [CC-AOS: Cost- and Horizon-Conditioned Amortized Backward Induction for Finite-Horizon Optimal Stopping](https://mlsanity.com/paper/2607.22774/) (2026)
- [Primal--Dual Alternating Neural Learning for Timely Classification with Performance Guarantees](https://mlsanity.com/paper/2608.23480/) (2026)

## Abstract (arXiv)

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