Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2501.18059

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

Akinori F. Ebihara·Taiki Miyagawa·Kazuyuki Sakurai·Hitoshi Imaoka
View original source ↗Read source document ↗Export metadata ↓

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

Explore related subjects

cs.LGcs.AI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Akinori F. Ebihara, Taiki Miyagawa, Kazuyuki Sakurai, Hitoshi Imaoka. 2025-01-29. Learning the Optimal Stopping for Early Classification within Finite Horizons via Sequential Probability Ratio Test. https://arxiv.org/abs/2501.18059

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Analysis of Regularized Learning in Banach Spaces for Linear-functional Data

This article delves into the study of the theory of regularized learning in Banach spaces for linear-functional data. It encompasses discussions on representer theorems, pseudo-approximation theorems, and convergence theorems. Regularized learning is designed to minimize regularized empirical risks over a Banach space. The empirical risks are calculated by utilizing training data and multi-loss functions. The input training data are composed of linear functionals in a predual space of the Banach space to capture discrete local information from multimodal data and multiscale models. Through the regularized learning, approximations of the exact solution to an unidentified or uncertain original problem are globally achieved. In the convergence theorems, the convergence of the approximate solutions to the exact solution is established through the utilization of the weak* topology of the Banach space. The theorems of regularized learning are utilized in the interpretation of classical machine learning, such as support vector machines and artificial neural networks.

cs.LG↗
arXiv2026

On Minimal Depth in Neural Networks

Understanding the relationship between the depth of a neural network and its representational capacity is a central problem in deep learning theory. In this work, we develop a geometric framework to analyze the expressivity of ReLU networks with the notion of depth complexity for convex polytopes. The depth of a polytope recursively quantifies the number of alternating convex hull and Minkowski sum operations required to construct it. This geometric perspective serves as a rigorous tool for deriving depth lower bounds and understanding the structural limits of deep neural architectures. We establish lower and upper bounds on the depth of polytopes, as well as tight bounds for classical families. These results yield two main consequences. First, we provide a purely geometric proof of the expressivity bound by Arora et al. (2018), confirming that $\lceil \log_2(n+1)\rceil$ hidden layers suffice to represent any continuous piecewise linear (CPWL) function. Second, we prove that, unlike general ReLU networks, convex polytopes do not admit a universal depth bound. Specifically, the depth of cyclic polytopes in dimensions $n \geq 4$ grows unboundedly with the number of vertices. This result implies that Input Convex Neural Networks (ICNNs) cannot represent all convex CPWL functions with a fixed depth, revealing a sharp separation in expressivity between ICNNs and standard ReLU networks.

cs.LG↗
arXiv2026

ELEMENT: Episodic and Lifelong Exploration via Maximum Entropy

Reinforcement learning agents depend on reward signals whose density is rarely under the designer's control, and when such signals are absent, an agent must generate its own drive to explore. State entropy maximization offers a principled objective for this, but existing methods break down at scale in two ways: the intrinsic reward vanishes once a state has been visited, discouraging revisits to the very gateways that lead onward, and estimating entropy over millions of accumulated observations becomes computationally prohibitive. We address both with Episodic and Lifelong Exploration via Maximum Entropy (ELEMENT), a multiscale intrinsically motivated framework for reward-free exploration that transfers to downstream tasks. ELEMENT couples lifelong entropy maximization with a complementary episodic term acting on a faster timescale. For the episodic term, we derive average episodic state entropy, an intrinsic reward that is the exact minimizer of a tractable upper bound on the reward-decomposition objective; for the lifelong term, we propose a $k$NN graph-based estimator that keeps entropy tractable without forgetting. ELEMENT consistently outperforms state-of-the-art intrinsic reward baselines on state coverage and unsupervised pre-training. Videos, code, and supplementary material: https://sites.google.com/view/element-rl.

cs.LG↗