# Toward Asymptotic Optimality: Sequential Unsupervised Regression of Density Ratio for Early Classification

arXiv: https://arxiv.org/abs/2302.09810 · HTML: https://mlsanity.com/paper/2302.09810/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 28 references that are in the map.
- First posted on arXiv 20 Feb 2023.

## Builds on (most cited first, 10 of 10)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [Learning Deep Models for Face Anti-Spoofing: Binary or Auxiliary Supervision](https://mlsanity.com/paper/1803.11097/) (2018)
- [Telescoping Density-Ratio Estimation](https://mlsanity.com/paper/2006.12204/) (2020)
- [Approaches and Applications of Early Classification of Time Series: A Review](https://mlsanity.com/paper/2005.02595/) (2020)
- [Training Neural Networks for Likelihood/Density Ratio Estimation](https://mlsanity.com/paper/1911.00405/) (2019)
- [Stop&Hop: Early Classification of Irregular Time Series](https://mlsanity.com/paper/2208.09795/) (2022)
- [The Power of Log-Sum-Exp: Sequential Density Ratio Matrix Estimation for Speed-Accuracy Optimization](https://mlsanity.com/paper/2105.13636/) (2021)

## Built on by (most cited first, 2 of 2)

- [Early Classification of Time Series: A Survey and Benchmark](https://mlsanity.com/paper/2406.18332/) (2024)
- [Alert: Learning Trigger Functions for Early Classification of Time Series using Deep-RL](https://mlsanity.com/paper/2502.06584/) (2025)

## Abstract (arXiv)

Theoretically-inspired sequential density ratio estimation (SDRE) algorithms are proposed for the early classification of time series. Conventional SDRE algorithms can fail to estimate DRs precisely due to the internal overnormalization problem, which prevents the DR-based sequential algorithm, Sequential Probability Ratio Test (SPRT), from reaching its asymptotic Bayes optimality. Two novel SPRT-based algorithms, B2Bsqrt-TANDEM and TANDEMformer, are designed to avoid the overnormalization problem for precise unsupervised regression of SDRs. The two algorithms statistically significantly reduce DR estimation errors and classification errors on an artificial sequential Gaussian dataset and real datasets (SiW, UCF101, and HMDB51), respectively. The code is available at: https://github.com/Akinori-F-Ebihara/LLR_saturation_problem.