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

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