Early Classification of Time Series in Non-Stationary Cost Regimes

Early Classification of Time Series (ECTS) addresses decision-making problems in which predictions must be made as early as possible while maintaining high accuracy. Most existing ECTS methods assume that the time-dependent decision costs governing the learning objective are known, fixed, and correctly specified. In practice, however, these costs are often uncertain and may change over time, leading to mismatches between training-time and deployment-time objectives. In this paper, we study ECTS under two practically relevant forms of cost non-stationarity: drift in the balance between misclassification and decision delay costs, and stochastic realizations of decision costs that deviate from the nominal training-time model. To address these challenges, we revisit representative ECTS approaches and adapt them to an online learning setting. Focusing on separable methods, we update only the triggering model during deployment, while keeping the classifier fixed. We propose several online adaptations and baselines, including bandit-based and RL-based approaches, and conduct controlled experiments on synthetic data to systematically evaluate robustness under cost non-stationarity. Our results demonstrate that online learning can effectively improve the robustness of ECTS methods to cost drift, with RL-based strategies exhibiting strong and stable performance across varying cost regimes.

💡 Research Summary

This paper tackles a largely overlooked aspect of Early Classification of Time Series (ECTS): the non‑stationarity of decision costs at deployment time. Traditional ECTS methods assume that the misclassification cost C_m and the delay cost C_d are known, fixed, and correctly specified during training. In many real‑world scenarios, however, the relative importance of accuracy versus earliness drifts over time, or the actual costs are stochastic realizations that differ from the nominal values used for training. The authors formalize two practical forms of cost non‑stationarity: (i) drift in the balance between C_m and C_d, and (ii) stochastic cost realizations drawn from a predefined distribution. They introduce a deployment‑time index u to denote successive incoming series, allowing C_m^{(u)} and C_d^{(u)} to vary with u.

The study focuses on separable ECTS approaches, where a classifier is trained offline once on complete series under nominal costs, and only the triggering model (the decision‑time policy) is adapted online. This design reflects the intuition that cost changes affect when to act rather than what the underlying class is. The paper adapts several well‑known separable methods to the online setting:

1. Prob‑Threshold – a confidence‑based trigger that fires when the maximum posterior probability exceeds a threshold. The online version maintains a set of candidate thresholds, estimates their average loss from recent feedback, and applies exponential decay to forget older observations.

2. Economy – an expectation‑based trigger that predicts when the current expected total cost is lower than any future expected cost. Online updates recompute expected costs using the latest observed losses.

3. Calimera – a backward‑induction regression model that predicts future costs and selects the current time step with minimal predicted total cost. The authors replace batch regression with online stochastic gradient updates.

4. Alert – a reinforcement‑learning (RL) based trigger that treats the prefix of the series as the state and the actions “predict” or “wait” as the policy. The reward is the realized loss L^{(u)}. Because RL receives immediate reward after each decision, it supports instant updates.

5. Bandit‑based triggers – each candidate threshold is treated as an arm. The authors employ UCB or Thompson Sampling with discounting or sliding‑window mechanisms to cope with drifting costs.

Three update regimes are distinguished: delayed updates (feedback only after the full series is observed), instant updates (feedback immediately after stopping, as in RL), and no updates (fixed policy based on a priori cost information).

Experiments are conducted on synthetic time‑series data where the authors can precisely control cost drift (linear, stepwise, or periodic changes in the C_m/C_d ratio) and stochastic cost variability (costs sampled from normal, beta, or mixture distributions). Evaluation metrics include average loss, classification accuracy, average decision time, and sensitivity to cost changes.

Results show that methods that keep a static trigger are highly vulnerable to cost drift, with loss increasing sharply as the cost balance shifts. Decay‑enhanced Prob‑Threshold and the bandit‑based triggers adapt better but still suffer under abrupt or highly noisy cost changes. The RL‑based Alert consistently achieves the lowest average loss across all drift and stochastic scenarios, maintaining a stable trade‑off between accuracy and earliness even when costs change rapidly or are highly variable.

The contribution of the paper is threefold: (1) it provides the first systematic study of ECTS under non‑stationary costs; (2) it proposes a suite of online adaptation strategies—including bandit and RL approaches—that require updating only the trigger while leaving the classifier untouched; (3) it demonstrates through controlled experiments that online adaptation, especially RL‑based policies, markedly improves robustness to cost drift and stochasticity.

The work opens several avenues for future research: integrating data‑drift detection with cost‑drift adaptation, learning predictive models of future costs to inform the trigger, and validating the proposed methods on real‑world datasets such as medical monitoring, predictive maintenance, or industrial defect detection. By showing that online trigger adaptation can bridge the gap between training‑time assumptions and deployment‑time realities, the paper makes a significant step toward practical, cost‑aware early decision‑making systems.