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Dynamic Perturbed Adaptive Method for Infinite Task-Conflicting Time Series

Jiang You, Xiaozhen Wang, Arben Cela

TL;DR

The paper addresses time series where the same inputs can map to different outputs due to shifting objectives. It introduces a dynamic perturbed adaptive trunk–branch architecture that maintains a slow-evolving trunk and task-specific, reinitialized branches, enabling continual test-time adaptation without explicit task labels and with theoretical guarantees. It proves that the dynamic model has strictly higher expressivity than static networks and LoRA, and establishes exponential convergence under the Polyak–Łojasiewicz condition along with sublinear dynamic regret. Empirically, it demonstrates superior adaptability on a synthetic benchmark with rapid task shifts, achieving fast test-time adaptation and progressive learning compared to strong baselines.

Abstract

We formulate time series tasks as input-output mappings under varying objectives, where the same input may yield different outputs. This challenges a model's generalization and adaptability. To study this, we construct a synthetic dataset with numerous conflicting subtasks to evaluate adaptation under frequent task shifts. Existing static models consistently fail in such settings. We propose a dynamic perturbed adaptive method based on a trunk-branch architecture, where the trunk evolves slowly to capture long-term structure, and branch modules are re-initialized and updated for each task. This enables continual test-time adaptation and cross-task transfer without relying on explicit task labels. Theoretically, we show that this architecture has strictly higher functional expressivity than static models and LoRA. We also establish exponential convergence of branch adaptation under the Polyak-Lojasiewicz condition. Experiments demonstrate that our method significantly outperforms competitive baselines in complex and conflicting task environments, exhibiting fast adaptation and progressive learning capabilities.

Dynamic Perturbed Adaptive Method for Infinite Task-Conflicting Time Series

TL;DR

The paper addresses time series where the same inputs can map to different outputs due to shifting objectives. It introduces a dynamic perturbed adaptive trunk–branch architecture that maintains a slow-evolving trunk and task-specific, reinitialized branches, enabling continual test-time adaptation without explicit task labels and with theoretical guarantees. It proves that the dynamic model has strictly higher expressivity than static networks and LoRA, and establishes exponential convergence under the Polyak–Łojasiewicz condition along with sublinear dynamic regret. Empirically, it demonstrates superior adaptability on a synthetic benchmark with rapid task shifts, achieving fast test-time adaptation and progressive learning compared to strong baselines.

Abstract

We formulate time series tasks as input-output mappings under varying objectives, where the same input may yield different outputs. This challenges a model's generalization and adaptability. To study this, we construct a synthetic dataset with numerous conflicting subtasks to evaluate adaptation under frequent task shifts. Existing static models consistently fail in such settings. We propose a dynamic perturbed adaptive method based on a trunk-branch architecture, where the trunk evolves slowly to capture long-term structure, and branch modules are re-initialized and updated for each task. This enables continual test-time adaptation and cross-task transfer without relying on explicit task labels. Theoretically, we show that this architecture has strictly higher functional expressivity than static models and LoRA. We also establish exponential convergence of branch adaptation under the Polyak-Lojasiewicz condition. Experiments demonstrate that our method significantly outperforms competitive baselines in complex and conflicting task environments, exhibiting fast adaptation and progressive learning capabilities.
Paper Structure (18 sections, 3 theorems, 47 equations, 5 figures, 1 table)

This paper contains 18 sections, 3 theorems, 47 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Work
  3. Method: A Dynamic Perturbed Adaptive Trunk--Branch Architecture
  4. Expressivity of the Trunk–Branch Architecture
  5. Static Model.
  6. Dynamic Model.
  7. Convergence of trunk Updates
  8. Dynamic Regret
  9. Dataset
  10. Data Generation Protocol
  11. Dataset Variants
  12. Experiment Design
  13. Limitations
  14. Conclusions and future work
  15. Proof of Theorem \ref{['thm:expressivity']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mathcal{H}_{\mathrm{static}}$ and $\mathcal{H}_{\mathrm{dyn}}$ be as defined above, with total parameter budget $P$. Suppose we partition $[0,T]$ into $K$ episodes, and for each episode $i = 1,\dots,K$, define Let $n_i$ be the parameter allocation per episode, with $\sum_{i=1}^K n_i = P$. Then with strict inequality whenever $\max_i d_{n_i}(\mathcal{F}_i) < d_P(\mathcal{H}_{\mathrm{static}

Figures (5)

  • Figure 1: a) Illustration of Kernel U-Net. b) We note $U$ the Trunk and $V$ is the Branch for constructing Dynamic Kernel. Trunk evolves slowly through all tasks, and the Branch is reinitialized at each new task. c) In detail when training a dynamic kernel, the phase 1 allows the Branch to adapt the task context, and the pahse 2 allows the model predicting the following values.
  • Figure 2: Illustation of synthetic dataset : (a) Dataset S1; (b) Dataset S2; (c) Dataset S3.
  • Figure 3: Prediction result of Dynamic K-U-Net on Dataset S1. The model consistently captures the target trajectories across all samples, validating its ability to adapt to structured task shifts with minimal error.
  • Figure 4: Prediction result of Dynamic K-U-Net on Dataset S2. Dynamic K-U-Net accurately tracks complex task mappings with asymmetric or partially conflicting objectives, demonstrating robust task-specific adaptation.
  • Figure 5: Prediction result of Dynamic K-U-Net on Dataset S3. Despite increasing ambiguity and overlap between subtasks, the model retains high predictive fidelity, highlighting its adaptability in dynamic and uncertain environments.

Theorems & Definitions (8)

  • Definition 1: Kolmogorov $n$-width
  • Theorem 1: Width Reduction via Time Partitioning
  • Remark 1: Optimal Allocation
  • Theorem 2: Linear Convergence under PL Condition
  • Theorem 3: Sublinear Dynamic Regret
  • proof
  • proof
  • proof