Modeling Dynamic Mixtures of Time-Delay Systems from Streaming Time Series

Matrix-valued time series arise in a wide range of applications, such as spatio-temporal data from medical imaging and geophysics. Existing methods are mainly designed for static settings and lack adaptability to streaming and time-varying environments. Adaptive filtering techniques have also been largely limited to data with scalar or vector values, leaving adaptive forecasting for matrix-valued time series inadequately understood. To bridge these gaps, we develop an adaptive tensor regression framework that includes Matrix-on-Matrix (MoM) and Tensor-on-Matrix (ToM) formulations for streaming matrix-valued prediction. The two formulations differ in whether to directly model matrix-valued outputs or to exploit temporal structure via higher-order tensor representations. For the proposed tensor regression framework, we develop stochastic gradient descent (SGD) algorithms for online learning. We show that stacking multiple responses across time into higher-order tensors improves performance; in particular, the ToM achieves lower steady-state error and stronger denoising capability than MoM, motivating our focus on the ToM model. We further characterize the tracking behavior of SGD under time-varying dynamics. From a statistical perspective, we establish fixed-time recovery guarantees for ToM under general low-dimensional structures, including sparsity, low-rankness, and their joint sparselow-rank models.

In standard reinforcement learning (RL) settings, the interaction between the agent and the environment is typically modeled as a Markov decision process (MDP), which assumes that the agent observes the system state instantaneously, selects an action without delay, and executes it immediately. In real-world dynamic environments, such as cyber-physical systems, this assumption often breaks down due to delays in the interaction between the agent and the system. These delays can vary stochastically over time and are typically unobservable when deciding on an action. Existing methods deal with this uncertainty conservatively by assuming a known fixed upper bound on the delay, even if the delay is often much lower. In this work, we introduce the interaction layer, a general framework that enables agents to adaptively handle unobservable and time-varying delays. Specifically, the agent generates a matrix of possible future actions, anticipating a horizon of potential delays, to handle both unpredictable delays and lost action packets sent over networks. Building on this framework, we develop a model-based algorithm, Actor-Critic with Delay Adaptation (ACDA), which dynamically adjusts to delay patterns. Our method significantly outperforms state-of-the-art approaches across a wide range of locomotion benchmark environments, including real-world measured delays.

Non-stationary time series forecasting is challenged by evolving distribution shifts that static models struggle to capture. While Mixture-of-Experts (MoE) architectures offer a promising paradigm for decoupling complex drift patterns, existing approaches are limited by fixed expert pools and memoryless routing, hampering their ability to adapt to abrupt regime shifts. To address this, we propose Dynamic TMoE, a framework that unifies architectural evolution with temporal continuity during learning phase. By detecting distribution shifts via Maximum Mean Discrepancy (MMD), we dynamically instantiate heterogeneous experts and prune redundant ones to optimize capacity. Additionally, a temporal memory router leverages recurrent states and an anomaly repository to ensure stable, context-aware expert selection without requiring test-time updates. Experiments on nine benchmarks demonstrate state-of-the-art performance, reducing MSE by 10.4% and MAE by 7.8%. Code is available at https://github.com/andone-07/Dynamic-TMoE.