Adaptation Under Non-Stationary Dynamics

• Adaptation under non-stationary dynamics is a field focused on developing systems that dynamically adjust to evolving environments and statistical shifts.
• It employs techniques such as continuous reparameterization, sliding-window approaches, and change-point detection to balance outdated and current data.
• Empirical validations in areas like power systems, reinforcement learning, and communication networks demonstrate its effectiveness in mitigating performance degradation.

Adaptation under non-stationary dynamics refers to the design and analysis of learning systems, algorithms, and controllers that can maintain or quickly recover performance when the statistical properties, physical mechanisms, or operational environment underlying a system evolve over time. This challenge is pervasive across time series forecasting, reinforcement learning, distributed systems, and real-world control domains such as energy grids, robotics, and communication networks, where exogenous or endogenous drivers induce time-varying dynamics. Adaptation strategies in non-stationary contexts integrate mechanisms for detecting regime changes, dynamically rebalancing past and present information, leveraging external auxiliary signals, and preserving model flexibility and plasticity across multiple forms of temporal variability.

1. The Non-Stationarity Problem: Sources and Implications

Non-stationary dynamics arise when the joint distribution over system states, actions, observations, or rewards drifts due to internal evolution, environmental perturbations, operational changes, or policy interventions. In power systems, this manifests as shifts in generation and demand due to weather, renewables, and climate events, which introduce abrupt and gradual changes in load profiles and coupling between exogenous variables and system trajectories (Li et al., 23 May 2025). In network systems, non-stationarity appears as abrupt bandwidth fluctuations, congestion events, or technology transitions, which can invalidate previously learned resource allocation policies (He et al., 2 May 2025). In machine learning models, real-world deployment frequently exposes models to continuously evolving data distributions, making adaptation critical to avoid catastrophic forgetting or performance collapse (Park et al., 2023).

Key challenges of adaptation under non-stationarity include:

• Model misspecification when stationary assumptions fail.
• Inadequate reactivity to abrupt or gradual context shifts.
• Poor utilization of asynchronous or sparse exogenous signals.
• Loss of model plasticity, impairing relearning and resilience to new patterns.

2. Methodological Foundations for Adaptation

A wide variety of methodological paradigms for adaptation have been developed in response to non-stationarity, including:

a) Continuous Parameterization and Environment-Driven Hypernetworks:

ExARNN approaches non-stationarity by integrating environmental variables (e.g., weather, time) as “meta-knowledge” for on-the-fly dynamic reparameterization of model internals. Specifically, a hierarchical hypernetwork driven by a Neural Controlled Differential Equation (NCDE) supports continuous fusion of exogenous measurements into RNN parameters, enabling time-adaptive forecasting even when auxiliary data are sparse and asynchronous (Li et al., 23 May 2025).

b) Sliding-Window and Restarting Schemes:

In sequential decision-making and time series forecasting, sliding window estimators and periodic resetting of static-optimized algorithms are used to limit the influence of outdated data and promote fast adaptation to new regimes. Crucially, the optimal window size or restart frequency is adapted to the rate of change in the underlying dynamics, balancing bias from stale data against variance from reduced sample sizes (Cheung et al., 20191307.54492310.18304).

c) Change-Point Detection and Model Switching:

Online statistical tests such as Cumulative Sum (CUSUM) statistics are deployed for high-confidence, low-delay detection of distribution shifts. Upon detection, systems can initiate new model instantiations, switch to a maintained pool of context-specific models, or update online regression/forecasting models tailored to the new regime (Alegre et al., 2021).

d) Layerwise and Componentwise Selectivity:

Auto-weighted adaptation schemes, such as FIM-based layer-wise learning rate selection, monitor the Fisher Information Matrix for each neural network layer to focus adaptation on model components most predictive for the current distribution, while freezing others to mitigate catastrophic forgetting and instability during continuous adaptation (Park et al., 2023).

e) Plasticity Preservation via Neuron Resets:

Neural network controllers risk entering “plasticity collapse,” with neurons persistently inactive in both forward and backward passes (silent neurons). Targeted reset strategies, as in the ReSiN algorithm, use joint forward- and backward-activity measures to identify and periodically reset such units, ensuring the network remains capable of relearning new patterns as the environment shifts (He et al., 2 May 2025).

3. Integration of Exogenous Signals and Mixed-Rate Data

Non-stationary systems frequently generate environmental or contextual auxiliary signals that improve adaptation, but these signals may arrive at irregular rates or with partial coverage relative to primary measurements. The ExARNN framework explicitly constructs a continuous-time latent embedding of exogenous vectors via a spline-interpolated NCDE, yielding a temporally-aligned feature at each target prediction step. This enables seamless adaptation even when, for instance, power measurements are recorded every 15 minutes while weather data are only available hourly—without manual upsampling or imputation (Li et al., 23 May 2025).

Such environment-driven parameter generation concretely connects fast and slow time scales, leveraging both high-frequency signals and slow, informative drifts in auxiliary data to dynamically modulate the forecasting or control policy.

4. Optimization and Regret Analyses under Non-Stationarity

Adaptation schemes are often judged by dynamic regret—the cumulative loss relative to a sequence of instantaneous or locally optimal policies—as opposed to static or stationary regret objectives. Key theoretical insights include:

• When the cumulative variation in system dynamics grows sublinearly (i.e., total variation budget $V_T = o(T)$), sublinear dynamic regret is achievable via appropriately windowed or restarted learners (Besbes et al., 2013).
• Upper bounds often take the form $R_T = O(V_T^\alpha T^{1-\alpha})$, with the exponent $\alpha$ depending on convexity, feedback, and noise (Besbes et al., 2013).
• Selecting look-back window size via empirical risk or stability principles guarantees minimax-optimal regret rates even without knowledge of the non-stationarity rate (Huang et al., 2023).
• Bandit and online optimization algorithms benefit from meta-algorithms (e.g., Bandit-over-Bandit) for online selection of windowing parameters (Cheung et al., 2019).

Table 1: Example Regret Rates under Non-Stationary Dynamics (Besbes et al., 2013)

Problem Structure	Variation Budget $V_T$	Minimax Dynamic Regret
Convex/Noisy Grad.	$o(T)$	$O(V_T^{1/3} T^{2/3})$
Strongly Convex	$o(T)$	$O(V_T^{1/2} T^{1/2})$

These analyses justify window-based or reinitialization schemas that attempt to match the local stationarity horizon of the evolving process.

5. Empirical Validation and Componentwise Analyses

Empirical studies validate the efficacy and necessity of adaptive mechanisms:

• ExARNN achieves superior mean absolute percentage error (MAPE) and MSE on non-stationary power datasets compared to vanilla RNNs, RNNs with time encoding, ODE-RNNs, and NCDEs without dynamic parameterization, substantiating the need for both continuous fusion of environment data and dynamic parameter adaptation (Li et al., 23 May 2025).
• Plasticity-aware resets (ReSiN) in adaptive bitrate streaming double average throughput and quality-of-experience compared to standard PPO or output-activity-only reset strategies by preventing long-term degeneration of the neural controller (He et al., 2 May 2025).
• Fisher information-based layer-freezing/attenuation reduces error accumulation in continual test-time adaptation, showing the critical role of structure-aware selectivity (Park et al., 2023).

Componentwise ablation reveals:

• ODE-based hidden state evolution alone is insufficient; controlled integration of exogenous data and parametric hypernetworks are both crucial for robust adaptation.
• Purely output-based dormancy indices are inadequate as plasticity detectors; joint forward–backward statistics are required (He et al., 2 May 2025).

6. Domains of Application and Generalization

Adaptation under non-stationary dynamics is foundational across diverse domains:

• Power systems forecasting with mixed-frequency signals (Li et al., 23 May 2025).
• Reinforcement learning for network resource control, robotics, and streaming media in highly dynamic environments (Akgül et al., 3 Mar 2025He et al., 2 May 2025Alegre et al., 2021).
• Continual test-time adaptation on evolving vision benchmarks (Park et al., 2023Shkodrani et al., 2018).
• Bandit decision-making under temporal drift (Cheung et al., 2019Gornet et al., 14 Jun 2024).

Notably, frameworks generalize across batch (offline, retraining), streaming (online adaptation), and episodic (RL) settings, with design principles carrying over through notions of windowing, statistical testing, dynamic parameterization, and plasticity maintenance.

7. Outlook: Algorithmic and Theoretical Frontiers

The adaptation landscape is evolving towards higher granularity and responsiveness:

• Hierarchical models that hierarchically integrate multiple external factors with different rates of change.
• Incorporation of causal structure and latent change factors in reinforcement learning for factorized, interpretable adaptation (Feng et al., 2022).
• Online change-point detection framed in calibrated statistical decision theory to minimize adaptation delay and false alarms (Alegre et al., 2021).
• Combination of continual adaptation and computational efficiency, employing selective updating under resource constraints (Park et al., 2023).

Remaining open challenges include theoretical characterization of plasticity maintenance, scaling continuous adaptation to high-dimensional end-to-end systems, adaptive integration of mixed-rate and mixed-modal exogenous information, and formal guarantees under adversarial non-stationarity.

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References

• "ExARNN: An Environment-Driven Adaptive RNN for Learning Non-Stationary Power Dynamics" (Li et al., 23 May 2025).
• "Understanding and Exploiting Plasticity for Non-stationary Network Resource Adaptation" (He et al., 2 May 2025).
• "Layer-wise Auto-Weighting for Non-Stationary Test-Time Adaptation" (Park et al., 2023).
• "A Stability Principle for Learning under Non-Stationarity" (Huang et al., 2023).
• "Non-stationary Stochastic Optimization" (Besbes et al., 2013).
• "Hedging the Drift: Learning to Optimize under Non-Stationarity" (Cheung et al., 2019).
• "Minimum-Delay Adaptation in Non-Stationary Reinforcement Learning via Online High-Confidence Change-Point Detection" (Alegre et al., 2021).
• "Efficient Contextual Bandits in Non-stationary Worlds" (Luo et al., 2017).
• "Factored Adaptation for Non-Stationary Reinforcement Learning" (Feng et al., 2022).
• "Dynamic Adaptation on Non-Stationary Visual Domains" (Shkodrani et al., 2018).
• "Overcoming Non-stationary Dynamics with Evidential Proximal Policy Optimization" (Akgül et al., 3 Mar 2025).
• "An Adaptive Method for Contextual Stochastic Multi-armed Bandits with Rewards Generated by a Linear Dynamical System" (Gornet et al., 14 Jun 2024).