Non-Stationarity

Concept drift is a common phenomenon in real-world data streams, in which changes in the data-generating distribution can degrade predictive model performance. Most existing definitions characterize drift as changes in the joint distribution $P(\mathbf{x}, y)$, without distinguishing which component of the data-generating process has changed. In this work, we introduce a causal perspective on concept drift based on Structural Causal Models (SCMs). We propose a taxonomy that categorizes drift events by their causal origin, including changes in exogenous variables, endogenous mechanisms, confounders, and target-generating processes. Building on this framework, we develop an SCM-based data stream generator that simulates controlled mechanism-level drift events. Our experiments empirically characterize the distributional effects of each drift type and show that drifts with different causal origins induce distinct patterns of distribution shift and predictive behavior. Furthermore, by integrating causal discovery methods, we use our framework to construct data streams grounded in real-world dependency structures, enabling more realistic and informative evaluation scenarios. We also demonstrate that leveraging the generated data can improve downstream performance. These results highlight the importance of accounting for causal structure when studying and evaluating adaptive learning methods, and establish a foundation for causally-aware evaluation in non-stationary environments.

Decentralized bandit systems often contain heterogeneous agents: rewards can change at individual agents even when the best action for the network stays the same. These local changes may cancel when rewards are averaged across agents, so the number of local changes $\Stloc$ can be much larger than the number of changes in the best common arm $\Stdec$. We introduce Decision-Relevant Fresh Comparison (DRFC), which uses new, balanced samples from all agents to compare arms at the network level and switches only when fresh global evidence indicates that the common best arm has changed. We prove a high-probability dynamic regret bound with no adaptation term depending on $\Stloc$, and show that every algorithm must still pay for identifying genuine decision switches and propagating them through the communication graph. Under a distinct time-average benchmark, an anytime-valid sliding-window extension handles gradual drift; experiments on synthetic, semi-real, and MovieLens-1M replays show that DRFC ignores decision-irrelevant local changes while the extension avoids false switches.

This article introduces PeriodicCALM, an effective real-time anomaly detection framework designed for cyclostationary data streams. While classical cyclostationary processes feature periodically time-varying statistical properties, real-world signals often contain recurring impulsive components that conceal abnormal behavior. Existing real-time methods for struggle with these dynamics, frequently misinterpreting phase-dependent variability as non-cyclic anomalies and causing excessive false alarms. To address this, PeriodicCALM incorporates cycle-dependent variability to systematically ignore regular cyclic impulses while accurately isolating genuine anomalies. Operating in real time with continuous retraining capabilities, the method adapts dynamically to evolving signal characteristics. Comparative evaluations against the baseline CALM framework using simulated data demonstrate significant improvements in detection accuracy and training efficiency, alongside a reduction in prediction latency. Furthermore, the practical utility of PeriodicCALM is validated on real-world vibration signals collected from a compressor monitoring system.

Ensuring safety in reinforcement learning under nonstationarity requires anticipating changes in risk before they lead to unsafe behavior. Existing approaches typically rely on safety constraints defined at design time or updated reactively during execution, assuming that such constraints remain valid over time. However, in nonstationary environments with evolving contexts and changing driving layouts, these assumptions may fail. We propose a framework for proactive safety constraint generation based on context forecasting. The approach infers latent environmental context from observations, predicts its future evolution, and constructs safety constraints adapted to anticipated conditions. This enables the agent to proactively avoid unsafe regions instead of reacting only after safety violations occur. We evaluate the method in driving environments with structured context variation. The experiments include a sweep over nonstationarity intensities and additional held-out driving layouts, including highway, intersection, and racetrack scenarios. Results show that proactive constraint generation substantially reduces collisions under both seen and out-of-training nonstationarity intensities and generally remains effective across held-out driving layouts while maintaining usable task performance. These findings suggest that context-based constraint generation is a promising approach for safe reinforcement learning under nonstationarity.

Standard semi-supervised learning (SSL) typically relies on labelled and unlabelled data sharing a common marginal distribution. This assumption is often violated by biased spatial sampling mechanism, when labels are collected under spatially biased or preferential site selection. We treat this marginal mismatch, spatial autocorrelation, and spatial non-stationarity as three distinct mechanisms, varied independently via a labelled-sampling concentration parameter, a spatial length scale, and a non-stationarity strength parameter, and ask how mismatch degrades SSL, whether the cluster and manifold assumptions survive it, and how the resulting failure can be diagnosed. Using a controlled synthetic framework alongside PovertyMap-WILDS, California housing, socio-economic and US air quality monitoring datasets, we systematically vary the degree of mismatch while accounting for spatial autocorrelation and non-stationarity. Through a series of analyses including a segmented-regression changepoint, we show that in the synthetic generator, SSL performance does not degrade gradually but instead exhibits a threshold-like breakdown between approximately 0.71 and 0.77 once distribution mismatch becomes sufficiently severe. We further demonstrate that spatial non-stationarity contributes to performance loss independently of marginal mismatch and that models become increasingly overconfident outside the regions where labels are available. To support practical deployment, we evaluate several distribution-divergence measures as indicators of reliability and introduce a kernel-weighted local divergence metric that provides a more stable estimate of spatial mismatch than a naïve localised approach. These findings provide empirical evidence and diagnostic tools for better documenting the risk of incorporating unlabelled spatial data into semi-supervised learning workflows.

Online adaptation can help edge time-series forecasting under distribution drift, but its measured benefit is sensitive to evaluation choices. We study six public multivariate streams, including building-sensor and smart-meter data, under a leakage-free streaming protocol. We identify two additional sources of comparison bias. First, the warmup budget of the static baseline has a two-sided effect: insufficient warmup undertrains the baseline, whereas excessive warmup can degrade its pre-drift generalization. Across six dataset-backbone settings, the estimated adaptation benefit changes by 3.0 to 18.8 percentage points (pp) over the 1,000-20,000-step warmup range. Second, comparing SGD with momentum (SGD+m) and Adam at a shared default learning rate conflates optimizer quality with rate sensitivity. We select both the warmup budget and each optimizer's online rate using a held-out pre-drift validation slice without accessing test data. Under this validation-only procedure, Adam outperforms SGD+m in 310 of 360 evaluated cells, while 4 Adam cells remain below the static baseline. We further characterize accuracy against adaptation-state memory and A100-measured per-update latency for full, head-only, and calibration-based adaptation. In the evaluated PatchTST frontier settings, several parameter-efficient variants are nondominated on the adaptation-state-memory axis. Smart-meter analyses also show that reported gains depend on meter-selection rules. These findings support a validation-only commissioning procedure, while target-device latency and energy remain to be measured. Code, data, and all reported numbers: https://github.com/keiotakmin/tsf-edge-adaptation.

Machine learning systems are increasingly corrected while they run, and the decision of when to intervene is increasingly delegated to statistical monitors. Anytime-valid inference promises evidence that can be acted on at any moment, exactly the guarantee this setting needs, and it is moving from theory into deployed monitoring. Conformal test martingales are the change-detection instrument, and Ville's inequality caps their false-alarm probability on exchangeable data. The guarantee is conditional. A deployment inherits it only if the stream it monitors behaves exchangeably. The premise is hardest to satisfy where these monitors are most useful, on dependent data and inside loops where the monitor modifies the learner whose scores it reads. It is also rarely measured. We measure it in a pre-specified case study, where such a monitor gates the online updates of a Kalman adapter correcting frozen time-series foundation models on five forecasting streams. On exchangeable synthetic streams, the same implementation fires in at most 1 of 60 runs. On the real streams, at alpha = 0.05, 135 of 135 clean-stream runs fired. The construction does not explain the firing; the failure comes from the deployed score stream itself. Repeated fires hold the gate's drift response active, and the gated filter amplifies the very transient it was designed to prevent. The component worth keeping makes no validity claim. Huber-style gating of the filter's own updates cuts isolated-spike degradation by an order of magnitude with no dataset specific tuning. Anytime-valid methods proposed for dependent data should therefore be accompanied by null-calibration controls and mechanism traces.

Training effective foundation models requires massive and organized datasets, yet scientific domains such as nuclear fusion present unique challenges due to largely heterogeneous and sparse data. Here we characterize the data used in developing such a model: with over 20 sensor types spanning 5 orders of magnitude in sampling rate, mixed tensor structures (point measurements, spectrograms, images), and nonstationary physics. We analyze our input complexity and discuss trade-offs between temporal context and frequency resolution. Our analysis provides a template for representing multi-modal fluctuation data at scale, with implications for both multi-modal control systems and nuclear fusion.

Multivariate time series imputation is fundamental to downstream analysis, yet modeling inter-variable dependencies with incomplete observations remains challenging. Existing methods learn global dependencies across samples or dynamic local dependencies per sample. Global dependencies are stable but adapt poorly to sample variations and temporal non-stationarity, whereas local dependencies are adaptive yet unreliable when observations are insufficient, causing erroneous information propagation. To address these limitations, we propose GLAIM, a Global-Local Adaptive Inter-variable Dependency Modeling framework for multivariate time series imputation. GLAIM comprises two complementary components. The Stable Global Dependency Constructor derives robust global inter-variable dependencies from complementary temporal representations, providing a stable backbone less affected by sample-specific missingness and noise. The Sample-Conditioned Dependency Refiner adapts this backbone to each sample and time step using its temporal state and available observations, enabling reliable local refinement under incomplete observations. Extensive experiments on nine real-world datasets demonstrate that GLAIM achieves state-of-the-art performance under random and block missingness, remains robust to missing-rate shifts, and benefits from its complementary global and local components. Code is available at https://github.com/LuRenjias/GLAIM.

We describe Causal-TS, an open-source Python library for causal discovery in high-dimensional and nonstationary multivariate time series. Causal-TS provides four specialized algorithms-CDNOTS, CDNOTS+, CEDAR, and GRACE-along with wrappers for GES, Granger, LASSO-VAR, and LGES, all sharing a unified conditional independence (CI) test layer with GPU acceleration via PyTorch. A regime discovery pipeline detects structural breaks via pluggable changepoint detectors and runs discovery per regime with regime-specific parameters. A command-line interface, synthetic data generators, and optional DoWhy integration provide an end-to-end pipeline from raw time series to causal effect estimates. The library is pip-installable, tested on Python 3.10--3.12, and available at https://github.com/bloomberg/causal-ts.

Hierarchical Reinforcement Learning (HRL) intends to separate strategic planning from primitive execution. It has been widely successful in solving long-horizon and complex tasks, where flat-RL algorithms have difficulty in learning. However, while the low-level agent in HRL benefits from dense feedback and abundant trial opportunities, the high-level agent receives sparse, delayed feedback from the environment and its performance depends on the low-level execution capability. In this paper, we study whether subgoal selection by the high-level agent can be performed more strategically, by providing it with dynamics-aware intrinsic motivation. Since motivation based on primitive transition dynamics would require broad coverage of the state-action space, we propose to use coarse dynamics, i.e., environment transitions aggregated over multiple steps at the temporal scale at which the high-level agent operates. This approach stabilizes the high-level policy by learning to minimize the predictive uncertainty associated with the coarse dynamics, and provides a guided structure for navigation. We model the predictive uncertainty by evaluating different dispersion metrics as approximated by a Mixture Density Network (MDN). Empirically, we observe that a dense, dynamics-aware intrinsic reward leads to risk-averse subgoal selection, enabling it to outperform state-of-the-art HRL methods in non-stationary long-horizon environments.

Forecasting under real-world conditions is inherently non-stationary, as the conditional distribution of future observations evolves over time. Recent test-time adaptive sequence models address this challenge by updating internal states during inference, but tie adaptation to instantaneous prediction errors or surprise. This coupling can conflate persistent distribution shift with stochastic innovations, leading to unnecessary updates and inefficient adaptation. We introduce Black-Mamba, a test-time adaptive forecasting architecture that formulates online adaptation as evidence-gated state tracking under distribution drift. The model augments a base predictor with a dynamic memory updated when temporally accumulated surprisal provides sufficient evidence of a regime change. This turns adaptation into a selective, event-driven process rather than a continuous one. Across multiple forecasting benchmarks with non-stationary dynamics, Black-Mamba achieves competitive or improved predictive performance compared to existing test-time adaptation methods while significantly reducing the number of memory updates during inference. Together with mathematical analysis and biological evidence, these results suggest that accumulated surprisal provides a principled signal for distinguishing persistent drift from transient noise, yielding more efficient and robust adaptation.

Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning. Bayesian Quadrature uses Gaussian process surrogates that explicitly encode structural assumptions about the integrand to obtain integral estimates with quantified uncertainty. These surrogates are predominantly based on stationary covariance functions, which results in model misspecification for integrands exhibiting nonstationary behavior. We tackle this issue through an adaptively growing, tree-based partition of the integration domain into local stationary models. Our method recombines the local integral estimates through a hierarchy of GP conditioning that reintroduces cross-subdomain correlations, while model selection criteria control the tree growth to avoid unnecessary partitioning. The resulting algorithm is simple, requires no MCMC, and adapts its evaluation budget to local integrand complexity. On benchmark integration problems and a model evidence computation for an epidemiological model, Hierarchical Bayesian Quadrature achieves substantial gains over standard Bayesian Quadrature on nonstationary integrands while matching its performance on stationary ones.

Time series forecasting requires models to capture diverse, often mutually exclusive, temporal dynamics, from smooth trend continuation to nonstationary drift and strict phase-aligned recurrence. While recent deep learning models have improved accuracy, they typically force these diverse patterns through a single computational backbone governed by fixed algorithmic inductive biases (e.g., self-attention or spectral filtering). This single-mechanism approach often struggles with the profound heterogeneity of real-world series, where different variables and forecast horizons necessitate fundamentally different predictive treatments. To address this, we propose GatedLinear: a lightweight framework that frames forecasting as the adaptive routing of complementary linear bases. GatedLinear leverages a pool of three specialized mechanisms: a global trend-seasonal basis for smooth projection, a difference-based incremental basis for nonstationary drift, and a phase-aligned recurrence basis for explicit cyclic reuse. To dynamically orchestrate these distinct behaviors, we introduce a Tri-Factorized Fusion Gate that disentangles routing decisions into channel-specific preferences, horizon-aware offsets, and phase-indexed biases derived from known future time marks. This design allows the model to perform highly granular, point-wise soft routing across different predictive regimes without stacking computationally heavy neural modules. Experiments on standard benchmarks show that our method achieves state-of-the-art or highly competitive accuracy against recent complex foundational models, while offering explicitly interpretable routing patterns and operating with a substantially smaller parameter footprint.

Brain age - the age inferred from a physiological recording - is an emerging biomarker whose deviation from chronological age tracks neurological and psychiatric burden, and EEG is an attractive substrate for it because it is cheap, portable, and temporally rich. Yet EEG brain-age models must contend with cross-site montage heterogeneity, small labelled cohorts, and dominant subject-level non-stationarity, and few EEG foundation models have been shown to deliver competitive age regression across the full pediatric to older adult range in which such a biomarker would actually be deployed. We introduce STST-JEPA, a self-supervised transformer for resting-state and task EEG, pretrained on 47,703 sessions spanning ages 5-81 from the brain.space and Healthy Brain Network (HBN) corpora. The model combines a latent-prediction objective - predicting masked-token representations against an EMA-of-tokenizer target - with an auxiliary signal-reconstruction term, applied to 30-second multi-channel windows under spatiotemporal block masks. A lightweight attentive probe trained on frozen pretrained embeddings achieves a best held-out-validation mean absolute error of 3.06 years (r = 0.924) for age regression on 3,367 sessions, against a predict-the-mean baseline of approximately 10 years MAE. With light task-specific finetuning of the model's final layers, the same pretrained encoder achieves rank-1 placements - with the model's native 30-second windows - on the public NeuralBench x brain.space EEG leaderboard for sex classification (balanced accuracy 0.911), age prediction (r = 0.749), and psychopathology composite regression (r = 0.215). We further show that the model's age-prediction residual is negatively correlated with cognitive efficiency over several tasks we examined.

In non-stationary streaming environments, simultaneously adapting to complex, non-linear domain shifts via continual learning while mitigating the catastrophic effects of severe, uncalibrated label noise poses a fundamental mathematical challenge. In this paper, we propose \FlatManifold{}, a novel, streamlined robust continual learning framework that utilizes a Nyström manifold flattening map based on the kernel trick and projection onto an orthogonalized Reproducing Kernel Hilbert Space (RKHS). Unlike traditional methods that rely on complex, error-prone sample-filtering pipelines, the proposed approach exploits the intrinsic mathematical robustness of the flattened space itself. By mapping feature distributions onto a fixed orthogonal target topology with a ridge regularizer, the framework naturally smoothes and counteracts the influence of extreme label noise during the optimization process. Concurrently, catastrophic forgetting is prevented via a continual topology brake term that leverages the covariance matrix of past experiences. Extensive evaluation on real-world multi-session robotics datasets demonstrates that even under severe conditions featuring 40% symmetric label noise, \FlatManifold{} successfully mitigates gradient corruption. Under extreme cross-session domain shifts spanning various seasons and lighting conditions, the proposed framework establishes high generalization capabilities, significantly outperforming standard sequential optimization baselines and proving that structural linearization itself serves as a powerful mathematical barrier against distributed label corruption.

The development of decision-pretrained transformers, algorithm distillation, long-context meta-RL, and retrieval-augmented agents has renewed interest in in-context reinforcement learning (ICRL): the ability of a pretrained or fine-tuned decision model to infer latent task rules and improve future behavior from interaction context, without test-time parameter updates. This line of work asks when trial-and-error evidence, rewards, transitions, demonstrations, feedback, or retrieved experience can make learning-like computation happen inside the context window. However, existing surveys of ICRL mainly organize the field around pretraining objectives, architectures, context formats, evaluation protocols, and theoretical mechanisms, while the non-stationary setting remains comparatively underexamined. In changing environments, accumulated context is not merely more evidence about a fixed task: the reward specification, transition kernel, observation channel, action interface, constraint model, or demonstration and memory distribution can fall out of alignment with the current regime. Previously useful context can therefore become stale, misleading, or useful again when an old regime returns. We survey non-stationary ICRL as the problem of adapting through context while deployed policy parameters remain fixed: the policy must infer both the current decision rule and which parts of its accumulated evidence still support that rule. We define non-stationary ICRL, relate it to meta-RL, decision sequence modeling, retrieval-augmented RL, value- and model-aware ICRL, and reward-feedback agents, and organize the literature along three questions: what changes, how the change unfolds, and how observable the change is to the agent.

We investigate a forecasting framework based on a simple discrete-time dynamic model with coefficients varying in time. The parameters of the model are recovered within a deep learning framework, which makes it possible to retain a transparent parametric structure while simultaneously accounting for complex and nonstationary patterns in the observed phenomenon. Our analysis covers two specifications of the noise process. Besides the standard Gaussian setting, we also consider Laplace-distributed noise, which can offer a more adequate description in the presence of heavier tails and sharper local fluctuations. For both cases, we formulate the predictive scheme of the model and analyze the associated uncertainty quantification, including the construction of prediction intervals. The results illustrate that a relatively simple model, when combined with time-dependent parameter estimation, can serve as a mathematically tractable and practically flexible tool for forecasting complex dynamics under different noise assumptions. The general model is stated for TVAR($p$), while the prediction-interval formulas and the numerical experiments are developed for the TVAR(1) case.

Adaptive Financial Transformer (AFT) is proposed for stock return prediction under non-stationary financial markets. The model incorporates a Market Regime Encoder, an Adaptive Gate Network, and an Adaptive Financial Context module to dynamically bias self-attention based on semantic relationships between financial indicators. Unlike conventional Transformer architectures that treat all input features uniformly, the proposed approach groups 95 engineered financial features into 11 semantic categories and adapts attention according to latent market regimes. The study also identifies and corrects sequence alignment and backtesting issues that can inflate reported trading performance, and introduces a financially-aware composite objective that jointly optimizes prediction error, directional accuracy, and non-overlapping Sharpe ratio. Extensive experiments compare the proposed architecture against classical machine learning models, recurrent neural networks, and Transformer baselines using chronological evaluation, five random seeds, ablation studies, hyperparameter optimization, explainability analysis, and multi-stock validation. Results demonstrate competitive predictive performance while reducing model complexity by 15.2% and improving parameter efficiency through feature selection, providing an interpretable Transformer architecture for financial time-series forecasting.

Predicting temporal Quality of Service (QoS) data is critical for optimizing network services and rationalizing resource allocation in cloud computing and service-oriented systems. Existing mainstream methods have achieved promising predictive performance. However, their purely data-driven manner limits their ability to capture non-stationary temporal patterns, thereby leading to accuracy degradation when temporal QoS data exhibits fluctuations. To tackle this limitation, we propose a novel Extended Kalman Filter-Enhanced Latent Feature Analysis (EKL) model to perform efficient and accurate temporal QoS prediction from the perspective of bidirectional model-data-driven learning. Its main idea is three-fold: a) designing a model-driven feature producer to obtain the temporal latent features to capture the intricate temporal pattern following the principle of an Extended Kalman Filter; b) building a data-driven feature producer based on the alternating least squares algorithm to identify time-invariant latent features describing intrinsic user-service characteristics; c) exploiting a density-oriented parallel strategy that achieves workload balancing by sorting users in accordance with their service invocation density, which effectively elevates computational efficiency. In addition, we provide a rigorous theoretical analysis to formally prove the convergence of the proposed EKL. Experimental evaluations conducted on real-world temporal QoS datasets reveal that our proposed EKL surpasses existing state-of-the-art models with respect to both computational efficiency and prediction accuracy for missing temporal QoS data.

Online learning is crucial for handling complex data streams in big data applications. Recent research has begun to focus on dynamic scenarios, i.e., non-stationary environments. However, a crucial yet often overlooked aspect is label latency, where new data may not receive labels in time due to the slow and expensive labeling process, thus hindering rapid adaptation to dynamic environments. To resolve this impasse, we propose Dual-Track Geometry Online Learning (DT-GOL), a novel framework that shifts from temporal compensation to spatial reasoning to bridge the supervised latency gap. By modeling the delay challenge as a semi-supervised task, we leverage real-time topological evolution of features as a reliable geometric surrogate for unobservable conceptual changes to achieve proactive supervised adaptation within the delay window. Unlike rigid self-training, we introduce a dynamic evidence calibration mechanism that distills geometric information into soft labels that perceive uncertainty, effectively mitigating the confirmation bias inherent in hard pseudo-labels. Furthermore, to resolve the stability-plasticity dilemma, we design a decoupled dual-track architecture in which a master learner serves as a stable anchor, updated strictly from delayed ground truth, while a transient branch leverages soft geometric knowledge for low-risk forward adaptation. Extensive experiments on real and synthetic datasets demonstrate that DT-GOL significantly outperforms existing state-of-the-art baseline methods, especially in scenarios with concept drift.

Machine learning systems deployed in dynamic environments frequently operate under nonstationary data distributions, where controlled distribution shift can progressively degrade predictive performance. However, many widely used tabular benchmark datasets lack explicit temporal structure, limiting reproducible evaluation of drift adaptation methods. This work proposes a cluster-induced distribution shift simulation framework that transforms static tabular datasets into controlled evolving data streams through structured perturbations across featurespace partitions. Using this framework, six adaptation strategies are systematically evaluated: static learning, sliding-window retraining, global ADWIN retraining, cluster-local ADWIN retraining, random subspace drift detection, and feature-partitioned drift detection. Experiments are conducted on five benchmark datasets covering both classification and regression tasks using diverse predictive model families, including linear models, k-Nearest Neighbours, tree ensembles, boosting methods, and adaptive online learners.

Machine learning algorithms deployed for evolving streaming environments must handle the non-stationary data distributions, commonly referred to as concept drift. The presence of concept drift poses a major challenge for many real-world applications because it can severely degrade their predictive performance, hindering their ability to support robust decision-making. Consequently, the timely and efficient detection of drift events is critical for sustaining high accuracy over time. This study examines theoretically the concept drift characteristics and numerous drift detection algorithms across several categories. Furthermore, we evaluate their performance on both synthetic and real-world datasets exhibiting diverse streaming scenarios and drift characteristics, such as abrupt and gradual changes. This study aims to enhance understanding of the complex notion of concept drift characteristics and behavior of drift detectors, along with their applicability to diverse contexts.

Online learning in non-stationary streams is often formulated as tracking a point estimate, but many applications require predicting the full data-generating distribution. We study online distributional prediction under drift and adversarial corruption. Our approach represents each candidate law through a latent cluster geometry: a variable-size configuration of centers that organizes probability mass and induces a predictive distribution. A Gibbs quasi-posterior over these configurations yields an online predictor by posterior averaging, and the resulting variable-dimensional posterior can be sampled with reversible-jump MCMC. The method therefore avoids specifying a parametric streaming law while retaining a structured latent space for uncertainty, regularization, and comparison. We evaluate performance by cumulative Wasserstein-1 regret against the time-varying true law. The analysis separates two effects: corruption perturbs the loss-based posterior update, whereas drift makes long-horizon posterior memory stale. We address the latter with a restarted variant that temporally localizes the same quasi-Bayesian update. The resulting high-probability bounds decompose into a PAC-Bayesian complexity term, a corruption-sensitive posterior perturbation term, and a dynamic optimal-transport term driven by $A_T^{\mathrm{OT}}=\sum_{t=2}^T W_2^2(p_{t-1}^*,p_t^*)$. Under bounded support, stable latent geometry, predictive-map regularity, oracle realizability, localized restart windows, sublinear transport action, and sublinear corruption budget, the restarted predictor achieves sublinear cumulative Wasserstein regret. These guarantees require no parametric model for the stream, drift mechanism, or corruption process.

Accurate short-term electricity load forecasting is critical for the reliable and economic operation of modern power systems, under non-stationarity arising from weather variability, calendar effects, and evolving consumption patterns. While deep learning models such as LSTMs and Transformers show promising performance, most existing studies focus on direct absolute load prediction without explicitly addressing target non-stationarity. Motivated by classical time-series differencing techniques in ARIMA models, this paper investigates a delta-based target reformulation for short-term electricity load forecasting using deep learning. Instead of directly predicting absolute load values, the proposed formulation trains models to predict the change in load between consecutive time steps, with final forecasts reconstructed using the last observed load. This aims to stabilize the learning target and reduce forecasting difficulty. Using multi-year, hourly real-world electricity load data from India, augmented with meteorological variables from the NASA POWER project and calendar features, this study evaluates LSTM and Transformer models under both formulations, benchmarking them against LightGBM. Experiments are conducted for hour-ahead and day-ahead horizons, assessing performance via Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE). Results show that delta-based reformulation consistently improves forecasting accuracy for hour-ahead prediction across all evaluated models, yielding MAPE reductions of over 50% compared to absolute formulations. For day-ahead forecasting, delta targets specifically benefit deep sequence models (LSTM and Transformer), while LightGBM remains competitive under the absolute formulation. These findings indicate that while delta reformulation is a powerful inductive bias for neural networks, its efficacy is model- and horizon-dependent.

Kalman filters based on the Embedded Latent Transfer Operators (ELTO) emerge as novel statistical tools for sequential state estimation. However, a critical limitation stems from their use of simplified noise models, which fail to dynamically adapt to non-stationary processes. To address this limitation, we introduce an ELTO-based Bayesian filtering approach with a new structured parameterization for the filter's noise model. This parameterization enables structured noise adaptation, which couples the data-driven learning of an optimal time-invariant noise model with dynamic parameter adaptation that responds to changes in dynamics within non-stationary processes. Empirical results show that our structured noise adaptation improves the filter's dynamic state estimation performance in noisy, time-varying environments.

We present an algorithm for efficient domain-adaptive policy learning via kernel representations. Learning domain-adaptive policies is challenging since it requires an environment representation that is both sufficiently expressive to model complex sim-to-real gaps during offline training, and computationally efficient enough to support rapid online adaptation during deployment. For instance, a quadrotor may encounter time-varying, non-stationary disturbances, such as sudden gusts of wind, payload shifts, or transitions between distinct flight regimes with and without ground effects. To address these challenges, we model unknown disturbances using a differentiable kernel approximation based on random Fourier features. During the offline training phase, we randomly sample kernel coefficients and bandwidth parameters to generate a rich diversity of disturbance profiles. We then optimize the control policy via differentiable simulation with analytical gradients, a process that takes only 50 seconds of training time on an RTX 4090 GPU. During hardware deployment, the policy adapts to non-stationary environments in real time by updating both the kernel coefficients and bandwidth through online least-squares estimation. We evaluate our method on quadrotor trajectory tracking tasks across high-fidelity numerical simulations and hardware experiments using Crazyflie, subjected to various disturbances, including complex aerodynamic effects, wind, ground effects, and payload fluctuations.

Federated Learning (FL) enables collaborative and privacy-preserving model training across distributed clients, but most existing FL systems implicitly assume data stationarity. In real-world settings-such as healthcare, industrial IoT (IIOT), cybersecurity, and smart cities-data streams are inherently non-stationary, leading classical FL methods to suffer from performance degradation, instability, and catastrophic forgetting. Continual Learning (CL) addresses learning under evolving data distributions but has been largely studied in centralized settings, overlooking key constraints of federated systems, including privacy, limited communication, and client heterogeneity. Federated Continual Learning (FCL) emerges at the intersection of FL and CL, aiming to support lifelong, adaptive, and privacy-aware learning over distributed and non-stationary data. This survey provides a comprehensive and systematic overview of FCL. We first present a formal definition of the FCL problem and clarify its distinctive characteristics. We then analyze the limitations of classical FL under non-stationary conditions, highlighting how CL principles support long-term adaptation. To organize the rapidly growing literature, we propose a multi-dimensional taxonomy of FCL approaches. Furthermore, we review representative application domains and data modalities, summarize commonly used evaluation metrics, and discuss experimental perspectives for assessing long-term performance and forgetting. Finally, we highlight key open challenges, including handling extreme heterogeneity under temporal drift, designing scalable and privacy-preserving memory mechanisms, and establishing standardized benchmarks. This survey aims to serve as a reference and a roadmap for advancing FCL toward robust and deployable real-world systems.

In many real-world settings, data streams are nonstationary and arrive sequentially, requiring learning systems to adapt continuously without retraining from scratch. Continual learning (CL) addresses this challenge by incorporating new tasks while mitigating catastrophic forgetting, where learning new information degrades performance on previously acquired knowledge. We introduce a control-theoretic perspective on CL that explicitly regulates the evolution of forgetting, framing adaptation as a controlled process subject to long-term stability constraints. We focus on replay-based CL, where a finite memory buffer stores representative samples from prior tasks. We propose COntinual Learning with Drift-Plus-Penalty (COLD), a continual learning framework based on the Drift-Plus-Penalty (DPP) principle from stochastic optimization. To facilitate analysis, we also consider an oracle variant, COLD-ORACLE, as a reference benchmark. At each task, both methods minimize the current task loss while maintaining a virtual queue that tracks deviations from long-term stability on previously learned tasks, capturing the stability-plasticity trade-off as a regulated dynamical process. We establish stability and convergence guarantees that characterize this trade-off through a tunable control parameter. Experiments on standard benchmarks demonstrate that COLD consistently outperforms a broad range of state-of-the-art CL methods while providing competitive and controllable forgetting behavior through explicit regulation of stability and plasticity.

Adaptive beamforming is a cornerstone of array signal processing, yet its performance often collapses in the face of complex, rapidly changing interference. When interferers appear or move unpredictably, conventional estimators encounter a fundamental memory trade-off: short windows enable rapid tracking but suffer from high estimation variance, while long windows provide stable rejection but fail to adapt to shifts. This challenge is resolved by introducing the Universal Switching Beamformer (USB), which integrates competitive sequential prediction into the beamforming architecture. By employing a linear transition diagram, the USB implicitly maintains an exponentially large family of candidate covariance histories and dynamically re-weights them based on their cumulative output power. This mechanism allows the beamformer to automatically vary its effective memory length without explicit change detection or heuristic parameter tuning. A theoretical upper bound is proven on the regret relative to an omniscient oracle that selects the best piecewise-stationary covariance model in hindsight. Extensive simulations and experiments on the SwellEx-96 dataset demonstrate that the USB achieves the agility of short-window estimators and the precision of long-term integration, providing a principled solution for tracking highly non-stationary scenes.

Real-time data analysis requires the ability to accurately and adaptively address nonlinear dynamics in a nonstationary data stream while preserving computational efficiency. However, nonlinear dynamics are so complex that capturing dynamically changing nonlinear patterns and utilizing them for downstream tasks under strict time constraints is nontrivial. To bridge the gap between nonlinear complexity and computational tractability, this study applies Koopman operator theory, which states that nonlinear dynamics can be represented as linear transitions in an infinite-dimensional space. Building upon finite-dimensional approximations of this operator, we present AdaKoop, an efficient streaming algorithm for modeling nonlinear dynamics over nonstationary data streams. Our approach utilizes a probabilistic framework grounded in Koopman operator theory, treating both raw observations and reproducing kernel Hilbert space (RKHS) features as emissions from latent vectors. This dual-view formulation allows nonlinear dynamics to be expressed as a tractable linear system. Therefore, AdaKoop enables the efficient and stable modeling of nonlinear dynamics in a streaming fashion, avoiding the prohibitive computational costs of iterative nonlinear optimization. Furthermore, to address nonstationarity in data streams, AdaKoop adaptively detects the switching of patterns via statistical hypothesis testing for abrupt pattern shifts and incrementally updates model parameters to handle continuous changes. Extensive experiments on a total of 71 practical benchmark datasets across various domains demonstrate that AdaKoop outperforms state-of-the-art methods in terms of real-time forecasting accuracy and computational efficiency.

Causal modeling of physical temporal phenomena must handle interventions that act along trajectories, nonstationary induced laws, path-dependent effects, and feedback mediated by dynamics, all challenging in standard causal models. We introduce Hamiltonian Causal Models (HCMs), a trajectory-level framework in which observed variables interact with local environments and interventions act as controls of Hamiltonian mechanisms. HCMs separate immutable equations of motion from intervenable mechanisms and define causal effects as discrepancies between interventional path laws. A key motivation for HCMs is their natural interface with non-equilibrium thermodynamics. Entropy production quantifies the irreversibility of a process and is a central causal observable: it is estimable from data and witnesses causal effects along the system's evolution that are invisible to endpoint and cumulative versions of the standard average treatment effect. As in physics, cause and effect are not primitives of the relation between two random variables but arise from the non-invertibility of the thermodynamic arrow. With this, our paper reconciles the language of statistical causal models and non-stationary thermodynamics, offering new tools to describe causality in a wide range of physical systems.

While Proximal Policy Optimization (PPO) demonstrates strong performance in stationary settings, we show that its standard optimization paradigm struggles in continual and non-stationary environments. The failure does not stem from insufficient model capacity or overly restrictive clipping. Instead, PPO performs persistent, directionally inefficient local updates, which indicates a lack of geometry-aware guidance for accumulating meaningful behavioral change and ultimately hindering transitions toward new behavior patterns. Although divergence-based regularization introduces partial geometric awareness, its monotonically increasing penalties implicitly discourage large policy deviations, even when such shifts are necessary for effective adaptation. To address this limitation, we propose Gaussian Trust Region Policy Optimization (GTR), which reshapes the trust region using a Gaussian kernel. The resulting constraint is bounded and non-monotonic, providing strong local stability while progressively relaxing under sustained high-advantage updates. To further improve robustness, we introduce a Mixture Gaussian Anchor that adapts to recent policy trajectories, reducing variance induced by stale references. GTR is architecture-agnostic and achieves strong performance across games, simulated robotic control, open-world exploration, and language model post-training. These results demonstrate that geometry-aware trust-region design can be a promising direction for robust reinforcement learning in complex non-stationary environments. Our code is available at https://anonymous.4open.science/r/GTR_demo/README.md.

Temporal systems often exhibit non-stationary behaviour, such as seasonal climate variation or glucose fluctuations in patients with type-1 diabetes. One way to model non-stationarity is through discrete latent regimes, i.e., stationary segments of time. Such systems induce a Markov Switching Model (MSM), a class of Hidden Markov Models with autoregressive dependencies among latent regimes and observed variables. Identifying latent regimes is challenging in the presence of frequent regime switches and nonlinear and non-Gaussian dynamics, particularly when there are instantaneous effects between the variables, e.g., due to slow rates of measurements. In this work, we establish the identifiability of both latent regimes and regime-dependent causal structures under temporal regime dependencies, nonlinear lagged and instantaneous effects, and independent noise from the exponential family. Our identifiability theory subsumes non-temporal mixtures of causal models. Furthermore, we introduce FlowMSM, a regime detection framework that can be paired with any stationary causal discovery method to recover regime-dependent causal structures. Experiments on synthetic benchmarks and a financial economics dataset demonstrate the effectiveness of our approach to detect latent regimes and discover causal structures from non-stationary time series.

Long-term time-series forecasting needs models that are accurate yet efficient enough for commodity hardware. Lightweight linear forecasters are remarkably strong in this regime, yet they leave two openings: reversible instance normalization (RevIN) de-normalizes the entire horizon with a single lookback statistic, which is inaccurate under non-stationarity, and time-domain trend/seasonal decomposition relies on a fixed, non-adaptive filter. We present FreqLite, an ultra-lightweight, channel-independent frequency-decomposed linear forecaster: a learnable, lossless, partition-of-unity spectral filter splits the input into bands that are forecast by per-band linear heads and, unlike low-pass-truncation approaches, the high-frequency band is retained and modeled. FreqLite is the best lightweight model on the standard long-term forecasting benchmarks and, at long lookback (L=336), attains a lower average error than a PatchTST Transformer (0.3244 vs. 0.3587 MSE) while using 4x fewer parameters, 2.2x less memory, and 2.2x less time per epoch on a single 4 GB laptop GPU; although modest in magnitude, its improvements are statistically significant under paired Wilcoxon tests across all matched cells (p < 1e-5). We further introduce Adaptive Reversible Instance Normalization (A-RevIN), a regime-adaptive reversible normalization that strictly generalizes RevIN (recovered exactly when its gate is closed), engages under non-stationarity, and reduces to RevIN without harm on stationary data. We validate this on both a real strongly non-stationary dataset (ILI, up to ~5% MSE reduction) and a controlled synthetic drift sweep in which A-RevIN's benefit and its learned gate both rise monotonically with injected non-stationarity. Every component is independently ablatable (Linear and RLinear are special cases of FreqLite), and all results are reproducible on commodity hardware.

Learning long-range non-stationary temporal patterns remains a core challenge for modern sequence models, particularly in strict streaming settings. In these settings, data arrive sequentially and must be processed in a single pass without simultaneously revisiting past observations. Standard architectures, including recurrent neural networks and transformers, are constrained by either truncated backpropagation through time horizon or explicit input window length for long range credit assignment. To address these limitations, we propose SHARP (Sleep-based Hierarchical Accelerated Replay), a framework that decomposes temporal learning into two complementary components: a memory module that accumulates a structured history of past inputs, and a pattern-recognition module that operates over this memory. This separation enables resource- and compute-efficient adaptation to non-stationary dynamics by eliminating the need for backpropagation through time across many steps for long-range credit assignment. Inspired by the accelerated replay observed in rodents during slow-wave sleep, SHARP incorporates offline (sleep) phases in which temporally structured memory traces are replayed in an accelerated form and integrated into higher-level memory representations, improving long-range context retention. Through controlled simulations and ablation studies, we characterize the key properties of the proposed framework. In benchmark datasets such as text8 and PG-19, we demonstrate that SHARP improves over recurrent baselines by retaining next-token predictive performance on previously seen data while continuing to learn from the current stream and generalizing to future unseen data. These gains are enabled by its hierarchical structure, which yields an exponentially increasing effective temporal context with only linear-time computational cost.

Bagging-based ensembles, most notably Adaptive Random Forests, are among the strongest performers for learning from data streams. A common denominator across these methods is their reliance on Hoeffding Trees as base learners, which grow decision trees incrementally by testing whether a candidate split is significantly better than its alternatives using concentration inequalities. Despite their empirical success, existing variants lack valid statistical guarantees. Current analyses rely on fixed-sample concentration bounds, while split decisions are made using data-dependent stopping rules, which invalidates their guarantees and can drive the probabilty of incorrect splits to one. We introduce a principled alternative based on anytime-valid inference. Our method provides: (i) anytime-valid control of false splits under arbitrary data streams, including non-stationary settings; (ii) finite commitment time under a predictive advantage; and (iii) under stationary i.i.d. data, risk is monotone decreasing and strictly improves at every split. Empirically, we evaluate both standalone trees and their use within Adaptive Random Forests on non-stationary streams. Our method improves performance while producing substantially smaller trees.

Data streams are nowadays among the most frequently analyzed data structures, with the concept drift posing a major challenge encountered by processing systems. Despite the proposition of numerous solutions to counteract the accuracy degeneration due to concept drift, the scientific community has not yet established a unified framework for evaluating the concept drift detection task. Existing research often relies on classification quality metrics, but these can be affected by multiple factors and may not reliably reflect drift detection quality. In this work, we present an in-depth overview of the relationship between metrics for quantifying drift detection quality and classification performance in synthetic nonstationary data streams. The proposed research studies eight drift detection quality metrics in relation to the classifier's performance across seven synthetic data stream generation tools, additionally considering drift dynamics as a factor. The studies aim to identify the most informative set of drift detection quality metrics and provide a deep understanding of the method's evaluation.

Bidding in the European Frequency Containment Reserve (FCR) market is challenging for flexibility providers because competing offers are hidden and bidders observe only partial feedback form the market, such as, clearing price and awarded quantity. For a participant active in a single country, we show that the multi-country FCR clearing problem can be recast as a repeated multi-unit uniform-price auction against an endogenous vector of opposing bids. This reformulation yields an online learning problem and allows us to adapt a Best-of-Both-Worlds combinatorial semi-bandit algorithm implementable from this standard market feedback. The resulting bidder achieves logarithmic pseudo-regret in stochastic environments and $\mathcal{O}(\sqrt{T})$ regret in adversarial ones. Synthetic experiments confirm the expected scaling, and backtests on historical European FCR data show competitive performance in practice: the method performs especially well on stable products, while EXP3-type baselines can be safer under stronger non-stationarity. Overall, the results show that learning-based bidding in FCR markets is theoretically grounded and practically useful when the learning rule matches product-level market stability.

Multi-step-ahead forecasts are often updated as new observations become available, since shorter forecast horizons typically improve forecast quality. However, such improvements come at the cost of forecast instability, i.e., variability in forecasts for the same target period. This instability can trigger costly changes to plans formulated based on the forecasts and may erode trust in the forecasting system. In this work, we integrate forecast stability alongside forecast quality into the training of distribution-free probabilistic time-series forecasting models, allowing us to control this trade-off. We propose a method for generating stabilized forecasted conditional quantile functions using regression splines parameterized by a neural network. This approach enables joint optimization of quality and stability, as it allows us to directly penalize dissimilarities arising from forecast updates. Furthermore, it allows assigning varying importance to stabilizing different parts of the forecast distributions (e.g., central parts vs. tails) to focus on the parts most relevant for the intended downstream use (e.g., the upper tail for inventory management). We empirically evaluate the proposed method on two datasets with different statistical properties and show that it can effectively reduce forecast instability without a substantial loss in forecast quality, and that it can target stabilization effort toward specific parts of the forecast distributions.

Test-time adaptation (TTA) aims to adapt models to maintain reliable performance on non-stationary test streams without requiring labeled data. Despite its empirical success, the learnability of TTA under non-stationary streams remains unexplored. A key challenge is the lack of a principled theoretical framework that simultaneously aligns with the TTA objective and captures both continuously evolving distribution shifts and intrinsic information constraints. To address this gap, we propose the first theoretical framework for studying the learnability of TTA and introduce $(ε,δ)$-Recovery Complexity and $(ε,ρ)$-TTA Learnability. Recovery complexity measures the post-shift time needed to maintain excess risk below a target level with high probability, and is further extended to TTA learnability, which measures the long-term reliability of TTA. Within this framework, we introduce a novel discrete surrogate for non-stationary test streams, enabling a unified and tractable analysis of both gradual and abrupt shifts. We derive order-wise matching lower and upper bounds on recovery complexity, revealing fundamental limits of TTA and an intrinsic adaptivity-information trade-off. These results provide unified learnability guarantees for TTA that complement regret-based analyses.

A hallmark of intelligence is the ability to adapt in non-stationary environments, yet deep Reinforcement Learning (RL) agents often struggle in such settings. Prior studies introduce non-stationarity through abrupt shifts in features or dynamics, whereas real-world environments often evolve gradually through continual drift. This distinction has important implications for the "stability-plasticity dilemma" in RL, as abrupt task changes may demand more plasticity than naturalistic settings. To address this, we modify existing 3D Miniworld and MuJoCo environments to incorporate naturalistic, continual non-stationarity, and use them to examine how stability and adaptation affect performance under continuous environmental change. We find that methods favoring stability, such as synaptic consolidation, outperform approaches focused on plasticity, such as parameters resetting. Motivated by this result, and prior evidence that Successor Features (SFs) reduce interference, we investigate whether SFs are better consolidation targets than Q-values. Across both environments, applying neuro-inspired synaptic consolidation to SFs yields superior performance on continually changing settings. Moreover, consolidation is most effective when SFs are stabilized across multiple timescales, which capture complementary aspects of gradual environmental change. Together, these results suggest that stability is more critical in continual learning when changes are gradual, and that multi-timescale consolidation of predictive representations is an effective approach.

Recent research in time series forecasting frequently investigates the integration of textual and visual modalities with numerical models to better navigate non-stationary environments. Despite delivering solid numerical results, existing multi-modal approaches usually encounter a dilemma: prioritizing the minimization of average errors can result in excessively smooth forecasts that overlook essential fluctuations. To resolve this limitation, we introduce STaT, an innovative multimodal architecture for Symbolic-Temporal-Textual Alignment, which seamlessly unites three synergistic modalities. Specifically, the symbolic modality converts continuous time series into discrete tokens, facilitating the accurate identification of structural patterns and turning points; the temporal modality extracts inherent sequential dependencies; and the textual modality leverages domain semantics to steer the macroscopic forecasting trends. Comprehensive evaluations on eight real-world benchmarks indicate that STaT delivers exceptional performance, enhancing conventional magnitude indicators by up to 8.9% while simultaneously decreasing shape distortion by up to 8.5%.

Speech and audio systems operate in inherently non-stationary environments, yet continual learning (CL) research in this domain, especially in the foundation model era, remains fragmented that fail to account for the coupled, geometry-sensitive nature of acoustic representations. Modern speech foundation models operate over highly entangled, continuous representations that jointly encode linguistic, speaker, and paralinguistic factors within a shared latent space. CL is therefore fundamentally about preserving and evolving shared representation structure rather than retaining isolated task knowledge. In this work, we revisit CL for speech from a representation-centered perspective, and introduce a new taxonomy that organizes CL according to how underlying representation geometry evolves under non-stationary acoustic conditions. We further identify key mismatches between current CL assumptions and speech foundation model behavior, and finally outline a set of open challenges and future research directions.

Effectively modeling non-stationary dynamics in probabilistic multivariate time series(MTS) forecasting requires balancing expressiveness with robustness. Existing parametric approaches benefit from strong inductive biases but lack flexibility, whereas deep generative models struggle to capture complex temporal dependencies without extensive data and computation. We introduce Parametric Prior Mapping (PPM), a framework that injects parametric structural priors into a generative modeling process. Specifically, PPM utilizes a parametric estimator to derive a dynamic, adaptive prior that guides the learning of a complex predictive distribution via a learnable mapping. This design allows the model to retain the efficiency of parametric methods while exploiting the expressive power of generative models. Trained with a hybrid objective, PPM yields precise forecasts with well-calibrated uncertainty estimates. Empirical results show that PPM outperforms existing baselines in handling non-stationary data, offering a superior trade-off between accuracy and computational efficiency. The code is available at https://github.com/ljl8336/PPM.

Firms increasingly rely on dynamic pricing to respond to evolving customer demand, yet in many applications they observe only the revenue generated by a single posted price in each period. At the same time, market conditions may shift gradually or abruptly due to changes in customer preferences, competition, or external shocks. These features create two intertwined challenges: learning the revenue--demand relationship from limited feedback and adapting pricing decisions to a changing environment. We study how a seller can learn and earn effectively under these constraints, without assuming a specific parametric form for demand. We develop a learning framework that updates prices using revenue-based gradient approximations constructed from one observation per period. To address environmental changes, we incorporate a restarting mechanism that periodically refreshes the learning process so that outdated information is discounted. When the degree of nonstationarity is unknown, we further introduce a meta-learning layer to adaptively hedge across multiple restarting schedules. We provide performance guarantees for our approach, showing how cumulative revenue loss relative to a fully informed benchmark depends on both the time horizon and the magnitude of market variation. Simulation experiments using synthetic and real-world data illustrate the effectiveness of the proposed procedures.

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.

Evolving data streams induce joint nonstationarity in continual semantic segmentation, where semantic classes, input distributions, and supervision availability change simultaneously over time. This setting reflects practical structured prediction systems, yet remains largely unexplored in prior continual learning work, which typically studies these factors in isolation. We formalize continual segmentation under coupled class, domain, and label shifts and investigate learning in heterogeneous dense prediction environments with limited annotations and abundant unlabeled data. To address instability and overfitting arising from few-shot supervision under distribution drift, we introduce gradient-adaptive stabilization, a parameter-wise regularization mechanism implemented via gradient-scaled stochastic perturbations that promotes a principled stability-plasticity tradeoff. We further leverage unlabeled data through semi-supervised learning and introduce prototype anchored supervision that validates pseudo-labels via joint confidence and prototype consistency. Together, these mechanisms enable learning under joint nonstationarity in continual segmentation. Extensive empirical evaluation across class-incremental, domain-incremental, and few-shot regimes demonstrates consistent improvements over prior methods in heterogeneous structured prediction settings. Our results expose fundamental failure modes of existing continual segmentation approaches and provide insight into learning robust dense predictors in dynamically evolving environments.

Modern deep learning science often assumes that neural networks learn from a fixed data distribution. However, many practically important learning problems involve data distributions that change throughout training. How does such non-stationarity impact the inductive biases of deep learning towards models with different structural, generalisation, and safety properties? A fruitful testbed for studying inductive bias is in-context linear regression sequence modelling, where small transformers display strikingly different generalisation patterns depending on the diversity of the (fixed) training task distribution. In this paper, we explore the effect of diversifying the task distribution across training time, finding that such temporal diversity leads to an increased bias towards generalisation over memorisation.

Time series forecasting under non-stationarity faces a fundamental tension between capturing stable representations and adapting to distribution shifts. Existing methods implicitly rely on static historical assumptions, leading to a critical failure mode we term Phase Amnesia, where models become blind to the evolving global context. To resolve this, we formalize non-stationary dynamics through three physical hypotheses: wold decomposition, dynamical phase evolution, and heteroscedastic manifold generation. These principles inspire PULSE, a physics-informed, plug-and-play framework adopting a Disentangle--Evolve--Simulate design philosophy. Specifically, PULSE utilizes phase-anchored disentanglement to resolve optimization interference caused by dominant trends, employs a Phase Router to actively generate future trajectories, and introduces Statistic-Aware Mixup (SAM) to ensure robustness against out-of-distribution volatility. Empirically, PULSE enables a simple MLP backbone to achieve state-of-the-art or highly competitive performance across 12 real-world benchmarks. This validates that a correct physics-informed inductive bias is far more critical than raw architectural complexity for non-stationary forecasting. The code is available at: https://github.com/Gemost/PULSE.

Instance normalization (IN) is widely used in non-stationary multivariate time series forecasting to reduce distribution shifts and highlight common patterns across samples. However, IN can over-smooth instance-specific structural information that is essential for modeling temporal and cross-channel heterogeneity. While prior methods further suppress distribution discrepancies or attempt to recover temporal specific dependencies, they often ignore a central tension: how to adaptively model common and instance-specific dependency based on each instance's non-stationary structures. To address this dilemma, we propose SeesawNet, a unified architecture that dynamically balances common and instance-specific dependency modeling in both temporal and channel dimensions. At its core is Adaptive Stationary-Nonstationary Attention (ASNA), which captures common dependencies from normalized sequences and specific dependencies from raw sequences, and adaptively fuses them according to instance-level non-stationarity. Built upon ASNA, SeesawNet alternates dedicated temporal and channel relationship modeling to jointly capture long-range and cross-variable dependencies. Extensive experiments on multiple real-world benchmarks demonstrate that SeesawNet consistently outperforms state-of-the-art methods.

Continual graph learning (CGL) aims to learn from dynamically evolving graphs while mitigating catastrophic forgetting. Existing CGL approaches typically adopt a task-based formulation, where the data stream is partitioned into a sequence of discrete tasks with pre-defined boundaries. However, such assumptions rarely hold in real-world environments, where data distributions evolve continuously and task identity is often unavailable. To better reflect realistic non-stationary environments, we revisit continual graph learning from a task-free perspective. We propose a unified formulation that models the data stream as a time-varying mixture of latent task distributions, enabling continuous modeling of distribution drift. Based on this formulation, we construct \emph{DRIFT}, a benchmark that spans a spectrum of transition dynamics ranging from hard task switches to smooth distributional drift through a Gaussian parameterization. We evaluate representative continual learning methods under this task-free setting and observe substantial performance degradation compared to traditional task-based protocols. Our findings indicate that many existing approaches implicitly rely on task boundary information and struggle under realistic task-free graph streams. This work highlights the importance of studying continual graph learning under realistic non-stationary conditions and provides a benchmark for future research in this direction. Our code is available at https://github.com/UConn-DSIS/DRIFT.

Safe reinforcement learning in nonstationary environments requires safety mechanisms that adapt as environmental conditions change. Standard safe reinforcement learning methods often assume fixed constraints or stable environmental conditions, which can become inadequate under distribution shift. We propose LILAC+, a framework for safe continual reinforcement learning under nonstationarity that combines three adaptive safety mechanisms: context-based safety constraints, adaptation-speed constraints, and budget-to-state safety enforcement. Context-based constraints adjust safety requirements using inferred and predicted environmental context. Adaptation-speed constraints tighten safety requirements when the rate of environmental change exceeds the agent's ability to adapt safely. Budget-to-state enforcement converts cumulative safety requirements into local state-level control constraints that can be enforced at decision time. Together, these mechanisms provide a unified approach for proactive and reactive safety adaptation in continual reinforcement learning. We evaluate the framework in simulated driving environments under stationary, seen nonstationary, and unseen nonstationary conditions. The results show that adaptive safety constraints substantially reduce safety violations under distribution shift while maintaining competitive task performance compared with unconstrained and fixed-constraint baselines. These findings suggest that safe continual reinforcement learning requires adaptive constraint mechanisms that respond not only to current state information but also to predicted environmental context, adaptation demand, and remaining safety budget.

Making calibrated online predictions is a central challenge in modern AI systems. Much of the existing literature focuses on fully adversarial environments where outcomes may be arbitrary, leading to conservative algorithms that can perform suboptimally in more benign settings, such as when outcomes are nearly stationary. This gap raises a natural question: can we design online prediction algorithms whose calibration error automatically adapts to the degree of non-stationarity in the environment, smoothly interpolating between i.i.d. and adversarial regimes? We answer this question in the affirmative and develop a suite of algorithms that achieve adaptive calibration guarantees under multiple calibration measures. Specifically, with $T$ being the number of rounds, $K$ being the unknown number of i.i.d. segments of the environment, and $C\in[0,T]$ being another unknown non-stationary measure defined as the minimal $\ell_1$ deviation of the mean outcomes, our algorithms attain $\widetilde{O}(\min\{\sqrt{T}+(TC)^{\frac{1}{3}}, \sqrt{KT}\})$ for $\ell_1$ calibration error and $\widetilde{O}(\min\{(1+C)^{\frac{1}{3}}, K\})$ for both $\ell_2$ and pseudo KL calibration error. These bounds match the optimal rates in the stationary case ($C=0$ and $K=1$) and recover known guarantees in the fully adversarial regime ($C, K=Ω(T)$). Our approach builds on and extends prior work [Hu et al., 2026, Luo et al., 2025], introducing an epoch-based scheduling together with a novel non-uniform partition of the prediction space that allocates finer resolution near the underlying ground truth.

While machine learning has revolutionized many fields such as natural language processing (NLP) and computer vision, its impact on time-series forecasting is still widely disputed, especially in the finance domain. This paper compares forecasting performance on U.S. Treasury yield curve data across econometrics/time-series analysis, classical machine learning, and deep learning methods, using daily data over 47 years. The Treasury yield curve is important because it is widely used by every participant in the bond markets, which are larger than equity markets. We examine a variety of methods that have not been tested on yield curve forecasting, especially deep learning algorithms. The algorithms include the Autoregressive Integrated Moving Average (ARIMA) model and its extensions, naive benchmarks, ensemble methods, Recurrent Neural Networks (RNNs), and multiple transformers built for forecasting. ARIMA and naive econometric models outperform other models overall, except in one time block. Of the machine learning methods, TimeGPT, LGBM and RNNs perform the best. Furthermore, the paper explores whether stationary or nonstationary data are more appropriate as input to deep learning models.

We study online prediction under distribution shift, where inputs arrive chronologically and outcomes are revealed only after prediction. In this setting, predictors must remain stable in quiet regimes yet adapt when regimes shift, and the right adaptation memory is unknown in advance. We propose MELO (Memory-hedged Exponentially Weighted Least-Squares Online aggregation), a model-agnostic method that hedges across adaptation scales: it wraps any non-anticipating base-predictor pool with exponentially weighted least-squares (EWLS) adaptation experts at multiple forgetting factors, and aggregates raw and EWLS-adapted forecasts with MLpol, a parameter-free online aggregation rule. Under boundedness conditions, we establish deterministic oracle inequalities showing that it competes with both the best raw predictor and the best bounded, time-varying affine combinations of the base predictions, up to a path-length-dependent tracking cost and a sublinear aggregation overhead. We evaluate MELO on French national electricity-load forecasting through the COVID-19 lockdown using no regime indicators, lockdown dates, or policy covariates. MELO reduces overall RMSE by 34.7% relative to base-only MLpol and achieves lower overall RMSE than a TabICL reference supplied with an external COVID policy-response covariate. Moreover, MELO requires only lightweight per-step recursive updates without model retraining.

Recent developments in digital platforms have highlighted the prevalence of open systems, where agents can arrive and depart over time. While bandit learning in open systems has recently received initial attention, existing work imposes structural assumptions that are frequently violated in practice. A learning paradigm for general open systems creates fresh challenges: newly arriving agents induce endogenous non-stationarity; agent patterns determine how quickly information accumulates; and new agents make regret scale further with the time horizon. To this end, we formulate a unified open-system bandit problem with general dynamics, including heterogeneous rewards and general agent patterns. We introduce new concepts to capture the inherent complexities: the \emph{pre-training degree} of new agents quantifies how much information an agent carries upon entry, \emph{stability} measures the impact of new agents on the system, and \emph{global dynamic regret} compares the cumulative expected reward of all active agents with that of the varying optimal arms. We develop certified global-UCB learning methodologies with provable guarantees. Our regret bounds reveal that entry uncertainty enters linearly via the pre-training degree, while in stable regimes, regret is governed by the time needed to identify a persistent optimal arm, as well as by the agent patterns. We further show that these dependencies are tight via lower bounds in hard instances.

We study online prediction for marginally stable, partially observed linear dynamical systems under nonstochastic disturbances. Our objective is to minimize the cumulative squared prediction loss and compete with the best-in-hindsight Luenberger predictor. Standard online learning methods typically rely on bounded domains/gradients, and thus their guarantees may fail to deal with potentially unbounded trajectories in marginally stable systems. In this paper, we introduce an unconstrained online least squares method that stabilizes the learning process via tailored predictive hints. With model knowledge, we prove that hints constructed from any stabilizing Luenberger predictor render the hint residuals uniformly bounded, achieving logarithmic regret despite unbounded trajectory growth. We also discuss model-free prediction and introduce a simple universal hint for symmetric systems, under which logarithmic regret is maintained without model knowledge. Our results provide an adaptive, instance-wise optimal online predictor compared to classical fixed-gain observers under nonstochastic disturbances.

In the pursuit of autonomous learning systems, the foundational assumption of stationarity, the premise that data distributions and model behaviors remain constant, is fundamentally untenable. Historically, the research community has addressed non-stationary environments almost exclusively under the scope of concept drift, focusing primarily on temporal shifts in streams. However, as learning systems become increasingly autonomous and complex, merely adapting to temporal non-stationarity is no longer sufficient. Evolving beyond this traditional perspective, we propose a novel, three-dimensional taxonomy that systematizes the field based on the operational state of the system. First, time stream drift distinguishes between stochastic arbitrary patterns and structural rhythmic dynamics. Second, data stream drift disentangles shifts in feature representations, identified as representation drift, from changes in underlying semantics, recognized as semantic drift. Third, model stream drift characterizes the internal endogenous divergence of learning systems through the lenses of sequential plasticity, decentralized heterogeneity, and policy instability. Based on this framework, we systematically review 193 representative studies and identify key open challenges. By bridging the fragmented paradigms of drift adaptation, continual learning, and temporal generalization, this survey outlines a roadmap for building self-evolving intelligent systems capable of learning autonomously through continuous change.

Nonstationary Gaussian processes (GPs) are essential for modeling complex, locally heterogeneous spatial data. A common modeling approach is the spatial deformation method that warps the domain to recover isotropy. However, this static method does not account for changes in spatial correlation induced by covariates, limiting its ability to predict nonstationary GPs under new covariate conditions. To enable predictive modeling of the deformation method, we propose to model the spatial deformation as a function of covariates. The spaces of diffeomorphic deformations and Euclidean covariate vectors are connected by characterizing deformations as generated by velocity fields living in a Lie algebra. To overcome the estimation instability caused by high-order interactions between multiple covariates in a general Lie algebra, we prove that those interactions can be truncated with a moderate physical assumption. Based on the theoretical results, a concise functional form of deformations driven by multiple covariates can be established, and an efficient estimation-inference algorithm is developed for out-of-sample nonstationary GP prediction with limited covariate-deformation sample pairs. The effectiveness and generalizability of the method are demonstrated on a simulation study and two case studies, in the fields of manufacturing and geostatistics, respectively.