recursive least squares

To address the performance degradation of conventional recursive least squares (RLS) algorithms under strong harmonic interference in power grid event estimation, this paper proposes a novel RLS algorithm featuring a second-order update mechanism. The method integrates exponential and instantaneous forgetting strategies, reconstructs the parameter update formulation using second-order gradient information, and establishes new theoretical properties regarding the convergence of both the inverse information matrix and the parameter vector—enabling superior adaptive forgetting design. Compared with classical first-order RLS, the proposed algorithm achieves significantly improved tracking accuracy and faster convergence in dynamic harmonic environments. Its effectiveness and robustness are validated across multiple typical grid events, including voltage sags and resonance transients. The approach provides a new paradigm for real-time state estimation in high-interference scenarios.

This paper addresses the online Kalman prediction problem for unknown, non-explosive linear stochastic systems, where conventional recursive least squares (RLS)-based methods suffer from overfitting and performance degradation due to ill-conditioned regression matrices. We propose a model-free online prediction framework that—uniquely—employs an exponential forgetting mechanism not merely for data weighting, but to dynamically balance regression model structure. Integrated with refined recursive updates and a novel error decomposition analysis, we rigorously establish a regret upper bound of $O(log^3 N)$, markedly improving upon the $O(sqrt{N})$ bound of RLS-type methods. The theoretical analysis guarantees stable learning under minimal assumptions. Empirical evaluations demonstrate over 37% reduction in prediction error. Our work provides both tight theoretical guarantees and an efficient implementation for high-accuracy, real-time state estimation in the absence of prior system knowledge.

In reinforcement learning, adaptive interaction data—where the behavior policy is nonstationary—invalidates standard estimators, undermining asymptotic normality for off-policy counterfactual policy evaluation and dynamic treatment effect (DTE) inference. To address this, we propose a weighted Z-estimation framework that constructs time-varying adaptive weights to stabilize heteroskedasticity, achieving, for the first time in the RL off-policy setting, both consistent and asymptotically normal DTE estimation. Our approach integrates dynamic causal inference with asymptotic statistical theory, enabling rigorous hypothesis testing and construction of uniformly valid confidence regions. Simulation studies and real-world RL experiments demonstrate substantial improvements in confidence interval coverage and statistical power. The method provides the first solution for structural parameter inference under adaptive experimentation that simultaneously offers theoretical guarantees—namely consistency, asymptotic normality, and uniform validity—and empirical robustness.

This paper addresses real-time identification of discrete-time state-dependent switching systems. We propose a two-time-scale adaptive algorithm: on the slow time scale, an online deterministic annealing mechanism estimates the unknown mode-switching signal and progressively identifies the number of modes; on the fast time scale, recursive least squares updates parameters of local models. To our knowledge, this is the first work to incorporate deterministic annealing into online switching system identification, thereby eliminating the need for prior knowledge of the number of modes—a key limitation of conventional methods—while ensuring both identifiability and computational efficiency. Theoretical analysis, grounded in stochastic approximation theory, accommodates both input–output and state-space modeling frameworks and supports piecewise-affine structures. Simulation results demonstrate the algorithm’s convergence, robustness, and low-latency adaptability, achieving high modeling accuracy while significantly improving real-time efficiency.

This paper addresses asymptotically efficient parameter identification for stochastic dynamic systems under output observation saturation—a prevalent class of nonlinearities. To overcome the limitations of existing methods, which rely on restrictive assumptions such as input signal periodicity or independence, we establish, for the first time, asymptotic achievability of the Cramér–Rao lower bound (CRLB) without such assumptions. We propose an adaptive Newton-type algorithm based on the negative log-likelihood function and a two-stage design, applicable to general stochastic feedback systems. Rigorous convergence analysis—leveraging stochastic Lyapunov theory and martingale limit theorems—establishes strong consistency, asymptotic normality, and mean-square error convergence to the CRLB. Numerical experiments demonstrate that the proposed method significantly outperforms existing algorithms in the same class.

This work addresses the challenge of simultaneously achieving real-time responsiveness, stability, and tracking performance in nonstationary data streams, where conventional adaptive algorithms often fall short. Focusing on the Momentum Least Mean Squares (MLMS) algorithm, the study establishes, for the first time, rigorous stability conditions, tracking error bounds, and dynamic regret bounds under time-varying stochastic linear systems. The analysis overcomes the significant theoretical difficulty posed by second-order products of stochastic matrices induced by the momentum term. By integrating tools from stochastic vector difference equations, nonstationary time series modeling, and online learning theory, the paper demonstrates that MLMS exhibits both rapid adaptability and robust tracking capability. Empirical evaluations on synthetic and real-world data streams confirm that MLMS significantly outperforms the classical LMS algorithm.

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.

This work addresses the inconsistency among multi-granularity forecasts in online hierarchical time series prediction by proposing a reconciliation method that explicitly models hierarchical relationships through a graph structure. The approach characterizes forecast residuals using a matrix normal distribution and formulates a multivariate linear regression framework, integrating ridge regression, Bayesian estimation, and shrinkage principles. An efficient online recursive inference mechanism is developed to enable adaptive forecast reconciliation and uncertainty quantification. The method is validated on a district heating load forecasting task, demonstrating its effectiveness. To support practical deployment, the authors release PyOnlineForecast, an open-source toolkit for online hierarchical forecasting.

This study addresses online learning of the Kalman filter for output and state estimation in partially observable linear dynamical systems with unknown system models. The authors propose a unified algorithmic framework based on online optimization, incorporating a stochastic querying mechanism to handle limited observability. Their theoretical analysis establishes, for the first time, that sublinear regret in state estimation is unattainable without queries, yet a √T regret bound becomes achievable with a finite number of stochastic queries, revealing a fundamental trade-off between query complexity and regret. The proposed algorithm attains a logarithmic regret bound (log T) for output estimation and a √T regret bound for state estimation. Numerical experiments corroborate the theoretical findings and demonstrate the algorithm’s empirical effectiveness.

This paper addresses the challenge of modeling time-varying parameters in financial and macroeconomic time series. We propose an implicit score-driven filtering framework that jointly maximizes the observed log-density and penalizes parameter deviations from predictions, enabling robust time-varying estimation in nonlinear dynamic systems. Our key contribution is the first introduction of an implicit update mechanism: leveraging optimization theory for log-concave observation densities, it preserves full posterior density information while ensuring filter stability and mean-squared error contraction—eliminating the sensitivity to learning-rate tuning inherent in explicit methods. The algorithm integrates implicit stochastic gradient updates, weighted ℓ₂ regularization, and one-step-ahead prediction. Empirical results demonstrate substantial improvements in estimation accuracy and stability across diverse financial and macroeconomic datasets, with strong robustness under model misspecification.