Regret Guarantees for Model-Free Cooperative Filtering under Asynchronous Observations

Predicting the output of a dynamical system from streaming data is fundamental to real-time feedback control and decision-making. We first derive an autoregressive representation that relates future local outputs to asynchronous past outputs. Building on this structure, we propose an online least-squares algorithm to learn this autoregressive model for real-time prediction. We then establish a regret bound of O(log^3 N) relative to the optimal model-based predictor, which holds for marginally stable systems. Moreover, we provide a sufficient condition characterized via a symplectic matrix, under which the proposed cooperative online learning method provably outperforms the optimal model-based predictor that relies solely on local observations. From a technical standpoint, our analysis exploits the orthogonality of the innovation process under asynchronous data structure and the persistent excitation of the Gram matrix despite delay-induced asymmetries. Overall, these results offer both theoretical guarantees and practical algorithms for model-free cooperative prediction with asynchronous observations, thereby enriching the theory of online learning for dynamical systems.

💡 Research Summary

This paper tackles the challenge of real-time prediction for linear stochastic dynamical systems in a practical setting where multiple sources of data are available asynchronously due to communication delays, and no prior knowledge of the system model is assumed. The authors develop a novel model-free cooperative filtering framework that integrates local past observations with delayed external observations to predict future local outputs.

The core contributions are fourfold. First, the authors derive an autoregressive (AR) representation that explicitly links future local outputs to past local and delayed external outputs. A key technical insight is proving that the innovation processes within this AR model remain orthogonal despite the asymmetries introduced by the time delays (Theorem 1). This preserved orthogonality is crucial for the subsequent regret analysis.

Second, building on this AR structure, they propose an online least-squares algorithm to learn the AR parameters in real-time. The primary theoretical guarantee is a regret bound of O(log^3 N) for this algorithm, where regret is defined as the cumulative excess prediction error compared to the optimal model-based cooperative predictor that has full knowledge of the system and noise statistics (Theorem 2). This logarithmic regret bound holds for marginally stable systems and signifies that the model-free learner’s performance converges to near-optimal at a sublinear rate. Establishing this bound requires overcoming significant analytical hurdles posed by the asymmetric data structure, leading to new proofs for the persistent excitation of the delayed Gram matrix.

Third, the paper goes beyond matching a benchmark and provides a sufficient condition under which the proposed cooperative method provably outperforms the optimal predictor that uses only local observations. Using analysis based on a symplectic matrix associated with the Riccati equation, the authors characterize conditions on the external sensor quality (Ce, Re) and the delay d such that, with high probability and for a sufficiently long time horizon, the online cooperative filter achieves a lower prediction error than the best possible local filter (Theorem 3, Corollary 5.1). This result formally quantifies when delayed external information leads to a fundamental performance improvement.

Fourth, from a technical standpoint, the analysis develops new tools to handle the asymmetry induced by asynchronous observations, which renders standard analytical techniques from single-stream online learning inapplicable. This includes establishing high-probability lower bounds on the minimum eigenvalue of the asymmetric and delay-parameterized Gram matrix, ensuring its persistent excitation.

In summary, this work provides the first model-free cooperative filtering algorithm with logarithmic regret guarantees under asynchronous observations. It bridges the gap between classical model-based estimation theory with delays and modern online learning, offering both a practical algorithm and rigorous theoretical foundations for cooperative prediction in networked systems with heterogeneous data sources and communication latencies. The results enrich the theory of online learning for dynamical systems and have potential applications in areas like distributed sensor networks, traffic flow prediction, and power grid estimation.