Distributed adaptive estimation for stochastic large regression models

Abstract: This paper studies the distributed adaptiveestimation problems for stochastic large regression modelswith an infinite number of parameters. By constructing a re-cursive local cost function, we propose a novel distributedrecursive least squares algorithm to estimate the unknownsystem parameters, where the growth rate of regressors'dimension is characterized by a non-decreasing positivefunction. The almost sure convergence of the proposedalgorithm is established under a cooperative excitationcondition, which incorporates the temporal information andthe spatial information to reflect the cooperative effectamong multiple agents. Moreover, we analyze the predic-tion error by establishing the asymptotic upper boundof the accumulated regret without any excitation condi-tions. The main difficulty of theoretical analysis lies in howto analyze properties of the product of non-independentand non-stationary random matrices, whose dimensionschange over time simultaneously. Some techniques, suchas stochastic Lyapunov function, double-array martingaletheory and algebraic graph theory, are employed to dealwith the above issue. Our theoretical results are derivedwithout imposing independence or stationarity assump-tions on the regression vectors, thereby not excluding thecorrelated feedback signals.

• The paper introduces a novel distributed adaptive estimation framework that scales to infinite-dimensional regression models under minimal statistical assumptions.
• It employs a dimension-adaptive recursive least squares algorithm with diffusion-type aggregation over time-varying, directed networks to ensure almost sure parameter convergence.
• It provides rigorous non-asymptotic regret bounds and proves vanishing prediction regret even with weak excitation conditions on high-dimensional regressors.

Distributed Adaptive Estimation for Stochastic Large Regression Models

Problem Context and Motivation

The estimation of parameters in high-dimensional (potentially infinite-dimensional) stochastic regression models is of central importance in modern system identification and learning, given the rise of complex systems such as deep neural networks and LLMs. The infeasibility of centralized estimation—due to prohibitive computational and storage costs—drives the need for distributed, online algorithms, especially in settings such as sensor or multi-agent networks. Traditional distributed estimation theory relies heavily on restrictive independence and stationarity assumptions, and almost all existing theoretical frameworks are limited to finite-dimensional cases. This paper addresses the deficit by developing a distributed adaptive estimation framework tailored to large-scale or infinite-dimensional parameter models under minimal statistical assumptions.

System Model and Technical Formulation

The focus is on a network of $n$ agents observing a stochastic large regression model of the form

$$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$

where ${\Theta} \in \mathbb{R}^{\infty \times m}$ is the unknown parameter matrix, $\varphi_{k,i} \in \mathbb{R}^\infty$ is an infinite-dimensional stochastic regressor, and $w_{k+1,i}$ is a martingale difference noise. The agents communicate over a time-varying directed digraph sequence, and the regressors may be temporally and spatially correlated, violating i.i.d./stationarity conditions.

A key technical consideration is that the dimension of the regressors can grow over time following a non-decreasing function $p_t$ (with $p_t = O(t)$), subsuming both truly infinite- and large-but-finite-dimensional settings as special cases.

Algorithmic Contributions

The main algorithmic innovation is a distributed recursive least squares (RLS) scheme adapted to grow the regressor dimension over time, allowing for local cost minimization on each agent with information exchange over the network. The design utilizes:

• Recursive Local Cost Functions: Each agent recursively accrues local prediction errors, combined via convex weights from its incoming neighbors' estimates according to the current network topology.
• Dimension-Adaptive Updates: Parameter and covariance updates are performed in a growing-dimensional space, requiring careful handling of matrix dimensions.
• Diffusion-Type Aggregation: Estimates and information matrices are updated via convex combinations across the network, generalizing classic diffusion RLS to the infinite-dimensional setting.

Algorithm initialization, update structure, and matrix operations are rigorously defined to facilitate the analysis of almost-sure convergence and regret bounds.

Theoretical Analysis

A principal challenge is the analysis of products of time-varying, non-stationary, and non-independent random matrices with increasing dimension over directed graphs. The analysis leverages:

• Stochastic Lyapunov Techniques: Construction of stochastic Lyapunov functions for the growing-dimensional error process.
• Double-Array Martingale Theory: Controls noise accumulation in the absence of independence assumptions, extending results involving maximal inequalities to arrays indexed by both time and model dimension.
• Algebraic Graph Theory: Utilized to handle products of adjacency matrices and guarantee sufficient mixing of information under minimal connectivity assumptions.

Assumptions

Compared to most of the literature, key statistical restrictions are relaxed:

• Independence and stationarity of the regressors are not assumed; the setting encompasses feedback-induced dependencies.
• Only a mild cooperative excitation condition is required: persistent but network-distributed excitation suffices, in contrast to standard PE conditions or i.i.d. assumptions.

Parameter Convergence

The authors establish almost sure strong consistency of the distributed RLS estimate under a cooperative excitation condition, formulated in terms of the network-wide information matrix's minimum eigenvalue and the growth rate of regressor dimension. Specifically, for an appropriate scaling of $p_t$ (e.g., $p_t = O(\log^\alpha t)$ or $p_t = O(t^\alpha)$ for $$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$0), convergence to the true parameter is achieved if the ratio of parameter dimension and accumulated network excitation decays to zero:

$$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$1

where $$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$2 is the minimum eigenvalue of the aggregated regressor covariance over the joint communication time windows.

This result directly generalizes known results for finite-dimensional distributed RLS, and also strengthens previous single-agent infinite-dimensional results (e.g., [Guo1991], [DAI2025]) by enabling cooperative estimation where information is deficient locally but sufficient globally.

Regret Analysis

A second major result is a non-asymptotic upper bound on the accumulated prediction regret. The average regret vanishes ($$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$3 per step) for certain regimes of regressor dimension growth, irrespective of excitation conditions. This means that, even without "persistent excitation," the online predictor remains competitive with the best offline predictor in hindsight, a property highly desirable in nonstationary and high-dimensional environments.

Numerical Performance and Comparative Claims

The paper makes several strong claims, supported by detailed non-asymptotic rates:

• The principal convergence rates scale with the regressor dimension $$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$4, the network excitation $$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$5, and noise accumulation, with tight upper bounds given explicit forms (e.g., regret is $$y^{\top}_{k+1,i} = \varphi^{\top}_{k,i}{\Theta} + w^{\top}_{k+1,i},$$6 in favorable regimes).
• The algorithm requires significantly weaker excitation compared to the classic PE or even block PE conditions often required for distributed RLS [Xie2021, Sayed2011Incre], and is robust to feedback-induced dependence in the regressors.
• The analysis holds under weak network connectivity assumptions (jointly strongly connected time windows, weight-balanced digraphs), encompassing realistic multi-agent topologies.

Implications and Future Directions

From a theoretical perspective, this work bridges a crucial gap in distributed estimation literature: it rigorously extends strong consistency, finite sample performance, and optimal regret results to the context of large- or infinite-dimensional parameter models over time-varying directed networks. By not relying on independence or stationarity, the results are applicable to settings with correlated, adversarial, or feedback-driven signals—ubiquitous in modern cyber-physical systems and networked learning scenarios.

Practically, the dimension-adaptive distributed RLS enables scalable online estimation in large-scale sensor and agent networks, federated learning architectures, and applications in control and reinforcement learning with high-dimensional bases. The algorithmic framework and analysis also lay a solid foundation for integrating privacy-preserving or communication-efficient extensions, such as those motivated by contemporary directions in federated and decentralized learning.

Open research directions include the study of adaptive selection strategies for regressor dimension, distributed nonlinear and multi-layer model identification, robustness to network or communication faults, and the incorporation of privacy constraints.

Conclusion

This paper presents rigorous algorithmic and theoretical advances for distributed adaptive estimation in stochastic large regression models, accommodating infinite-dimensional parameters, correlated regressors, and time-varying network topologies. The derived distributed RLS algorithm achieves almost sure parameter consistency under minimal excitation and statistical assumptions and guarantees vanishing regret in prediction, thus providing a foundational framework for high-dimensional online multi-agent learning and system identification (2604.25468).