Distributed Online Linear Regression

We study online linear regression problems in a distributed setting, where the data is spread over a network. In each round, each network node proposes a linear predictor, with the objective of fitting the \emph{network-wide} data. It then updates its predictor for the next round according to the received local feedback and information received from neighboring nodes. The predictions made at a given node are assessed through the notion of regret, defined as the difference between their cumulative network-wide square errors and those of the best off-line network-wide linear predictor. Various scenarios are investigated, depending on the nature of the local feedback (full information or bandit feedback), on the set of available predictors (the decision set), and the way data is generated (by an oblivious or adaptive adversary). We propose simple and natural distributed regression algorithms, involving, at each node and in each round, a local gradient descent step and a communication and averaging step where nodes aim at aligning their predictors to those of their neighbors. We establish regret upper bounds typically in ${\cal O}(T^{3/4})$ when the decision set is unbounded and in ${\cal O}(\sqrt{T})$ in case of bounded decision set.

💡 Research Summary

The paper addresses the problem of online linear regression in a distributed setting where data are spread across the nodes of a communication network. At each time step t, every node i holds a local covariate vector h_i(t) ∈ ℝ^m and an outcome z_i(t) ∈ ℝ. The goal is to maintain a local predictor x_i(t) that, when evaluated on the whole network, incurs a cumulative squared loss close to that of the best offline linear predictor y* (the solution of the network‑wide least‑squares problem). Performance is measured by regret, defined as the difference between the cumulative network‑wide loss of a node’s predictions and the loss of y*.

Two feedback models are considered. In the full‑information case each node observes both h_i(t) and z_i(t) after making its prediction. In the bandit case the node only observes the loss value at two nearby points and must estimate a gradient via a two‑point finite‑difference scheme. The network is modeled as a strongly connected directed graph G = (V, E) with a doubly stochastic weight matrix W_G that governs the averaging of neighboring estimates.

The authors propose two simple algorithms:

1. DOLR‑FIF (Full‑Information Feedback) – each node performs a local gradient descent step on its instantaneous loss with step size 1/(α_h T^β) (β = 3/4) and then averages the resulting intermediate vectors with its neighbors using the weights of W_G.

2. DOLR‑BF (Bandit Feedback) – each node queries the loss at x_i(t) ± ε u_i(t) (u_i(t) a random unit vector, ε = 1/√T), constructs an unbiased gradient estimator g_i,t, takes a similar gradient step with learning rate 1/(κ T^β) (κ chosen larger than 2 n m^2 α_h/(n−α_h)), and finally averages with neighbors.

Both algorithms are fully decentralized: each round requires only one vector transmission per edge. The analysis hinges on two key quantities: (i) the accumulated magnitude of the (possibly unbounded) gradients, and (ii) the cumulative disagreement among the nodes’ predictors, which is controlled by the second largest singular value σ₂(W_G) < 1. By carefully selecting β = 3/4, the authors bound the gradient accumulation and show that the disagreement term decays geometrically, leading to a regret upper bound of order O(T^{3/4}) when the decision set K = ℝ^m (unbounded). The constant C_G depends on σ₂(W_G), the number of nodes n, and the initial distance to y*.

When the decision set K is a bounded convex set, the algorithms can incorporate a projection onto K after each update. This yields the classic O(√T) regret, but the projection may be computationally expensive for complex K. To avoid costly projections, the authors adopt the “optimization with long‑term constraints” framework: they relax the constraints to a simple Euclidean ball that contains K, allowing cheap projections while penalizing constraint violations. In this setting they achieve O(√T) regret and O(T^{3/4}) cumulative constraint violation, both under full‑information and bandit feedback.

The paper also treats adaptive adversaries (where the data (h_i(t), z_i(t)) may depend on past predictions) and the special case where an exact solution y* exists such that the total loss is zero. In both scenarios, the same algorithms attain O(√T) regret even with bandit feedback.

A detailed communication‑regret trade‑off analysis is provided. The regret constant scales with σ₂(W_G) and n. For a complete graph σ₂ = 0, giving the smallest regret but requiring O(n²) messages per round. For random geometric graphs σ₂ ≈ 1 – Θ(log n / n), the regret grows roughly as n⁶·log²n while only O(n log n) messages are exchanged. For k‑regular expanders σ₂ is constant, leading to regret O(n⁴) with O(k n) messages. For a path graph σ₂ ≈ 1 – Θ(1/n²), regret scales as O(n⁸) with only O(n) messages. These results illustrate how network topology influences both learning performance and communication load.

Compared with prior work, this is the first paper to provide regret guarantees for distributed online linear regression without assuming bounded gradients or bounded decision sets. It extends classical online gradient descent and multi‑point bandit methods to a networked environment, and it integrates long‑term constraint techniques to handle complex feasible sets.

In summary, the authors present a clean, theoretically grounded approach to distributed online linear regression that achieves sublinear regret under realistic assumptions, quantifies the impact of network structure on performance, and opens avenues for future research on asynchronous updates, time‑varying graphs, privacy‑preserving extensions, and empirical validation.