Online Linear Regression with Paid Stochastic Features

2025-11-11

AI Summary

This paper studies online linear regression under noisy observations, where the learner can pay to reduce observation noise and must balance prediction error against payment cost to minimize cumulative regret under a joint loss. We propose the first “payment-adjustable noise” online learning framework and rigorously characterize the optimal regret rates: $O(sqrt{T})$ when the noise covariance is known, and $O(T^{2/3})$ when it is unknown. Technically, we derive uniform convergence bounds for the empirical loss over both the payment policy space and the linear predictor space, leveraging matrix martingale concentration inequalities. Our key contributions are: (i) establishing the fundamental optimal regret order for online linear regression with controllable noise; (ii) revealing how the information structure—specifically, whether the noise covariance is known—affects inherent learning difficulty; and (iii) providing principled algorithmic design guidelines that achieve these optimal rates.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We study an online linear regression setting in which the observed feature vectors are corrupted by noise and the learner can pay to reduce the noise level. In practice, this may happen for several reasons: for example, because features can be measured more accurately using more expensive equipment, or because data providers can be incentivized to release less private features. Assuming feature vectors are drawn i.i.d. from a fixed but unknown distribution, we measure the learner's regret against the linear predictor minimizing a notion of loss that combines the prediction error and payment. When the mapping between payments and noise covariance is known, we prove that the rate $sqrt{T}$ is optimal for regret if logarithmic factors are ignored. When the noise covariance is unknown, we show that the optimal regret rate becomes of order $T^{2/3}$ (ignoring log factors). Our analysis leverages matrix martingale concentration, showing that the empirical loss uniformly converges to the expected one for all payments and linear predictors.

Problem

• Online linear regression with feature noise that can be reduced through payments
• Learner pays to decrease noise in observed feature vectors for better predictions
• Analyzes regret against optimal predictor considering both prediction error and payment costs

Innovation

• Online linear regression with paid feature noise reduction
• Regret analysis with known and unknown noise covariance
• Matrix martingale concentration for empirical loss convergence