Online Linear Regression with Paid Stochastic Features

TL;DR

This paper investigates online linear regression where features are noisy and can be made more accurate at a cost, analyzing regret bounds under known and unknown noise-cost mappings.

Contribution

It introduces a framework for online linear regression with paid noise reduction, deriving optimal regret rates for known and unknown noise-cost relationships.

Findings

Optimal regret rate is √T when noise covariance is known.

Regret rate becomes T^{2/3} when noise covariance is unknown.

Matrix martingale concentration is used to analyze empirical loss convergence.

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.