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Online Linear Regression with Paid Stochastic Features

Nadav Merlis, Kyoungseok Jang, Nicolò Cesa-Bianchi

TL;DR

This work tackles online linear regression with noisy, paid-for features where observations are $\,\hat{x}_t = x_t + n_t(c_t)$ and predictions are linear. It develops two regimes: (i) known noise covariances $\Sigma_n(c)$, where a uniform-loss estimator leveraging data across all payments yields a regret of $\widetilde{\mathcal{O}}(\sqrt{T})$, and (ii) unknown covariances, where a grid-based optimistic-loss strategy gives $\widetilde{\mathcal{O}}(T^{2/3})$ regret; both results are shown via matrix martingale concentration to guarantee uniform convergence of the empirical loss. The paper also derives matching one-dimensional lower bounds and connects the unknown-covariance setting to Lipschitz-bandit frameworks. Overall, it demonstrates that paying to reduce feature-noise can be efficiently learned online, achieving near-optimal regret while balancing prediction accuracy and payment costs. The findings have implications for privacy-aware data collection and resource-allocation in experimental settings, where the cost of noise reduction can be managed to optimize predictive performance.

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.

Online Linear Regression with Paid Stochastic Features

TL;DR

This work tackles online linear regression with noisy, paid-for features where observations are and predictions are linear. It develops two regimes: (i) known noise covariances , where a uniform-loss estimator leveraging data across all payments yields a regret of , and (ii) unknown covariances, where a grid-based optimistic-loss strategy gives regret; both results are shown via matrix martingale concentration to guarantee uniform convergence of the empirical loss. The paper also derives matching one-dimensional lower bounds and connects the unknown-covariance setting to Lipschitz-bandit frameworks. Overall, it demonstrates that paying to reduce feature-noise can be efficiently learned online, achieving near-optimal regret while balancing prediction accuracy and payment costs. The findings have implications for privacy-aware data collection and resource-allocation in experimental settings, where the cost of noise reduction can be managed to optimize predictive performance.

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 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 (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.

Paper Structure

This paper contains 24 sections, 17 theorems, 113 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline.
  3. Related Work
  4. Errors-in-variables models.
  5. Local DP.
  6. Online learning with noisy features.
  7. Privacy and incentives.
  8. Additional Notations.
  9. Regularity assumptions.
  10. Optimality Criterion.
  11. Regret Definition
  12. Online Linear Regression with Known Noise Covariances
  13. Tractability of the loss minimization.
  14. Statistical efficiency.
  15. Extension to Unknown Noise Covariances
  16. ...and 9 more sections

Key Result

Lemma 0

For any arbitrary history-dependent prediction rule $\hat{y}_t:\mathbb{R}^d\mapsto \mathbb{R}$ and any arbitrary sequence $\nu_1,\dots,\nu_T\in\mathbb{R}^d$, both potentially depend on $I_{t-1}^o$, it holds that In particular, if the algorithm only outputs linear predictions $\hat{y}_t(\hat{x}_t) = \hat{x}_t^\top \nu_t$, then

Theorems & Definitions (43)

  • Remark 1
  • Claim 0
  • Claim 0
  • Remark 2
  • Lemma 0
  • Proposition 0
  • Remark 3
  • Theorem 0
  • Theorem 0
  • Corollary 0
  • ...and 33 more