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Optimal Algorithms in Linear Regression under Covariate Shift: On the Importance of Precondition

Yuanshi Liu, Haihan Zhang, Qian Chen, Cong Fang

TL;DR

This work establishes an information-theoretic minimax optimality result for high-dimensional linear regression under covariate shift, showing that the minimax estimator is a linear preconditioned transform of the source-domain estimator with an efficiently computable preconditioner. It introduces a convex program to select the optimal preconditioner $\boldsymbol{A}$ and proves a matching upper bound, thereby characterizing the minimax rate. Beyond the lower bound, the authors analyze ASGD and SGD with momentum, providing instance-based conditions under which ASGD attains optimality in a broad class of target distributions via a preconditioning interpretation. The study also uncovers emergent phenomena in learning curves under covariate shift and discusses practical implications for designing preconditioned optimization methods in covariate-shift settings.

Abstract

A common pursuit in modern statistical learning is to attain satisfactory generalization out of the source data distribution (OOD). In theory, the challenge remains unsolved even under the canonical setting of covariate shift for the linear model. This paper studies the foundational (high-dimensional) linear regression where the ground truth variables are confined to an ellipse-shape constraint and addresses two fundamental questions in this regime: (i) given the target covariate matrix, what is the min-max \emph{optimal} algorithm under covariate shift? (ii) for what kinds of target classes, the commonly-used SGD-type algorithms achieve optimality? Our analysis starts with establishing a tight lower generalization bound via a Bayesian Cramer-Rao inequality. For (i), we prove that the optimal estimator can be simply a certain linear transformation of the best estimator for the source distribution. Given the source and target matrices, we show that the transformation can be efficiently computed via a convex program. The min-max optimal analysis for SGD leverages the idea that we recognize both the accumulated updates of the applied algorithms and the ideal transformation as preconditions on the learning variables. We provide sufficient conditions when SGD with its acceleration variants attain optimality.

Optimal Algorithms in Linear Regression under Covariate Shift: On the Importance of Precondition

TL;DR

This work establishes an information-theoretic minimax optimality result for high-dimensional linear regression under covariate shift, showing that the minimax estimator is a linear preconditioned transform of the source-domain estimator with an efficiently computable preconditioner. It introduces a convex program to select the optimal preconditioner and proves a matching upper bound, thereby characterizing the minimax rate. Beyond the lower bound, the authors analyze ASGD and SGD with momentum, providing instance-based conditions under which ASGD attains optimality in a broad class of target distributions via a preconditioning interpretation. The study also uncovers emergent phenomena in learning curves under covariate shift and discusses practical implications for designing preconditioned optimization methods in covariate-shift settings.

Abstract

A common pursuit in modern statistical learning is to attain satisfactory generalization out of the source data distribution (OOD). In theory, the challenge remains unsolved even under the canonical setting of covariate shift for the linear model. This paper studies the foundational (high-dimensional) linear regression where the ground truth variables are confined to an ellipse-shape constraint and addresses two fundamental questions in this regime: (i) given the target covariate matrix, what is the min-max \emph{optimal} algorithm under covariate shift? (ii) for what kinds of target classes, the commonly-used SGD-type algorithms achieve optimality? Our analysis starts with establishing a tight lower generalization bound via a Bayesian Cramer-Rao inequality. For (i), we prove that the optimal estimator can be simply a certain linear transformation of the best estimator for the source distribution. Given the source and target matrices, we show that the transformation can be efficiently computed via a convex program. The min-max optimal analysis for SGD leverages the idea that we recognize both the accumulated updates of the applied algorithms and the ideal transformation as preconditions on the learning variables. We provide sufficient conditions when SGD with its acceleration variants attain optimality.

Paper Structure

This paper contains 44 sections, 44 theorems, 249 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation and Minimax Lower Bound
  4. Problem Setup
  5. Linear Regression under Covariate Shifts
  6. Assumptions
  7. Minimax Lower Bound
  8. Optimal Algorithm
  9. Optimality of ASGD
  10. ASGD Upper Bound
  11. Optimality under the Diagonal Dominant Condition
  12. Further Discussions
  13. Optimality Beyond the Diagonal Dominant Condition
  14. When is Emergence Possible?
  15. Conclusion
  16. ...and 29 more sections

Key Result

Theorem 1

Let $W=\left\{\boldsymbol{\mathbf{w}}^*\in\mathbb{R}^d:\left\Vert\boldsymbol{\mathbf{w}}^*\right\Vert_{\boldsymbol{\mathbf{M}}}^2\leq 1\right\}$ as in eq: constraint and $\boldsymbol{\mathbf{S}}$, $\boldsymbol{\mathbf{T}}$ are positive semi-definite matrices. Given probability $\tilde{P}\in\mathcal{ where $\boldsymbol{\mathbf{S}}'=\boldsymbol{\mathbf{M}}^{-1/2}\boldsymbol{\mathbf{S}}\boldsymbol{\m

Theorems & Definitions (100)

  • Remark
  • Remark
  • Definition 1
  • Remark : Prior knowledge of $\boldsymbol{\mathbf{S}}$ and $\boldsymbol{\mathbf{T}}$
  • Theorem 1: Lower Bound
  • Remark
  • Theorem 2: Upper Bound
  • Theorem 3: Matching Bounds
  • Theorem 4: Upper Bound of Accelerated SGD
  • Remark
  • ...and 90 more