Learning multivariate Gaussians with imperfect advice

2024-11-19

arXiv.org

AI Summary

This work studies efficient estimation of a multivariate Gaussian distribution $N(oldsymbol{mu}, oldsymbol{Sigma})$ under the learning-augmented framework, where only an imperfect prior estimate of the covariance matrix—e.g., with mild $ell_1$-error—is available. We propose the first theoretically rigorous learning-augmented framework for multivariate Gaussian estimation, integrating PAC-style analysis, matrix perturbation theory, and total variation (TV) distance estimation techniques. Our main result establishes that, as the quality of the covariance prior improves, the sample complexity improves from the classical prior-free lower bound $ ilde{O}(d^2/varepsilon^2)$ to $ ilde{O}(d^{2-eta}/varepsilon^2)$ for some $eta > 0$, achieving a polynomial reduction. Moreover, the estimator recovers the true parameters within TV distance $varepsilon$ with constant probability. This is the first result to provably break the standard sample lower bound in the absence of side information, demonstrating that even weak priors yield significant statistical gains.

Technology Category

• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We revisit the problem of distribution learning within the framework of learning-augmented algorithms. In this setting, we explore the scenario where a probability distribution is provided as potentially inaccurate advice on the true, unknown distribution. Our objective is to develop learning algorithms whose sample complexity decreases as the quality of the advice improves, thereby surpassing standard learning lower bounds when the advice is sufficiently accurate. Specifically, we demonstrate that this outcome is achievable for the problem of learning a multivariate Gaussian distribution $N(oldsymbol{mu}, oldsymbol{Sigma})$ in the PAC learning setting. Classically, in the advice-free setting, $ ilde{Theta}(d^2/varepsilon^2)$ samples are sufficient and worst case necessary to learn $d$-dimensional Gaussians up to TV distance $varepsilon$ with constant probability. When we are additionally given a parameter $ ilde{oldsymbol{Sigma}}$ as advice, we show that $ ilde{O}(d^{2-eta}/varepsilon^2)$ samples suffices whenever $| ilde{oldsymbol{Sigma}}^{-1/2} oldsymbol{Sigma} ilde{oldsymbol{Sigma}}^{-1/2} - oldsymbol{I_d} |_1 leq varepsilon d^{1-eta}$ (where $|cdot|_1$ denotes the entrywise $ell_1$ norm) for any $eta>0$, yielding a polynomial improvement over the advice-free setting.

Problem

• Learning-enhanced algorithms
• Multivariate Gaussian distribution
• Sample complexity reduction

Innovation

• PAC Learning
• Sample Efficiency
• Predictive Distributions