On Robustness of Principal Component Regression

Consider the setting of Linear Regression where the observed response variables, in expectation, are linear functions of the $p$-dimensional covariates. To achieve vanishing prediction error, the number of required samples scales faster than $p \sigma^2$, where $\sigma^2$ is a bound on the noise variance. In high-dimensional settings where $p$ is large but the covariates admit a low-dimensional representation (say $r \ll p$), Principal Component Regression (PCR) is an effective approach; here, the response variables are regressed with respect to the principal components of the covariates. The resulting number of required samples to achieve vanishing prediction error now scales faster than $r \sigma^2 (\ll p \sigma^2)$. Despite the tremendous utility of PCR, its ability to handle settings with noisy, missing, and mixed (discrete and continuous) valued covariates is not understood and remains an important open challenge. As our main contribution, we address this challenge by rigorously establishing that PCR is robust to noisy, sparse, and possibly mixed valued covariates. Specifically, under PCR, vanishing prediction error is achieved if the number of samples scales as $r \max(\sigma^2, \rho^{-4} \log^5(p))$, where $\rho$ denotes the fraction of observed (noisy) covariates. We establish generalization error bounds for PCR, which provides a systematic approach to select the number of components $r$ in a data-driven manner. The key to our result is a simple, but powerful equivalence between: (i) PCR; (ii) Linear Regression with covariate pre-processing via Hard Singular Value Thresholding (HSVT). From a technical standpoint, this work advances the state-of-the-art analysis for HSVT by establishing stronger guarantees with respect to the $\| \cdot \|_{2, \infty}$-error for the estimated matrix rather than the Frobenius norm as is standard in the matrix estimation/completion literature.