Near Optimal Heteroscedastic Regression with Symbiotic Learning

We consider the classical problem of heteroscedastic linear regression, where we are given $n$ samples $(\mathbf{x}_i, y_i) \in \mathbb{R}^d \times \mathbb{R}$ obtained from $y_i = \langle \mathbf{w}^{*}, \mathbf{x}_i \rangle + \epsilon_i \cdot \langle \mathbf{f}^{*}, \mathbf{x}_i \rangle$, where $\mathbf{x}_i \sim N(0,\mathbf{I})$, $\epsilon_i \sim N(0,1)$, and our task is to estimate $\mathbf{w}^{*}$. In addition to the classical applications of heteroscedastic models in fields such as statistics, econometrics, time series analysis etc., it is also particularly relevant in machine learning when data is collected from multiple sources of varying but apriori unknown quality, e.g., large model training. Our work shows that we can estimate $\mathbf{w}^{*}$ in squared norm up to an error of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2 \cdot \left(\frac{1}{n} + \left(\frac{d}{n}\right)^2\right)\right)$ and prove a matching lower bound (up to logarithmic factors). Our result substantially improves upon the previous best known upper bound of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2\cdot \frac{d}{n}\right)$. Our upper bound result is based on a novel analysis of a simple, classical heuristic going back to at least Davidian and Carroll (1987) and constitutes the first non-asymptotic convergence guarantee for this approach. As a byproduct, our analysis also provides improved rates of estimation for both linear regression and phase retrieval with multiplicative noise, which maybe of independent interest. The lower bound result relies on a careful application of LeCam's two point method, adapted to work with heavy tailed random variables where the relevant mutual information quantities are infinite (precluding a direct application of LeCam's method), and could also be of broader interest.