Information Hidden in Gradients of Regression with Target Noise

Second-order information—such as curvature or data covariance—is critical for optimisation, diagnostics, and robustness. However, in many modern settings, only the gradients are observable. We show that the gradients alone can reveal the Hessian, equalling the data covariance $\Sigma$ for the linear regression. Our key insight is a simple variance calibration: injecting Gaussian noise so that the total target noise variance equals the batch size ensures that the empirical gradient covariance closely approximates the Hessian, even when evaluated far from the optimum. We provide non-asymptotic operator-norm guarantees under sub-Gaussian inputs. We also show that without such calibration, recovery can fail by an $\Omega(1)$ factor. The proposed method is practical (a “set target-noise variance to $n$” rule) and robust (variance $\mathcal{O}(n)$ suffices to recover $\Sigma$ up to scale). Applications include preconditioning for faster optimisation, adversarial risk estimation, and gradient-only training, for example, in distributed systems. We support our theoretical results with experiments on synthetic and real data.