Estimating curvature information from gradients alone is a fundamental problem in machine learning. The challenge becomes particularly acute in settings where only aggregate mini-batch gradients are available, yet curvature information is still required for applications such as preconditioning in optimisation, sampling, and early stopping. Perhaps surprisingly, we show that a simple target-perturbation scheme suffices to recover such curvature information from gradients alone. Our method injects zero-mean noise into the targets, with variance calibrated to the local output curvature. The covariance of the resulting noisy batch gradients then recovers practical curvature matrices, without requiring per-sample gradients, second-order derivatives, or other internal network quantities. For a broad class of non-linear models, our scheme recovers the Generalised Gauss--Newton (GGN) matrix; for feed-forward ReLU networks, it additionally recovers the diagonal blocks of the population Hessian. We demonstrate that the resulting estimator is simple, lightweight, provably optimal, and easy to integrate into existing training pipelines. Experiments demonstrate accurate curvature recovery, faster optimisation when the estimator is used for preconditioning, and the utility of the estimated GGN for early stopping.

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