Hyperparameter optimization of neural networks can be elegantly formulated as a bilevel optimization problem. While research on bilevel optimization of neural networks has been dominated by implicit differentiation and unrolling, hypernetworks such as Self-Tuning Networks (STNs) have recently gained traction due to their ability to amortize the optimization of the inner objective. In this paper, we diagnose several subtle pathologies in the training of STNs. Based on these observations, we propose the Delta-STN, an improved hypernetwork architecture which stabilizes training and optimizes hyperparameters much more efficiently than STNs. The key idea is to focus on accurately approximating the best-response Jacobian rather than the full best-response function; we achieve this by reparameterizing the hypernetwork and linearizing the network around the current parameters. We demonstrate empirically that our Delta-STN can tune regularization hyperparameters (e.g. weight decay, dropout, number of cutout holes) with higher accuracy, faster convergence, and improved stability compared to existing approaches.