Which Optimizer Leaks Most? Optimizer Implicit Bias as a Training-Data Privacy Controller

Releasing a neural network checkpoint can leak information about its training data, and existing reconstruction results rely on gradient-descent training. We ask whether the choice of optimizer affects the ability to recover the training data. Extending work connecting Karush--Kuhn--Tucker conditions to training-data reconstruction, we show that the implicit biases of popular optimizers (SGD, Adam, Muon) leave very different fingerprints of the training data. We argue that the key distinction is the \emph{weight-specificity} of these implicit biases, namely how much continuous information about the final weights the optimizer-dependent target of the stationarity conditions encodes. The limiting stationarity conditions for models trained with SGD and Muon retain enough weight-specific information to reconstruct training examples in practice. For Adam in the $\ell_\infty$ implicit-bias regime, the optimizer-dependent part of these conditions encodes only a discrete pattern of maximal coordinates and their signs, and at nondegenerate KKT points the conditions are too few to identify the training set to first order. Experiments on MNIST and CIFAR-10 support these claims: models trained with Muon can yield stronger reconstructions at much lower training cost than models trained with SGD, although this advantage depends on the training regime and architecture, and models trained with Adam appear consistently more resistant to the evaluated attacks.