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

Data reconstruction attacks on trained neural networks aim to recover the data on which the network has been trained and pose a significant threat to privacy, especially if the training dataset contains sensitive information. Here, we propose a unified optimization formulation of the data reconstruction problem based on initial and trained parameter values, incorporating state-of-the-art proposals. We show that in the random feature model, this formulation provably leads to training data reconstruction with high probability, provided the network width is sufficiently large; this unprecedented finite-width result uses PAC-style bounds. Furthermore, when the data lies in a low-dimensional subspace, we show that the network width requirement for successful reconstruction can be relaxed, with bounds depending on the subspace dimension rather than the ambient dimension. For general neural network models and unknown data orientations, we propose an efficient reconstruction algorithm that approximates the low-dimensional data subspace through the change in the first-layer weights during training and uses only the last-layer weights for reconstruction, thus reducing the search space dimension and the required network width for high-quality reconstructions. Our numerical experiments on synthetic datasets and CIFAR-10 confirm that our subspace-aware reconstruction approach outperforms standard full-space techniques.

In this paper, we challenge the prevailing view that information dependency (including rote memorization) drives training data exposure to image reconstruction attacks. We show that extensive exposure can persist without rote memorization and is instead caused by a tunable connection to adversarial robustness. We begin by presenting three surprising results: (1) recent defenses that inhibit reconstruction by Model Inversion Attacks (MIAs), which evaluate leakage under an idealized attacker, do not reduce standard measures of information dependency (HSIC); (2) models that maximally memorize their training datasets remain robust to MIA reconstruction; and (3) models trained without seeing 97% of the training pixels, where recent information-theoretic bounds give arbitrarily strong privacy guarantees under standard assumptions, can still be devastatingly reconstructed by MIA. To explain these findings, we provide causal evidence that privacy under MIA arises from what the adversarial examples literature calls ``non-robust'' features (generalizable but imperceptible and unstable features). We further show that recent MIA defenses obtain their privacy improvements by unintentionally shifting models toward such features. To establish this causal relationship, we introduce Anti Adversarial Training (AT-AT), a training regime that intentionally learns non-robust features to obtain both superior reconstruction defense and higher accuracy than state-of-the-art defenses. Our results revise the prevailing understanding of training data exposure and reveal a new privacy-robustness tradeoff.

Adversarial training (AT) remains one of the most reliable empirical defenses against adversarial attacks. Its robustness critically depends on how the underlying min-max objective is optimized. In practice, Stochastic Gradient Descent (SGD) optimizer remains the default optimization choice for AT, whereas adaptive optimizers often improve standard training but may yield inferior robustness. Recently, the Muon optimizer, which orthogonalizes matrix-valued updates via an approximate polar decomposition, has achieved notable success in large-scale training at a memory cost comparable to SGD. This raises a security-relevant question: can orthogonalized optimization improve AT under strong and heterogeneous threat models? Focusing on this problem, we conduct a comprehensive theoretical and empirical study. Theoretically, we show that Muon imposes a spectral-norm stability ceiling on matrix updates, limiting uncontrolled spectral growth in the training dynamics without explicitly shrinking the learned weights. Empirically, across five architectures and three $\ell_p$ threat models ($\ell_\infty$, $\ell_1$, $\ell_2$) and their union, Muon is competitive with SGD on CNNs and substantially outperforms AdamW on both CNNs and ViTs. These results identify optimizer geometry as a security-relevant factor in adversarial training, while clarifying the empirical regimes in which orthogonalized updates are beneficial. Overall, our findings highlight optimizer design as a security-critical component of AT.