arXiv:2510.18410v3 Announce Type: replace-cross Abstract: Standard dropout applies a fixed, uniform noise rate to all units, ignoring available gradient information. We propose momentum adaptive gradient dropout (MAGDrop), a Bayesian linear layer that adapts per-unit weight noise by tracking an exponential moving average of gradient magnitudes. Units with large recent gradient magnitudes receive less noise to preserve learned features; units with below-average gradient magnitudes receive more noise to promote exploration. This adaptation transforms the classifier head into a data-dependent Bayesian predictor whose mean predictor has no additional inference cost. Our primary contribution is a certificate-aware low-rank posterior, MAGDrop-Subspace, that addresses the tension between strong accuracy and tight bounds from probably approximately correct (PAC)-Bayes theory. A full-head stochastic posterior incurs a Kullback--Leibler (KL) divergence of approximately 8,625 nats. By confining the posterior to an $r$-dimensional subspace constructed from an independent prior-construction sample $S_0$ and certifying on an independent sample $S$ with $S_0\perp S$, MAGDrop-Subspace reduces KL by approximately a factor of 800 while keeping accuracy close to the baseline. On CIFAR-10 with a 13-layer convolutional neural network and $r=50$, MAGDrop-Subspace achieves a PAC-Bayes Gibbs-risk bound of 0.1882 at 90\% confidence using binary KL inversion, compared with 0.4439 for the full-head baseline. Mean-predictor accuracy is 82.31\%, compared with 83.74\% for the baseline. Complementary accuracy experiments on ImageNet-1K show that MAGDrop reaches top-1 accuracy similar to the strong baseline: 78.57\% compared with 78.51\% on the base Vision Transformer with $16\times16$ patches, and 82.14\% compared with 82.23\% on ConvNeXt-Base.