The Adam optimizer is a popular choice in contemporary deep learning due to its strong empirical performance. However we observe that in privacy sensitive scenarios, the traditional use of Differential Privacy (DP) with the Adam optimizer leads to sub-optimal performance on several tasks. We find that this performance degradation is due to a DP bias in Adam's second moment estimator, introduced by the addition of independent noise in the gradient computation to enforce DP guarantees. This DP bias leads to a different scaling for low variance parameter updates, that is inconsistent with the behavior of non-private Adam, and Adam's sign descent interpretation. We propose the DP-AdamBC optimization algorithm, which corrects for the bias in the second moment estimation and retrieves the expected behaviour of Adam. Empirically, DP-AdamBC significantly improves the optimization performance of DP-Adam by up to 3.5% in final accuracy in image, text, and graph node classification tasks.

We propose Adaptive Randomized Smoothing (ARS) to certify the predictions of our test-time adaptive models against adversarial examples. ARS extends the analysis of randomized smoothing using $f$-Differential Privacy to certify the adaptive composition of multiple steps. For the first time, our theory covers the sound adaptive composition of general and high-dimensional functions of noisy inputs. We instantiate ARS on deep image classification to certify predictions against adversarial examples of bounded $L_{\infty}$ norm. In the $L_{\infty}$ threat model, ARS enables flexible adaptation through high-dimensional input-dependent masking. We design adaptivity benchmarks, based on CIFAR-10 and CelebA, and show that ARS improves standard test accuracy by 1 to 15\% points. On ImageNet, ARS improves certified test accuracy by up to 1.6% points over standard RS without adaptivity. Our code is available at [https://github.com/ubc-systopia/adaptive-randomized-smoothing](https://github.com/ubc-systopia/adaptive-randomized-smoothing).