Guarantees of confidentiality via Hammersley-Chapman-Robbins bounds

Protecting privacy during inference with deep neural networks is possible by adding noise to the activations in the last layers prior to the final classifiers or other task-specific layers. The activations in such layers are known as "features" (or, less commonly, as "embeddings" or "feature embeddings"). The added noise helps prevent reconstruction of the inputs from the noisy features. Lower bounding the variance of every possible unbiased estimator of the inputs quantifies the confidentiality arising from such added noise. Convenient, computationally tractable bounds are available from classic inequalities of Hammersley and of Chapman and Robbins -- the HCR bounds. Numerical experiments indicate that the HCR bounds are on the precipice of being effectual for small neural nets with the data sets, "MNIST" and "CIFAR-10," which contain 10 classes each for image classification. The HCR bounds appear to be insufficient on their own to guarantee confidentiality of the inputs to inference with standard deep neural nets, "ResNet-18" and "Swin-T," pre-trained on the data set, "ImageNet-1000," which contains 1000 classes. Supplementing the addition of noise to features with other methods for providing confidentiality may be warranted in the case of ImageNet. In all cases, the results reported here limit consideration to amounts of added noise that incur little degradation in the accuracy of classification from the noisy features. Thus, the added noise enhances confidentiality without much reduction in the accuracy on the task of image classification.