As training Deep Neural Networks (DNNs) becomes more expensive, the interest in protecting the ownership of the models with watermarking techniques increases. Uchida et al. proposed a digital watermarking algorithm that embeds the secret message into the model coefficients. However, despite its appeal, in this paper, we show that its efficacy can be compromised by the optimization algorithm being used. In particular, we found through a theoretical analysis that, as opposed to Stochastic Gradient Descent (SGD), the update direction given by Adam optimization strongly depends on the sign of a combination of columns of the projection matrix used for watermarking. Consequently, as observed in the empirical results, this makes the coefficients move in unison giving rise to heavily spiked weight distributions that can be easily detected by adversaries. As a way to solve this problem, we propose a new method called Block-Orthonormal Projections (BOP) that allows one to combine watermarking with Adam optimization with a minor impact on the detectability of the watermark and an increased robustness.
Keywords: Adam; deep neural networks; detectability; optimization algorithms; stochastic gradient descent; watermarking.