Private Adaptive Gradient Methods for Convex Optimization

While the success of stochastic gradient methods for solving empirical risk minimization has motivated their adoption across much of machine learning, increasing privacy risks in data-intensive tasks have made applying them more challenging [DMNS06]: gradients can leak users' data, intermediate models can compromise individuals, and even final trained models may be non-private without substantial care. This motivates a growing line of work developing private variants of stochastic gradient descent (SGD), where algorithms guarantee differential privacy by perturbing individual gradients with random noise [DJW13; ST13b; ACGMMTZ16; DJW18; BFTT19; FKT20]. Yet these noise addition procedures typically fail to reflect the geometry underlying the optimization problem, which in non-private cases is essential: for high-dimensional problems with sparse parameters, mirror descent and its variants [BT03; NJLS09] are essential, while in the large-scale stochastic settings prevalent in deep learning, AdaGrad and other adaptive variants [DHS11] provide stronger theoretical and practical performance. Even more, methods that do not adapt (or do not leverage geometry) can be provably sub-optimal, in that there exist problems where their convergence is much slower than adaptive variants that reflect appropriate geometry [LD19]. To address these challenges, we introduce Pagan (Private AdaGrad with Adaptive Noise), a new differentially private variant of stochastic gradient descent and AdaGrad.