Optimal Rates for $O(1)$-Smooth DP-SCO with a Single Epoch and Large Batches

Abstract

In this paper we revisit the DP stochastic convex optimization (SCO) problem. For convex smooth losses, it is well-known that the canonical DP-SGD (stochastic gradient descent) achieves the optimal rate of $O\left(\frac{LR}{\sqrt{n}} + \frac{LR \sqrt{p \log(1/\delta)}}{\epsilon n}\right)$ under $(\epsilon, \delta)$-DP, and also well-known that variants of DP-SGD can achieve the optimal rate in a single epoch. However, the batch gradient complexity (i.e., number of adaptive optimization steps), which is important in applications like federated learning, is less well-understood. In particular, all prior work on DP-SCO requires $\Omega(n)$ batch gradient steps, multiple epochs, or convexity for privacy. We propose an algorithm, Accelerated-DP-SRGD (stochastic recursive gradient descent), which bypasses the limitations of past work: it achieves the optimal rate for DP-SCO (up to polylog factors), in a single epoch using $\sqrt{n}$ batch gradient steps with batch size $\sqrt{n}$, and can be made private for arbitrary (non-convex) losses via clipping. If the global minimizer is in the constraint set, we can further improve this to $n^{1/4}$ batch gradient steps with batch size $n^{3/4}$. To achieve this, our algorithm combines three key ingredients, a variant of stochastic recursive gradients (SRG), accelerated gradient descent, and correlated noise generation from DP continual counting.