# Private Stochastic Convex Optimization: Optimal Rates in ℓ1 Geometry

AuthorsHilal Asi, Vitaly Feldman, Tomer Koren, Kunal Talwar

content type paperpublished July 2021

AuthorsHilal Asi, Vitaly Feldman, Tomer Koren, Kunal Talwar

Stochastic convex optimization over an $ℓ_1$-bounded domain is ubiquitous in machine learning applications such as LASSO but remains poorly understood when learning with differential privacy. We show that, up to logarithmic factors the optimal excess population loss of any $(\varepsilon, \delta)$-differentially private optimizer is $\sqrt{\log(d)/n}\; +$ $\sqrt{d}/\varepsilon n.$ The upper bound is based on a new algorithm that combines the iterative localization approach of FeldmanKoTa20 with a new analysis of private regularized mirror descent. It applies to $ℓ_p$ bounded domains for $p\in [1,2]$ and queries at most $n^{3/2}$ gradients improving over the best previously known algorithm for the $\ell_2$ case which needs $n^2$ gradients. Further, we show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $\sqrt{\log(d)/n}\; +$ $(\log(d)/\varepsilon n)^{2/3}.$ This bound is achieved by a new variance-reduced version of the Frank-Wolfe algorithm that requires just a single pass over the data. We also show that the lower bound in this case is the minimum of the two rates mentioned above.

*= Equal Contributors
Online prediction from experts is a fundamental problem in machine learning and several works have studied this problem under privacy constraints. We propose and analyze new algorithms for this problem that improve over the regret bounds of the best existing algorithms for non-adaptive adversaries. For approximate differential privacy, our algorithms achieve regret bounds of O(Tlogd+logd/ε)O(\sqrt{T \log d} + \log…

See paper details*=Equal Contributors
We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret O(ε−1log1.5d)O \big( \varepsilon^{-1} \log^{1.5}{d} \big)O(ε−1log1.5d) where ddd is the number of experts. This…

See paper details