Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry

TL;DR

This paper establishes optimal rates for differentially private stochastic convex optimization over $\, ext{l}_1$-bounded domains, introducing new algorithms that improve gradient query complexity and extend to smooth loss functions.

Contribution

The paper provides the first near-optimal excess loss bounds for private convex optimization in $\, ext{l}_1$ geometry, with new algorithms combining localization, mirror descent, and variance reduction.

Findings

Optimal excess loss rate of $\, ext{sqrt}( ext{log}(d)/n) + ext{sqrt}(d)/( ext{epsilon} n)$.

New algorithm reduces gradient queries to $n^{3/2}$ for $\, ext{l}_p$ domains, improving over previous $n^2$.

Variance-reduced Frank-Wolfe achieves near-optimal bounds for smooth loss functions.

Abstract

Stochastic convex optimization over an $\ell_1$-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~\citet{FeldmanKoTa20} with a new analysis of private regularized mirror descent. It applies to $\ell_p$ 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.